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**SAMPLE QUESTIONS**

Chapter 1 Functions

Functions

In Exercises 1–6, find the domain and range of each function.

- ƒ(x) = 18 + x2 2. ƒ(x) = 1 – 2x
- F(x) = 25x + 10 4. g(x) = 2×2 – 3x
- ƒ(t) = 4

3 – t

- G(t) = 5

t2 – 4

In Exercises 7 and 8, which of the graphs are graphs of functions of x,

and which are not? Give reasons for your answers.

- a.

x

y

0

b.

x

y

0

- a.

x

y

0

b.

x

y

0

Finding Formulas for Functions

- Express the area and perimeter of an equilateral triangle as a

function of the triangle’s side length x.

- Express the side length of a square as a function of the length d of

the square’s diagonal. Then express the area as a function of the

diagonal length.

- Express the edge length of a cube as a function of the cube’s

diagonal length d. Then express the surface area and volume of

the cube as a function of the diagonal length.

- A point P in the first quadrant lies on the graph of the function

ƒ(x) = 2x. Express the coordinates of P as functions of the

slope of the line joining P to the origin.

- Consider the point (x, y) lying on the graph of the line

2x + 4y = 5. Let L be the distance from the point (x, y) to the

origin (0, 0). Write L as a function of x.

- Consider the point (x, y) lying on the graph of y = 2x – 3. Let

L be the distance between the points (x, y) and (4, 0). Write L as a

function of y.

Functions and Graphs

Find the natural domain and graph the functions in Exercises 15–20.

- ƒ(x) = 5 – 2x 16. ƒ(x) = 1 – 2x – x2
- g(x) = 20 x 0 18. g(x) = 2-x
- F(t) = t> 0 t 0 20. G(t) = 1> 0 t 0
- Find the domain of y = x + 7

12 – 2×2 – 25

.

- Find the range of y = 2 + x2

x2 + 4 .

- Graph the following equations and explain why they are not

graphs of functions of x.

- 0 y 0 = x b. y2 = x2
- Graph the following equations and explain why they are not

graphs of functions of x.

- 0 x 0 + 0 y 0 = 1 b. 0 x + y 0 = 1

Piecewise-Defined Functions

Graph the functions in Exercises 25–28.

- ƒ(x) = e

x, 0 … x … 1

2 – x, 1 6 x … 2

- g(x) = e

1 – x, 0 … x … 1

2 – x, 1 6 x … 2

- F(x) = e

4 – x2, x … 1

x2 + 2x, x 7 1

- G(x) = e

1>x, x 6 0

x, 0 … x

Find a formula for each function graphed in Exercises 29–32.

- a.

x

y

0

1

2

(1, 1)

b.

t

y

0

2

1 2 3 4

- a.

x

y

2 5

2

(2, 1)

b.

_1

x

y

3

1 2

2

1

_2

_3

_1

(2, _1)

- a.

x

y

3

1

(_1, 1) (1, 1)

b.

x

y

1

2

(_2, _1) (1, _1) (3, _1)

- a.

x

y

0

1

T T2

(T, 1)

b.

t

y

0

A

T

_A

T2

3T

2

2T

The Greatest and Least Integer Functions

- For what values of x is
- :x; = -1? b. <x= = 0?
- What real numbers x satisfy the equation :x; = <x=?
- Does <-x= = -:x; for all real x? Give reasons for your answer.
- Graph the function

ƒ(x) = e

:x;, x Ú 0

<x=, x 6 0.

Why is ƒ(x) called the integer part of x?

Increasing and Decreasing Functions

Graph the functions in Exercises 37–46. What symmetries, if any, do

the graphs have? Specify the intervals over which the function is

increasing and the intervals where it is decreasing.

- y = -x3 38. y = –

1

x2

- y = –

1x

- y = 1

0 x 0

- y = 20 x 0 42. y = 2-x
- y = x3>8 44. y = -42x
- y = -x3>2 46. y = (-x)2>3

Even and Odd Functions

In Exercises 47–58, say whether the function is even, odd, or neither.

Give reasons for your answer.

- ƒ(x) = -8 48. ƒ(x) = x-5
- g(x) = 9×3 – 3 50. ƒ(x) = x2 + x
- g(x) = x3 + x 52. g(x) = x4 + 3×2 – 1
- g(x) = 1

x2 – 4

- g(x) = x

x2 – 1

- h(t) = 1

t3 + 1

- h(t) = _ t3 _
- h(t) = 5t – 1 58. h(t) = 2 _ t _ + 1

Theory and Examples

- The variable s is proportional to t, and s = 15 when t = 105.

Determine t when s = 40.

- Kinetic energy The kinetic energy K of a mass is proportional

to the square of its velocity y. If K = 12,960 joules when

y = 18 m>sec, what is K when y = 10 m>sec?

- The variables r and s are inversely proportional, and r = 7 when

s = 4. Determine s when r = 10.

- Boyle’s Law Boyle’s Law says that the volume V of a gas at

constant temperature increases whenever the pressure P decreases,

so that V and P are inversely proportional. If P = 14.7 lb>in2

when V = 1000 in3, then what is V when P = 23.4 lb>in2?

- A box with an open top is to be constructed from a rectangular

piece of cardboard with dimensions 14 in. by 22 in. by cutting out

equal squares of side x at each corner and then folding up the

sides as in the figure. Express the volume V of the box as a function

of x.

x

x

x

x

x

x

x

x

22

14

- The accompanying figure shows a rectangle inscribed in an isosceles

right triangle whose hypotenuse is 2 units long.

- Express the y-coordinate of P in terms of x. (You might start

by writing an equation for the line AB.)

- Express the area of the rectangle in terms of x.

x

y

_1 0 x 1

A

B

P(x, ?)

In Exercises 65 and 66, match each equation with its graph. Do not

use a graphing device, and give reasons for your answer.

- a. y = x4 b. y = x7 c. y = x10

x

y

f

g

h

0

- a. y = 5x b. y = 5x c. y = x5

x

y

f

h

g

0

- a. Graph the functions ƒ(x) = x>2 and g(x) = 1 + (4>x) to-

gether to identify the values of x for which

x

2 7 1 + 4x

.

- Confirm your findings in part (a) algebraically.
- a. Graph the functions ƒ(x) = 3>(x – 1) and g(x) = 2>(x + 1)

together to identify the values of x for which

3

x – 1 6 2

x + 1

.

- Confirm your findings in part (a) algebraically.
- For a curve to be symmetric about the x-axis, the point (x, y) must

lie on the curve if and only if the point (x, -y) lies on the curve.

Explain why a curve that is symmetric about the x-axis is not the

graph of a function, unless the function is y = 0.

- Three hundred books sell for $40 each, resulting in a revenue of

(300)($40) = $12,000. For each $5 increase in the price, 25

fewer books are sold. Write the revenue R as a function of the

number x of $5 increases.

- A pen in the shape of an isosceles right triangle with legs of

length x ft and hypotenuse of length h ft is to be built. If fencing

costs $2/ft for the legs and $8/ft for the hypotenuse, write the

total cost C of construction as a function of h.

- Industrial costs A power plant sits next to a river where the

river is 800 ft wide. To lay a new cable from the plant to a location

in the city 2 mi downstream on the opposite side costs $180

per foot across the river and $100 per foot along the land.

P x Q

Power plant

City

800 ft

2 mi

NOT TO SCALE

- Suppose that the cable goes from the plant to a point Q on the

opposite side that is x ft from the point P directly opposite the

plant. Write a function C(x) that gives the cost of laying the

cable in terms of the distance x.

- Generate a table of values to determine if the least expensive

location for point Q is less than 2000 ft or greater than 2000 ft

from point P.

- a.

x

y

3

1

(_1, 1) (1, 1)

b.

x

y

1

2

(_2, _1) (1, _1) (3, _1)

- a.

x

y

0

1

T T2

(T, 1)

b.

t

y

0

A

T

_A

T2

3T

2

2T

The Greatest and Least Integer Functions

- For what values of x is
- :x; = -1? b. <x= = 0?
- What real numbers x satisfy the equation :x; = <x=?
- Does <-x= = -:x; for all real x? Give reasons for your answer.
- Graph the function

ƒ(x) = e

:x;, x Ú 0

<x=, x 6 0.

Why is ƒ(x) called the integer part of x?

Increasing and Decreasing Functions

Graph the functions in Exercises 37–46. What symmetries, if any, do

the graphs have? Specify the intervals over which the function is

increasing and the intervals where it is decreasing.

- y = -x3 38. y = –

1

x2

- y = –

1x

- y = 1

0 x 0

- y = 20 x 0 42. y = 2-x
- y = x3>8 44. y = -42x
- y = -x3>2 46. y = (-x)2>3

Even and Odd Functions

In Exercises 47–58, say whether the function is even, odd, or neither.

Give reasons for your answer.

- ƒ(x) = -8 48. ƒ(x) = x-5
- g(x) = 9×3 – 3 50. ƒ(x) = x2 + x
- g(x) = x3 + x 52. g(x) = x4 + 3×2 – 1
- g(x) = 1

x2 – 4

- g(x) = x

x2 – 1

- h(t) = 1

t3 + 1

- h(t) = _ t3 _
- h(t) = 5t – 1 58. h(t) = 2 _ t _ + 1

Theory and Examples

- The variable s is proportional to t, and s = 15 when t = 105.

Determine t when s = 40.

- A point P in the first quadrant lies on the graph of the function

ƒ(x) = 2x. Express the coordinates of P as functions of the

slope of the line joining P to the origin.

- Consider the point (x, y) lying on the graph of the line

2x + 4y = 5. Let L be the distance from the point (x, y) to the

origin (0, 0). Write L as a function of x.

- Consider the point (x, y) lying on the graph of y = 2x – 3. Let

L be the distance between the points (x, y) and (4, 0). Write L as a

function of y.

Functions and Graphs

Find the natural domain and graph the functions in Exercises 15–20.

- ƒ(x) = 5 – 2x 16. ƒ(x) = 1 – 2x – x2
- g(x) = 20 x 0 18. g(x) = 2-x
- F(t) = t> 0 t 0 20. G(t) = 1> 0 t 0
- Find the domain of y = x + 7

12 – 2×2 – 25

.

- Find the range of y = 2 + x2

x2 + 4 .

- Graph the following equations and explain why they are not

graphs of functions of x.

- 0 y 0 = x b. y2 = x2
- Graph the following equations and explain why they are not

graphs of functions of x.

- 0 x 0 + 0 y 0 = 1 b. 0 x + y 0 = 1

Piecewise-Defined Functions

Graph the functions in Exercises 25–28.

- ƒ(x) = e

x, 0 … x … 1

2 – x, 1 6 x … 2

- g(x) = e

1 – x, 0 … x … 1

2 – x, 1 6 x … 2

- F(x) = e

4 – x2, x … 1

x2 + 2x, x 7 1

- G(x) = e

1>x, x 6 0

x, 0 … x

Find a formula for each function graphed in Exercises 29–32.

- a.

x

y

0

1

2

(1, 1)

b.

t

y

0

2

1 2 3 4

- a.

x

y

2 5

2

(2, 1)

b.

_1

x

y

3

1 2

2

1

_2

_3

_1

(2, _1)

- Kinetic energy The kinetic energy K of a mass is proportional

to the square of its velocity y. If K = 12,960 joules when

y = 18 m>sec, what is K when y = 10 m>sec?

- The variables r and s are inversely proportional, and r = 7 when

s = 4. Determine s when r = 10.

- Boyle’s Law Boyle’s Law says that the volume V of a gas at

constant temperature increases whenever the pressure P decreases,

so that V and P are inversely proportional. If P = 14.7 lb>in2

when V = 1000 in3, then what is V when P = 23.4 lb>in2?

- A box with an open top is to be constructed from a rectangular

piece of cardboard with dimensions 14 in. by 22 in. by cutting out

equal squares of side x at each corner and then folding up the

sides as in the figure. Express the volume V of the box as a function

of x.

x

x

x

x

x

x

x

x

22

14

- The accompanying figure shows a rectangle inscribed in an isosceles

right triangle whose hypotenuse is 2 units long.

- Express the y-coordinate of P in terms of x. (You might start

by writing an equation for the line AB.)

- Express the area of the rectangle in terms of x.

x

y

_1 0 x 1

A

B

P(x, ?)

In Exercises 65 and 66, match each equation with its graph. Do not

use a graphing device, and give reasons for your answer.

- a. y = x4 b. y = x7 c. y = x10

x

y

f

g

h

0

- a. y = 5x b. y = 5x c. y = x5

x

y

f

h

g

0

- a. Graph the functions ƒ(x) = x>2 and g(x) = 1 + (4>x) to-

gether to identify the values of x for which

x

2 7 1 + 4x

.

- Confirm your findings in part (a) algebraically.
- a. Graph the functions ƒ(x) = 3>(x – 1) and g(x) = 2>(x + 1)

together to identify the values of x for which

3

x – 1 6 2

x + 1

.

- Confirm your findings in part (a) algebraically.
- For a curve to be symmetric about the x-axis, the point (x, y) must

lie on the curve if and only if the point (x, -y) lies on the curve.

Explain why a curve that is symmetric about the x-axis is not the

graph of a function, unless the function is y = 0.

- Three hundred books sell for $40 each, resulting in a revenue of

(300)($40) = $12,000. For each $5 increase in the price, 25

fewer books are sold. Write the revenue R as a function of the

number x of $5 increases.

- A pen in the shape of an isosceles right triangle with legs of

length x ft and hypotenuse of length h ft is to be built. If fencing

costs $2/ft for the legs and $8/ft for the hypotenuse, write the

total cost C of construction as a function of h.

- Industrial costs A power plant sits next to a river where the

river is 800 ft wide. To lay a new cable from the plant to a location

in the city 2 mi downstream on the opposite side costs $180

per foot across the river and $100 per foot along the land.

P x Q

Power plant

City

800 ft

2 mi

NOT TO SCALE

- Suppose that the cable goes from the plant to a point Q on the

opposite side that is x ft from the point P directly opposite the

plant. Write a function C(x) that gives the cost of laying the

cable in terms of the distance x.

- Generate a table of values to determine if the least expensive

location for point Q is less than 2000 ft or greater than 2000 ft

from point P.

Choosing a Viewing Window

In Exercises 1–4, use graphing software to determine which of the

given viewing windows displays the most appropriate graph of the

specified function.

- ƒ(x) = x4 – 7×2 + 6x
- 3-1, 14 by 3-1, 14 b. 3-2, 24 by 3-5, 54
- 3-10, 104 by 3-10, 104 d. 3-5, 54 by 3-25, 154
- ƒ(x) = x3 – 4×2 – 4x + 16
- 3-1, 14 by 3-5, 54 b. 3-3, 34 by 3-10, 104
- 3-5, 54 by 3-10, 204 d. 3-20, 204 by 3-100, 1004
- ƒ(x) = 5 + 12x – x3
- 3-1, 14 by 3-1, 14 b. 3-5, 54 by 3-10, 104
- 3-4, 44 by 3-20, 204 d. 3-4, 54 by 3-15, 254
- ƒ(x) = 25 + 4x – x2
- 3-2, 24 by 3-2, 24 b. 3-2, 64 by 3-1, 44
- 3-3, 74 by 30, 104 d. 3-10, 104 by 3-10, 104

Finding a Viewing Window

In Exercises 5–30, find an appropriate graphing software viewing window

for the given function and use it to display its graph. The window

should give a picture of the overall behavior of the function. There is

more than one choice, but incorrect choices can miss important

aspects of the function.

- ƒ(x) = x4 – 4×3 + 15 6. ƒ(x) = x3

3 – x2

2 – 2x + 1

- ƒ(x) = x5 – 5×4 + 10 8. ƒ(x) = 4×3 – x4
- ƒ(x) = x29 – x2 10. ƒ(x) = x2(6 – x3)

T

T

- y = 2x – 3×2>3 12. y = x1>3(x2 – 8)
- y = 5×2>5 – 2x 14. y = x2>3(5 – x)
- y = 0 x2 – 1 0 16. y = 0 x2 – x 0
- y = x + 3

x + 2

- y = 1 – 1

x + 3

- ƒ(x) = x2 + 2

x2 + 1

- ƒ(x) = x2 – 1

x2 + 1

- ƒ(x) = x – 1

x2 – x – 6

- ƒ(x) = 8

x2 – 9

- ƒ(x) = 6×2 – 15x + 6

4×2 – 10x

- ƒ(x) = x2 – 3

x – 2

- y = sin 250x 26. y = 3 cos 60x
- y = cos a x

50

b 28. y = 1

10

sin a x

10b

- y = x + 1

10

sin 30x 30. y = x2 + 1

50

cos 100x

Use graphing software to graph the functions specified in Exercises 31–36.

Select a viewing window that reveals the key features of the function.

- Graph the lower half of the circle defined by the equation

x2 + 2x = 4 + 4y – y2.

- Graph the upper branch of the hyperbola y2 – 16×2 = 1.
- Graph four periods of the function ƒ(x) = – tan 2x.
- Graph two periods of the function ƒ(x) = 3 cot

x

2 + 1.

- Graph the function ƒ(x) = sin 2x + cos 3x.
- Graph the function ƒ(x) = sin3 x.

Exercises 1.4

1.5 E xponential Functions

Exponential functions are among the most important in mathematics and occur in a wide

variety of applications, including interest rates, radioactive decay, population growth, the

spread of a disease, consumption of natural resources, the earth’s atmospheric pressure, temperature

change of a heated object placed in a cooler environment, and the dating of fossils.

In this section we introduce these functions informally, using an intuitive approach. We give

a rigorous development of them in Chapter 7, based on important calculus ideas and results.

Exponential Behavior

When a positive quantity P doubles, it increases by a factor of 2 and the quantity becomes

2P. If it doubles again, it becomes 2(2P) = 22P, and a third doubling gives 2(22P) = 23P.

Continuing to double in this fashion leads us to consider the function ƒ(x) = 2x. We call

this an exponential function because the variable x appears in the exponent of 2x. Functions

such as g(x) = 10 x and h(x) = (1>2)x are other examples of exponential functions.

In general, if a _ 1 is a positive constant, the function

ƒ(x) = ax, a > 0

Sketching Exponential Curves

In Exercises 1–6, sketch the given curves together in the appropriate

coordinate plane and label each curve with its equation.

- y = 2x, y = 4x, y = 3-x, y = (1>5)x
- y = 3x, y = 8x, y = 2-x, y = (1>4)x
- y = 2-t and y = -2t 4. y = 3-t and y = -3t
- y = ex and y = 1>ex 6. y = -ex and y = -e-x

In each of Exercises 7–10, sketch the shifted exponential curves.

- y = 2x – 1 and y = 2-x – 1
- y = 3x + 2 and y = 3-x + 2
- y = 1 – ex and y = 1 – e-x
- y = -1 – ex and y = -1 –

Applying the Laws of Exponents

Use the laws of exponents to simplify the expressions in Exercises

11–20.

- 497 # 49-6.5 12. 91>3 # 91>6
- 44.2

43.7 14. 35>3

32>3

- 1641>1222 16. 11322222>2
- 223 # 723 18. 12321>2 # 121221>2
- a 2

22

b

4

- a26

3 b

2

Composites Involving Exponential Functions

Find the domain and range for each of the functions in Exercises

21–24.

- ƒ(x) = 1

2 + ex 22. g(t) = cos (e-t)

- g(t) = 21 + 3-t 24. ƒ(x) = 3

1 – e2x

Applications

In Exercises 25–28, use graphs to find approximate solutions.

- 2x = 5 26. ex = 4
- 3x – 0.5 = 0 28. 3 – 2-x = 0

In Exercises 29–36, use an exponential model and a graphing calculator

to estimate the answer in each problem.

- Population growth The population of Knoxville is 500,000

and is increasing at the rate of 3.75% each year. Approximately

when will the population reach 1 million?

- Population growth The population of Silver Run in the year

1890 was 6250. Assume the population increased at a rate of

2.75% per year.

- Estimate the population in 1915 and 1940.
- Approximately when did the population reach 50,000?
- Radioactive decay The half-life of phosphorus-32 is about

14 days. There are 6.6 grams present initially.

- Express the amount of phosphorus-32 remaining as a function

of time t.

- When will there be 1 gram remaining?
- If Jean invests $2300 in a retirement account with a 6% interest rate

compounded annually, how long will it take until Jean’s account

has a balance of $4150?

- Doubling your money Determine how much time is required

for an investment to double in value if interest is earned at the rate

of 6.25% compounded annually.

- Tripling your money Determine how much time is required

for an investment to triple in value if interest is earned at the rate

of 5.75% compounded continuously.

- Cholera bacteria Suppose that a colony of bacteria starts with

1 bacterium and doubles in number every half hour. How many

bacteria will the colony contain at the end of 24 hr?

- Eliminating a disease Suppose that in any given year the number

of cases of a disease is reduced by 20%. If there are 10,000

cases today, how many years will it take

- to reduce the number of cases to 1000?
- to eliminate the disease; that is, to reduce the number of cases

to less than 1?

Identifying One-to-One Functions Graphically

Which of the functions graphed in Exercises 1–6 are one-to-one, and

which are not?

1.

x

y

0

y _ _3x3

2.

x

y

_1 0 1

y _ x4 _ x2

- y

x

y _ 20 x 0

4.

x

y

y _ int x

5.

x

y

0

y _ 1x

6.

x

y

y _ x1_3

In Exercises 7–10, determine from its graph if the function is one-toone.

- ƒ(x) = e

3 – x, x 6 0

3, x Ú 0

- ƒ(x) = e

2x + 6, x … -3

x + 4, x 7 -3

- ƒ(x) = d

1 – x

2

, x … 0

x

x + 2

, x 7 0

- ƒ(x) = e

2 – x2, x … 1

x2, x 7 1

Graphing Inverse Functions

Each of Exercises 11–16 shows the graph of a function y = ƒ(x).

Copy the graph and draw in the line y = x. Then use symmetry with

respect to the line y = x to add the graph of ƒ -1 to your sketch. (It is

not necessary to find a formula for ƒ -1.) Identify the domain and

range of ƒ -1.

- 12.

x

y

0 1

1

y _ f (x) _ 1 , x _ 0

x2 + 1

x

y

0 1

1

y _ f (x) _ 1 _ , x > 0 1x

- 14.

x

y

0 p2

p2

_

1

_1

p2

p2

_

y _ f (x) _ sin x,

_ x _

p2

p2

_

y _ f (x) _ tan x,

< x <

x

y

0 p2

p2

_

- 16.

x

y

0

6

3

f (x) _ 6 _ 2x,

0 _ x _ 3

x

y

0

1

_1 3

_2

x + 1, _1 _ x _ 0

_2 + x, 0 < x < 3

f (x) _ 2

3

- a. Graph the function ƒ(x) = 21 – x2, 0 … x … 1. What symmetry

does the graph have?

- Show that ƒ is its own inverse. (Remember that 2×2 = x if

x Ú 0.)

- a. Graph the function ƒ(x) = 1>x. What symmetry does the

graph have?

- Show that ƒ is its own inverse.

Formulas for Inverse Functions

Each of Exercises 19–24 gives a formula for a function y = ƒ(x) and

shows the graphs of ƒ and ƒ -1. Find a formula for ƒ -1 in each case.

- ƒ(x) = x2 + 1, x Ú 0 20. ƒ(x) = x2, x … 0

x

y

1

0 1

y _ f (x)

y _ f –1(x)

x

y

1

0 1

y _ f –1(x)

y _ f (x)

- ƒ(x) = x3 – 1 22. ƒ(x) = x2 – 2x + 1, x Ú 1

x

y

1

_1 1

_1

y _ f (x)

y _ f –1(x)

x

y

1

0 1

y _ f (x)

y _ f –1(x)

- ƒ(x) = (x + 1)2, x Ú -1 24. ƒ(x) = x2>3, x Ú 0

x

y

0

1

_1

_1 1

y _ f (x)

y _ f –1(x)

x

y

0

1

1

y _ f –1(x)

y _ f (x)

Each of Exercises 25–36 gives a formula for a function y = ƒ(x). In

each case, find ƒ -1(x) and identify the domain and range of ƒ -1. As a

check, show that ƒ(ƒ -1(x)) = ƒ -1(ƒ(x)) = x.

- ƒ(x) = x5 26. ƒ(x) = x4, x Ú 0
- ƒ(x) = x3 + 1 28. ƒ(x) = (1>2)x – 7>2
- ƒ(x) = 1>x2, x 7 0 30. ƒ(x) = 1>x3, x _ 0
- ƒ(x) = x + 3

x – 2

- ƒ(x) =

2x

2x – 3

- ƒ(x) = x2 – 2x, x … 1 34. ƒ(x) = (2×3 + 1)1>5

(Hint: Complete the square.)

- ƒ(x) = x + b

x – 2

, b 7 -2 and constant

- ƒ(x) = x2 – 2bx, b 7 0 and constant, x … b

Inverses of Lines

- a. Find the inverse of the function ƒ(x) = mx, where m is a constant

different from zero.

- What can you conclude about the inverse of a function

y = ƒ(x) whose graph is a line through the origin with a nonzero

slope m?

- Show that the graph of the inverse of ƒ(x) = mx + b, where m

and b are constants and m _ 0, is a line with slope 1>m and

y-intercept -b>m.

- a. Find the inverse of ƒ(x) = x + 1. Graph ƒ and its inverse

together. Add the line y = x to your sketch, drawing it with

dashes or dots for contrast.

- Find the inverse of ƒ(x) = x + b (b constant). How is the

graph of ƒ -1 related to the graph of ƒ?

- What can you conclude about the inverses of functions whose

graphs are lines parallel to the line y = x?

- a. Find the inverse of ƒ(x) = -x + 1. Graph the line

y = -x + 1 together with the line y = x. At what angle do

the lines intersect?

- Find the inverse of ƒ(x) = -x + b (b constant). What angle

does the line y = -x + b make with the line y = x?

- What can you conclude about the inverses of functions whose

graphs are lines perpendicular to the line y = x?

Logarithms and Exponentials

- Express the following logarithms in terms of ln 2 and ln 3.
- ln 0.75 b. ln (4>9)
- ln (1>2) d. ln23

9

- ln 322 f. ln 213.5
- Express the following logarithms in terms of ln 5 and ln 7.
- ln (1>125) b. ln 9.8
- ln 727 d. ln 1225
- ln 0.056 f. (ln 35 + ln (1>7))>(ln 25)

Use the properties of logarithms to write the expressions in Exercises

43 and 44 as a single term.

- a. ln sin u – ln asin u

2 b b. ln (3×2 – 9x) + ln a 1

3xb

- 1

2

ln (4t4) – ln b

- a. ln sec u + ln cos u b. ln (8x + 4) – 2 ln c
- 3 ln23

t2 – 1 – ln (t + 1)

Find simpler expressions for the quantities in Exercises 45–48.

- a. eln 8.3 b. e-ln 6×6 c. eln 3x-ln 5y
- a. eln (x2+y2) b. e-ln 0.3 c. eln px-ln 2
- a. 2 ln 2e b. ln (ln ee) c. ln (e-x2-y2)
- a. ln (esec u) b. ln (e(ex)) c. ln (e2 ln x)

In Exercises 49–54, solve for y in terms of t or x, as appropriate.

- ln y = 4t + 5 50. ln y = -t + 5
- ln (y – 30) = 5t 52. ln (c – 2y) = t
- ln (y – 4) – ln 5 = x + ln x
- ln (y2 – 1) – ln (y + 1) = ln (sin x)

In Exercises 55 and 56, solve for k.

- a. e3k = 27 b. 35e8k = 175 c. ek>8 = a
- a. e5k = 1

4

- 80ek = 1 c. e(ln 0.8)k = 0.8

In Exercises 57–60, solve for t.

- a. e-0.3t = 64 b. ekt = 1

7

- e(ln 0.7)t = 0.8
- a. e-0.01t = 1000 b. ekt = 1

10

- e(ln 2)t = 1

2

- e2t = x6 60. e(x2)e(2x+1) = et

Simplify the expressions in Exercises 61–64.

- a. 5log5 7 b. 8log822 c. 1.3log1.3 75
- log4 16 e. log323 f. log4 a1

4b

- a. 2log2 3 b. 10log10 (1>2) c. plogp 7
- log11 121 e. log121 11 f. log3 a1

9b

- a. 2log4 x b. 9log3 x c. log2 (e(ln 2)(sin x))
- a. 25log5 (3×2) b. loge (ex) c. log4 (2ex sin x)

Express the ratios in Exercises 65 and 66 as ratios of natural logarithms

and simplify.

- a.

log11 x

log12 x

b.

log5 x

log125 x

c.

logx a

logx5 a

- a.

log9 x

log3 x

b.

log210 x

log22 x

c.

loga b

logb a

Arcsine and Arccosine

In Exercises 67–70, find the exact value of each expression.

- a. sin-1 a1

2b b. sin-1 a 1

22

b c. sin-1 a-23

2

b

- a. cos-1 a1

2b b. cos-1 a -1

22

b c. cos-1 a23

2

b

- a. arccos (-1) b. arccos (0)
- a. arcsin (-1) b. arcsin a-

1

22

b

Theory and Examples

- If ƒ(x) is one-to-one, can anything be said about g(x) = -ƒ(x)? Is

it also one-to-one? Give reasons for your answer.

- If ƒ(x) is one-to-one and ƒ(x) is never zero, can anything be said

about h(x) = 1>ƒ(x)? Is it also one-to-one? Give reasons for your

answer.

- Suppose that the range of g lies in the domain of ƒ so that the

composite ƒ _ g is defined. If ƒ and g are one-to-one, can anything

be said about ƒ _ g? Give reasons for your answer.

- If a composite ƒ _ g is one-to-one, must g be one-to-one? Give

reasons for your answer.

- Find a formula for the inverse function ƒ -1 and verify that

(ƒ _ ƒ -1)(x) = (ƒ -1 _ ƒ)(x) = x.

- ƒ(x) = 100

1 + 2-x b. ƒ(x) = 50

1 + 1.1-x

- The identity sin-1 x + cos-1 x = P>2 Figure 1.68 establishes

the identity for 0 6 x 6 1. To establish it for the rest of 3-1, 1],

verify by direct calculation that it holds for x = 1, 0, and -1.

Then, for values of x in (-1, 0), let x = -a, a 7 0, and apply

Eqs. (3) and (5) to the sum sin-1 (-a) + cos-1 (-a).

- Start with the graph of y = ln x. Find an equation of the graph

that results from

- shifting down 3 units.
- shifting right 1 unit.
- shifting left 1, up 3 units.
- shifting down 4, right 2 units.
- reflecting about the y-axis.
- reflecting about the line y = x.
- Start with the graph of y = ln x. Find an equation of the graph

that results from

- vertical stretching by a factor of 2.
- horizontal stretching by a factor of 3.
- vertical compression by a factor of 4.
- horizontal compression by a factor of 2.
- The equation x2 = 2x has three solutions: x = 2, x = 4, and one

other. Estimate the third solution as accurately as you can by

graphing.

- Could xln 2 possibly be the same as 2ln x for x 7 0? Graph the

two functions and explain what you see.

- Radioactive decay The half-life of a certain radioactive substance

is 36 hours. There are 12 grams present initially.

- Express the amount of substance remaining as a function of

time t.

- When will there be 1 gram remaining?
- Doubling your money Determine how much time is required

for a $6000 investment to double in value if interest is earned at

the rate of 3.75% compounded annually.

- Population growth The population of a city is 280,000 and is

increasing at the rate of 2.75% per year. Predict when the population

will be 560,000.

- Radon-222 The decay equation for a certain substance is

known to be y = y0 e-0.0462t, with t in days. About how long will it

take the substance in a sealed sample of air to fall to 73% of its

original value?

Chapter 2 Limits and Continuity

Exercises 2.1

Average Rates of Change

In Exercises 1–6, find the average rate of change of the function over

the given interval or intervals.

- ƒ(x) = 8×3 + 8
- 35, 74 b. 3-5, 54
- g(x) = x2 – 2x
- 31, 34 b. 3-2, 44
- h(t) = cot t
- 33p>4, 5p>44 b. 3p>3, 3p>24
- g(t) = 2 + cos t
- 30, p4 b. 3-p, p4
- R(u) = 23u + 1; 30, 54
- P(u) = u3 – 4u2 + 5u; 31, 24

Slope of a Curve at a Point

In Exercises 7–14, use the method in Example 3 to find (a) the slope

of the curve at the given point P, and (b) an equation of the tangent

line at P.

- y = x2 – 5, P(2, -1)
- y = 1 – 5×2, P(2, -19)
- y = x2 – 2x – 3, P(2, -3)
- y = x2 – 4x, P(1, -3)
- y = x3, P(2, 8)
- y = 2 – x3, P(1, 1)
- y = x3 – 12x, P(1, -11)
- y = x3 – 3×2 + 4, P(2, 0)

Instantaneous Rates of Change

- Speed of a car The accompanying figure shows the time-todistance

graph for a sports car accelerating from a standstill.

0 5

200

100

Elapsed time (sec)

Distance (m)

10 15 20

300

400

500

600

650

P

Q1

Q2

Q3

Q4

t

s

- Estimate the slopes of secants PQ1, PQ2, PQ3, and PQ4,

arranging them in order in a table like the one in Figure 2.6.

What are the appropriate units for these slopes?

- Then estimate the car’s speed at time t = 20 sec.
- The accompanying figure shows the plot of distance fallen versus

time for an object that fell from the lunar landing module a distance

80 m to the surface of the moon.

- Estimate the slopes of the secants PQ1, PQ2, PQ3, and PQ4,

arranging them in a table like the one in Figure 2.6.

- About how fast was the object going when it hit the surface?

t

y

0

20

Elapsed time (sec)

Distance fallen (m)

5 10

P

40

60

80

Q1

Q2

Q3

Q4

- The profits of a small company for each of the first five years of

its operation are given in the following table:

Year Profit in $1000s

2010 6

2011 27

2012 62

2013 111

2014 174

- Plot points representing the profit as a function of year, and

join them by as smooth a curve as you can

- What is the average rate of increase of the profits between

2012 and 2014?

- Use your graph to estimate the rate at which the profits were

changing in 2012.

- Make a table of values for the function F(x) = (x + 2)>(x – 2)

at the points x = 1.2, x = 11>10, x = 101>100, x = 1001>1000,

x = 10001>10000, and x = 1.

- Find the average rate of change of F(x) over the intervals

31, x4 for each x _ 1 in your table.

- Extending the table if necessary, try to determine the rate of

change of F(x) at x = 1.

- Let g(x) = 2x for x Ú 0.
- Find the average rate of change of g(x) with respect to x over

the intervals 31, 24, 31, 1.54 and 31, 1 + h4.

- Make a table of values of the average rate of change of g with

respect to x over the interval 31, 1 + h4 for some values of h

approaching zero, say h = 0.1, 0.01, 0.001, 0.0001, 0.00001,

and 0.000001.

- What does your table indicate is the rate of change of g(x)

with respect to x at x = 1?

- Calculate the limit as h approaches zero of the average rate of

change of g(x) with respect to x over the interval 31, 1 + h4.

- Let ƒ(t) = 1>t for t _ 0.
- Find the average rate of change of ƒ with respect to t over the

intervals (i) from t = 2 to t = 3, and (ii) from t = 2 to t = T.

- Make a table of values of the average rate of change of ƒ with

respect to t over the interval 32, T4 , for some values of T

approaching 2, say T = 2.1, 2.01, 2.001, 2.0001, 2.00001,

and 2.000001.

- What does your table indicate is the rate of change of ƒ with

respect to t at t = 2?

- Calculate the limit as T approaches 2 of the average rate of

change of ƒ with respect to t over the interval from 2 to T. You

will have to do some algebra before you can substitute T = 2.

- The accompanying graph shows the total distance s traveled by a

bicyclist after t hours.

0 1

10

20

30

40

2 3 4

Elapsed time (hr)

Distance traveled (mi)

t

s

- Estimate the bicyclist’s average speed over the time intervals

30, 14, 31, 2.54 , and 32.5, 3.54 .

- Estimate the bicyclist’s instantaneous speed at the times t = 12

,

t = 2, and t = 3.

- Estimate the bicyclist’s maximum speed and the specific time

at which it occurs.

- The accompanying graph shows the total amount of gasoline A in

the gas tank of an automobile after being driven for t days.

- Estimate the average rate of gasoline consumption over the

time intervals 30, 34, 30, 54, and 37, 104 .

- Estimate the instantaneous rate of gasoline consumption at

the times t = 1, t = 4, and t = 8.

- Estimate the maximum rate of gasoline consumption and the

specific time at which it occurs

Exercises 2.2

Limits from Graphs

- For the function g(x) graphed here, find the following limits or

explain why they do not exist.

- lim

xS1

g(x) b. lim

xS2

g(x) c. lim

xS3

g(x) d. lim

xS2.5

g(x)

3

x

y

2

1

1

y _ g(x)

- For the function ƒ(t) graphed here, find the following limits or

explain why they do not exist.

- lim

tS -2

ƒ(t) b. lim

tS -1

ƒ(t) c. lim

tS0

ƒ(t) d. lim

tS -0.5

ƒ(t)

t

s

1

0 1

s _ f (t)

_1

_2 _1

- Which of the following statements about the function y = ƒ(x)

graphed here are true, and which are false?

- lim

xS0

ƒ(x) exists.

- lim

xS0

ƒ(x) = 0

- lim

xS0

ƒ(x) = 1

- lim

xS1

ƒ(x) = 1

- lim

xS1

ƒ(x) = 0

- lim

xSc

ƒ(x) exists at every point c in (-1, 1).

- lim

xS1

ƒ(x) does not exist.

x

y

_1 1 2

1

_1

y _ f (x)

- Which of the following statements about the function y = ƒ(x)

graphed here are true, and which are false?

- lim

xS2

ƒ(x) does not exist.

- lim

xS2

ƒ(x) = 2

- lim

xS1

ƒ(x) does not exist.

- lim

xSc

ƒ(x) exists at every point c in (-1, 1).

- lim

xSc

ƒ(x) exists at every point c in (1, 3).

x

y

_1 1 2 3

1

_1

_2

y _ f (x)

Existence of Limits

In Exercises 5 and 6, explain why the limits do not exist.

- lim

xS0

x

0 x 0 6. lim

xS1

1

x – 1

- Suppose that a function ƒ(x) is defined for all real values of x

except x = c. Can anything be said about the existence of

limxSc ƒ(x)? Give reasons for your answer.

- Suppose that a function ƒ(x) is defined for all x in 3-1, 1]. Can

anything be said about the existence of limxS0 ƒ(x)? Give reasons

for your answer.

- If limxS1 ƒ(x) = 5, must ƒ be defined at x = 1? If it is, must

ƒ(1) = 5? Can we conclude anything about the values of ƒ at

x = 1? Explain.

- If ƒ(1) = 5, must limxS1 ƒ(x) exist? If it does, then must

limxS1 ƒ(x) = 5? Can we conclude anything about limxS1 ƒ(x)?

Explain.

Calculating Limits

Find the limits in Exercises 11–22.

- lim

xS -3

(x2 – 13) 12. lim

xS3

(-x2 + 8x – 7)

- lim

tS6

8(t – 5)(t – 7) 14. lim

xS -1

(2×3 – 5×2 + 3x + 5)

- lim

xS2

x + 2

x + 5

- lim

sS2>3

(8 – 3s)(2s – 1)

- lim

xS-1>4

16x(13x + 16)2 18. lim

yS2

y + 2

y2 + 5y + 6

- lim

yS -9

(18 – y)7>3 20. lim

zS4

2z2 – 10

- lim

hS0

5

25h + 4 + 4

- lim

hS0

24h + 1 – 1

h

Limits of quotients Find the limits in Exercises 23–42.

- lim

xS9

x – 9

x2 – 81

- lim

xS -3

x + 3

x2 + 4x + 3

- lim

xS 8

x2 – 2x – 48

x – 8

- lim

xS6

x2 – 4x – 12

x – 6

- lim

tS1

t2 + t – 2

t2 – 1

- lim

tS -1

t2 + 3t + 2

t2 – t – 2

- lim

xS -2

-2x – 4

x3 + 2×2 30. lim

yS0

5y3 + 8y2

3y4 – 16y2

- lim

xS1

x-1 – 1

x – 1

- lim

xS0

1

x – 1 + 1

x + 1

x

- lim

uS2

u4 – 16

u3 – 8

- lim

uS3

u4 – 81

u3 – 27

- lim

xS25

2x – 5

x – 25

- lim

xS4

4x – x2

2 – 2x

- lim

xS87

x – 87

2x + 13 – 10

- lim

xS -1

2×2 + 8 – 3

x + 1

- lim

xS2

2×2 + 12 – 4

x – 2

- lim

xS -2

x + 2

2×2 + 5 – 3

- lim

xS -24

23 – 2×2 – 47

x + 24

- lim

xS4

4 – x

5 – 2×2 + 9

Limits with trigonometric functions Find the limits in Exercises

43–50.

- lim

xS0

(2 sin x – 1) 44. lim

xSp>4

sin2 x

- lim

xS0

sec x 46. lim

xSp>3

tan x

- lim

xS0

1 + x + sin x

3 cos x

- lim

xS0

(x2 – 1)(2 – cos x)

- lim

xS -p

2x + 4 cos (x + p) 50. lim

xS0

27 + sec2 x

Using Limit Rules

- Suppose limxS0 ƒ(x) = 1 and limxS0 g(x) = -5. Name the

rules in Theorem 1 that are used to accomplish steps (a), (b), and

(c) of the following calculation.

lim

xS0

2ƒ(x) – g(x)

(ƒ(x) + 7)2>3 =

lim

xS0

(2ƒ(x) – g(x))

lim

xS0

(ƒ(x) + 7)2>3 (a)

=

lim

xS0

2ƒ(x) – lim

xS0

g(x)

alim

xS0

(ƒ(x) + 7)b

2>3 (b)

=

2 lim

xS0

ƒ(x) – lim

xS0

g(x)

alim

xS0

ƒ(x) + lim

xS0

7b

2>3 (c)

=

(2)(1) – (-5)

(1 + 7)2>3 = 7

4

- Let limxS1 h(x) = 5, limxS1 p(x) = 1, and limxS1 r(x) = 2.

Name the rules in Theorem 1 that are used to accomplish steps

(a), (b), and (c) of the following calculation.

lim

xS1

25h(x)

p(x)(4 – r(x)) =

lim

xS1

25h(x)

lim

xS1

(p(x)(4 – r(x)))

(a)

=

4lim

xS1

5h(x)

alim

xS1

p(x)b alim

xS1

(4 – r(x))b

(b)

=

45lim

xS1

h(x)

alim

xS1

p(x)b alim

xS1

4 – lim

xS1

r(x)b

(c)

=

2(5)(5)

(1)(4 – 2) = 5

2

- Suppose limxS3 ƒ(x) = 3 and limxS3 g(x) = -8. Find
- lim

xS3

ƒ(x)g(x) b. lim

xS3

3ƒ(x)g(x)

- lim

xS3

(ƒ(x) + 5g(x)) d. lim

xS3

ƒ(x)

ƒ(x) – g(x)

- Suppose limxS4 ƒ(x) = 0 and limxS4 g(x) = -3. Find
- lim

xS4

(g(x) + 3) b. lim

xS4

xƒ(x)

- lim

xS4

(g(x))2 d. lim

xS4

g(x)

ƒ(x) – 1

- Suppose limxSb ƒ(x) = 10 and limxSb g(x) = -3. Find
- lim

xSb

(ƒ(x) + g(x)) b. lim

xSb

6(x)

- lim

xSb

ƒ(x) # g(x) d. lim

xSb

ƒ(x)>g(x)

- Suppose that limxS -2 p(x) = 4, limxS -2 r(x) = 0, and

limxS -2 s(x) = -3. Find

- lim

xS -2

(p(x) + r(x) + s(x))

- lim

xS -2

p(x) # r(x) # s(x)

- lim

xS -2

(-4p(x) + 5r(x))>s(x)

Limits of Average Rates of Change

Because of their connection with secant lines, tangents, and instantaneous

rates, limits of the form

lim

hS0

ƒ(x + h) – ƒ(x)

h

occur frequently in calculus. In Exercises 57–62, evaluate this limit

for the given value of x and function ƒ.

- ƒ(x) = x2, x = 1
- ƒ(x) = x2, x = -2
- ƒ(x) = 3x – 4, x = 2
- ƒ(x) = 1>x, x = -2
- ƒ(x) = 2x, x = 7
- ƒ(x) = 23x + 1, x = 0

Using the Sandwich Theorem

- If 25 – 2×2 … ƒ(x) … 25 – x2 for -1 … x … 1, find

limxS0 ƒ(x).

- If 2 – x2 … g(x) … 2 cos x for all x, find limxS0 g(x).
- a. It can be shown that the inequalities

1 – x2

6 6 x sin x

2 – 2 cos x 6 1

hold for all values of x close to zero. What, if anything, does

this tell you about

lim

xS0

x sin x

2 – 2 cos x

?

Give reasons for your answer.

- Graph y = 1 – (x2>6), y = (x sin x)>(2 – 2 cos x), and

y = 1 together for -2 … x … 2. Comment on the behavior

of the graphs as xS 0.

- a. Suppose that the inequalities

1

2 – x2

24 6 1 – cos x

x2 6 1

2

hold for values of x close to zero. (They do, as you will see in

Section 9.9.) What, if anything, does this tell you about

lim

xS0

1 – cos x

x2 ?

Give reasons for your answer.

- Graph the equations y = (1>2) – (x2>24),

y = (1 – cos x)>x2, and y = 1>2 together for -2 … x … 2.

Comment on the behavior of the graphs as xS 0.

Estimating Limits

You will find a graphing calculator useful for Exercises 67–76.

- Let ƒ(x) = (x2 – 9)>(x + 3).
- Make a table of the values of ƒ at the points x = -3.1,

-3.01, -3.001, and so on as far as your calculator can go.

Then estimate limxS -3 ƒ(x). What estimate do you arrive at

if you evaluate ƒ at x = -2.9, -2.99, -2.999,cinstead?

- Support your conclusions in part (a) by graphing ƒ near

c = -3 and using Zoom and Trace to estimate y-values on

the graph as xS -3.

- Find limxS -3 ƒ(x) algebraically, as in Example 7.
- Let g(x) = (x2 – 2) >(x – 22).
- Make a table of the values of g at the points x = 1.4, 1.41,

1.414, and so on through successive decimal approximations

of 22. Estimate limxS22 g(x).

- Support your conclusion in part (a) by graphing g near

c = 22 and using Zoom and Trace to estimate y-values on

the graph as xS 22.

- Find limxS22 g(x) algebraically.
- Let G(x) = (x + 6)> (x2 + 4x – 12).
- Make a table of the values of G at x = -5.9, -5.99, -5.999,

and so on. Then estimate limxS -6 G(x). What estimate do

you arrive at if you evaluate G at x = -6.1, -6.01,

-6.001, cinstead?

- Support your conclusions in part (a) by graphing G and using

Zoom and Trace to estimate y-values on the graph as

xS -6.

- Find limxS -6 G(x) algebraically.
- Let h(x) = (x2 – 2x – 3) > (x2 – 4x + 3).
- Make a table of the values of h at x = 2.9, 2.99, 2.999, and

so on. Then estimate limxS3 h(x). What estimate do you

arrive at if you evaluate h at x = 3.1, 3.01, 3.001,c

instead?

- Support your conclusions in part (a) by graphing h near

c = 3 and using Zoom and Trace to estimate y-values on the

graph as xS 3.

- Find limxS3 h(x) algebraically.
- Let ƒ(x) = (x2 – 1) > ( 0 x 0 – 1).
- Make tables of the values of ƒ at values of x that approach

c = -1 from above and below. Then estimate limxS -1 ƒ(x).

T

T

- Support your conclusion in part (a) by graphing ƒ near

c = -1 and using Zoom and Trace to estimate y-values on

the graph as xS -1.

- Find limxS -1 ƒ(x) algebraically.
- Let F(x) = (x2 + 3x + 2) > (2 – 0 x 0 ).
- Make tables of values of F at values of x that approach

c = -2 from above and below. Then estimate limxS -2 F(x).

- Support your conclusion in part (a) by graphing F near

c = -2 and using Zoom and Trace to estimate y-values on

the graph as xS -2.

- Find limxS -2 F(x) algebraically.
- Let g(u) = (sin u)>u.
- Make a table of the values of g at values of u that approach

u0 = 0 from above and below. Then estimate limuS0 g(u).

- Support your conclusion in part (a) by graphing g near

u0 = 0.

- Let G(t) = (1 – cos t)>t2.
- Make tables of values of G at values of t that approach t0 = 0

from above and below. Then estimate limtS0 G(t).

- Support your conclusion in part (a) by graphing G near

t0 = 0.

- Let ƒ(x) = x1>(1-x).
- Make tables of values of ƒ at values of x that approach c = 1

from above and below. Does ƒ appear to have a limit as

xS 1? If so, what is it? If not, why not?

- Support your conclusions in part (a) by graphing ƒ near c = 1.
- Let ƒ(x) = (3x – 1)>x.
- Make tables of values of ƒ at values of x that approach c = 0

from above and below. Does ƒ appear to have a limit as

xS 0? If so, what is it? If not, why not?

- Support your conclusions in part (a) by graphing ƒ near c = 0.

Theory and Examples

- If x4 … ƒ(x) … x2 for x in 3-1, 14 and x2 … ƒ(x) … x4 for

x 6 -1 and x 7 1, at what points c do you automatically know

limxSc ƒ(x)? What can you say about the value of the limit at

these points?

- Suppose that g(x) … ƒ(x) … h(x) for all x _ 2 and suppose that

lim

xS2

g(x) = lim

xS2

h(x) = -5.

Can we conclude anything about the values of ƒ, g, and h at

x = 2? Could ƒ(2) = 0? Could limxS2 ƒ(x) = 0? Give reasons

for your answers.

- If lim

xS4

ƒ(x) – 5

x – 2 = 1, find lim

xS4

ƒ(x).

- If lim

xS -2

ƒ(x)

x2 = 1, find

- lim

xS -2

ƒ(x) b. lim

xS -2

ƒ(x)

x

- a. If lim

xS2

ƒ(x) – 5

x – 2 = 3, find lim

xS2

ƒ(x).

- If lim

xS2

ƒ(x) – 5

x – 2 = 4, find lim

xS2

ƒ(x).

- If lim

xS0

ƒ(x)

x2 = 1, find

- lim

xS0

ƒ(x)

- lim

xS0

ƒ(x)

x

- a. Graph g(x) = x sin (1>x) to estimate limxS0 g(x), zooming in

on the origin as necessary.

- Confirm your estimate in part (a) with a proof.
- a. Graph h(x) = x2 cos (1>x3) to estimate limxS0 h(x), zooming

in on the origin as necessary.

- Confirm your estimate in part (a) with a proof.

COMPUTER EXPLORATIONS

Graphical Estimates of Limits

In Exercises 85–90, use a CAS to perform the following steps:

- Plot the function near the point c being approached.
- From your plot guess the value of the limit.

Exercises 2.3

Centering Intervals About a Point

In Exercises 1–6, sketch the interval (a, b) on the x-axis with the

point c inside. Then find a value of d 7 0 such that for all

x, 0 6 0 x – c 0 6 d 1 a 6 x 6 b.

- a = 1, b = 7, c = 5
- a = 1, b = 7, c = 2
- a = -7>2, b = -1>2, c = -3
- a = -7>2, b = -1>2, c = -3>2
- a = 4>9, b = 4>7, c = 1>2
- a = 2.7591, b = 3.2391, c = 3

Finding Deltas Graphically

In Exercises 7–14, use the graphs to find a d 7 0 such that for all x

0 6 0 x – c 0 6 d 1 0 ƒ(x) – L 0 6 P.

- 14.

2.5

2

1.5

y

x

_1

L _ 2

f (x) _

c _ _1

P _ 0.5

16

_ 9

16

_25

0

“_x

2

y _

“_x

2

0

y

x

c _

L _ 2

P _ 0.01

y _ 1x

f (x) _ 1×12

2.01

2

1.99

12

1

2.01

1

1.99

NOT TO SCALE

Finding Deltas Algebraically

Each of Exercises 15–30 gives a function ƒ(x) and numbers L, c, and

P 7 0. In each case, find an open interval about c on which the inequality

0 ƒ(x) – L 0 6 P holds. Then give a value for d 7 0 such that for

all x satisfying 0 6 0 x – c 0 6 d the inequality 0 ƒ(x) – L 0 6 P

holds.

- ƒ(x) = x + 1, L = 5, c = 4, P = 0.01
- ƒ(x) = 2x – 2, L = -6, c = -2, P = 0.02
- ƒ(x) = 2x + 1, L = 1, c = 0, P = 0.1
- ƒ(x) = 2x, L = 1>2, c = 1>4, P = 0.1
- ƒ(x) = 219 – x, L = 3, c = 10, P = 1
- ƒ(x) = 2x – 7, L = 4, c = 23, P = 1
- ƒ(x) = 1>x, L = 1>4, c = 4, P = 0.05
- ƒ(x) = x2, L = 3, c = 23, P = 0.1
- ƒ(x) = x2, L = 4, c = -2, P = 0.5
- ƒ(x) = 1>x, L = -1, c = -1, P = 0.1
- ƒ(x) = x2 – 5, L = 11, c = 4, P = 1
- ƒ(x) = 120>x, L = 5, c = 24, P = 1
- ƒ(x) = mx, m 7 0, L = 2m, c = 2, P = 0.03
- ƒ(x) = mx, m 7 0, L = 3m, c = 3, P = c 7 0
- ƒ(x) = mx + b, m 7 0, L = (m>2) + b,

c = 1>2, P = c 7 0

- ƒ(x) = mx + b, m 7 0, L = m + b, c = 1,

P = 0.05

Using the Formal Definition

Each of Exercises 31–36 gives a function ƒ(x), a point c, and a positive

number P. Find L = lim

xSc

ƒ(x). Then find a number d 7 0 such

that for all x

0 6 0 x – c 0 6 d 1 0 ƒ(x) – L 0 6 P.

- ƒ(x) = 2 – 4x, c = 4, P = 0.03
- ƒ(x) = -3x – 2, c = -1, P = 0.03
- ƒ(x) = x2 – 9

x – 3

, c = 3, P = 0.03

- ƒ(x) = x2 + 6x + 5

x + 5

, c = -5, P = 0.05

- ƒ(x) = 21 – x, c = -3, P = 0.5
- ƒ(x) = 4>x, c = 2, P = 0.4

Prove the limit statements in Exercises 37–50.

- lim

xS4

(9 – x) = 5 38. lim

xS3

(3x – 7) = 2

- lim

xS9

2x – 5 = 2 40. lim

xS0

24 – x = 2

- lim

xS1

ƒ(x) = 1 if ƒ(x) = e

x2, x _ 1

2, x = 1

- lim

xS -2

ƒ(x) = 4 if ƒ(x) = e

x2, x _ -2

1, x = -2

- lim

xS1

1x

= 1 44. lim

xS23

1

x2 = 1

3

- lim

xS -3

x2 – 9

x + 3 = -6 46. lim

xS1

x2 – 1

x – 1 = 2

- lim

xS1

ƒ(x) = 2 if ƒ(x) = e

4 – 2x, x 6 1

6x – 4, x Ú 1

- lim

xS0

ƒ(x) = 0 if ƒ(x) = e

2x, x 6 0

x>2, x Ú 0

- lim

xS0

x sin

1x

= 0

- lim

xS0

x2 sin

1x

= 0

x

y

1

_1

_1 0 1

y _ x2

y _ _x2

y _ x2 sin1x

2p

2p

_

Theory and Examples

- Define what it means to say that lim

xS0

g(x) = k.

- Prove that lim

xSc

ƒ(x) = L if and only if lim

hS0

ƒ(h + c) = L.

- A wrong statement about limits Show by example that the

following statement is wrong.

The number L is the limit of ƒ(x) as x approaches c

if ƒ(x) gets closer to L as x approaches c.

Explain why the function in your example does not have the

given value of L as a limit as xS c.

- Another wrong statement about limits Show by example that

the following statement is wrong.

The number L is the limit of ƒ(x) as x approaches c if, given any

P 7 0, there exists a value of x for which 0 ƒ(x) – L 0 6 P.

Explain why the function in your example does not have the

given value of L as a limit as xS c.

- Grinding engine cylinders Before contracting to grind engine

cylinders to a cross-sectional area of 9 in2, you need to know how

much deviation from the ideal cylinder diameter of c = 3.385 in.

you can allow and still have the area come within 0.01 in2 of the

required 9 in2. To find out, you let A = p(x>2)2 and look for the

interval in which you must hold x to make 0 A – 9 0 … 0.01.

What interval do you find?

- Manufacturing electrical resistors Ohm’s law for electrical

circuits like the one shown in the accompanying figure states that

V = RI. In this equation, V is a constant voltage, I is the current

in amperes, and R is the resistance in ohms. Your firm has been

asked to supply the resistors for a circuit in which V will be 120

volts and I is to be 5 { 0.1 amp. In what interval does R have to

lie for I to be within 0.1 amp of the value I0 = 5?

V I R

_

+

T

When Is a Number L Not the Limit of ƒ(x) as x u c?

Showing L is not a limit We can prove that limxSc ƒ(x) _ L by

providing an P 7 0 such that no possible d 7 0 satisfies the condition

for all x, 0 6 0 x – c 0 6 d 1 0 ƒ(x) – L 0 6 P.

We accomplish this for our candidate P by showing that for each

d 7 0 there exists a value of x such that

0 6 0 x – c 0 6 d and 0 ƒ(x) – L 0 Ú P.

y

x

0 c _ d c c + d

L

L _ P

L + P

y _ f (x)

a value of x for which

0 < 0 x _ c 0 < d and 0 f (x) _ L 0 _ P

f (x)

- Let ƒ(x) = e

x, x 6 1

x + 1, x 7 1.

x

y

y _ x + 1

y _ x

y _ f (x)

1

1

2

- Let P = 1>2. Show that no possible d 7 0 satisfies the following

condition:

For all x, 0 6 0 x – 1 0 6 d 1 0 ƒ(x) – 2 0 6 1>2.

That is, for each d 7 0 show that there is a value of x such

that

0 6 0 x – 1 0 6 d and 0 ƒ(x) – 2 0 Ú 1>2.

This will show that limxS1 ƒ(x) _ 2.

- Show that limxS1 ƒ(x) _ 1.
- Show that limxS1 ƒ(x) _ 1.5.

Finding Deltas Algebraically

Each of Exercises 15–30 gives a function ƒ(x) and numbers L, c, and

P 7 0. In each case, find an open interval about c on which the inequality

0 ƒ(x) – L 0 6 P holds. Then give a value for d 7 0 such that for

all x satisfying 0 6 0 x – c 0 6 d the inequality 0 ƒ(x) – L 0 6 P

holds.

- ƒ(x) = x + 1, L = 5, c = 4, P = 0.01
- ƒ(x) = 2x – 2, L = -6, c = -2, P = 0.02
- ƒ(x) = 2x + 1, L = 1, c = 0, P = 0.1
- ƒ(x) = 2x, L = 1>2, c = 1>4, P = 0.1
- ƒ(x) = 219 – x, L = 3, c = 10, P = 1
- ƒ(x) = 2x – 7, L = 4, c = 23, P = 1
- ƒ(x) = 1>x, L = 1>4, c = 4, P = 0.05
- ƒ(x) = x2, L = 3, c = 23, P = 0.1
- ƒ(x) = x2, L = 4, c = -2, P = 0.5
- ƒ(x) = 1>x, L = -1, c = -1, P = 0.1
- ƒ(x) = x2 – 5, L = 11, c = 4, P = 1
- ƒ(x) = 120>x, L = 5, c = 24, P = 1
- ƒ(x) = mx, m 7 0, L = 2m, c = 2, P = 0.03
- ƒ(x) = mx, m 7 0, L = 3m, c = 3, P = c 7 0
- ƒ(x) = mx + b, m 7 0, L = (m>2) + b,

c = 1>2, P = c 7 0

- ƒ(x) = mx + b, m 7 0, L = m + b, c = 1,

P = 0.05

Using the Formal Definition

Each of Exercises 31–36 gives a function ƒ(x), a point c, and a positive

number P. Find L = lim

xSc

ƒ(x). Then find a number d 7 0 such

that for all x

0 6 0 x – c 0 6 d 1 0 ƒ(x) – L 0 6 P.

- ƒ(x) = 2 – 4x, c = 4, P = 0.03
- ƒ(x) = -3x – 2, c = -1, P = 0.03
- ƒ(x) = x2 – 9

x – 3

, c = 3, P = 0.03

- ƒ(x) = x2 + 6x + 5

x + 5

, c = -5, P = 0.05

- ƒ(x) = 21 – x, c = -3, P = 0.5
- ƒ(x) = 4>x, c = 2, P = 0.4

Prove the limit statements in Exercises 37–50.

- lim

xS4

(9 – x) = 5 38. lim

xS3

(3x – 7) = 2

- lim

xS9

2x – 5 = 2 40. lim

xS0

24 – x = 2

- lim

xS1

ƒ(x) = 1 if ƒ(x) = e

x2, x _ 1

2, x = 1

- lim

xS -2

ƒ(x) = 4 if ƒ(x) = e

x2, x _ -2

1, x = -2

- lim

xS1

1x

= 1 44. lim

xS23

1

x2 = 1

3

- lim

xS -3

x2 – 9

x + 3 = -6 46. lim

xS1

x2 – 1

x – 1 = 2

- lim

xS1

ƒ(x) = 2 if ƒ(x) = e

4 – 2x, x 6 1

6x – 4, x Ú 1

- lim

xS0

ƒ(x) = 0 if ƒ(x) = e

2x, x 6 0

x>2, x Ú 0

- lim

xS0

x sin

1x

= 0

x

y

y _ x sin1x

1p

_

1p

1

2p

_

1

2p

90

- lim

xS0

x2 sin

1x

= 0

x

y

1

_1

_1 0 1

y _ x2

y _ _x2

y _ x2 sin1x

2p

2p

_

Theory and Examples

- Define what it means to say that lim

xS0

g(x) = k.

- Prove that lim

xSc

ƒ(x) = L if and only if lim

hS0

ƒ(h + c) = L.

- A wrong statement about limits Show by example that the

following statement is wrong.

The number L is the limit of ƒ(x) as x approaches c

if ƒ(x) gets closer to L as x approaches c.

Explain why the function in your example does not have the

given value of L as a limit as xS c.

- Another wrong statement about limits Show by example that

the following statement is wrong.

The number L is the limit of ƒ(x) as x approaches c if, given any

P 7 0, there exists a value of x for which 0 ƒ(x) – L 0 6 P.

Explain why the function in your example does not have the

given value of L as a limit as xS c.

- Grinding engine cylinders Before contracting to grind engine

cylinders to a cross-sectional area of 9 in2, you need to know how

much deviation from the ideal cylinder diameter of c = 3.385 in.

you can allow and still have the area come within 0.01 in2 of the

required 9 in2. To find out, you let A = p(x>2)2 and look for the

interval in which you must hold x to make 0 A – 9 0 … 0.01.

What interval do you find?

- Manufacturing electrical resistors Ohm’s law for electrical

circuits like the one shown in the accompanying figure states that

V = RI. In this equation, V is a constant voltage, I is the current

in amperes, and R is the resistance in ohms. Your firm has been

asked to supply the resistors for a circuit in which V will be 120

volts and I is to be 5 { 0.1 amp. In what interval does R have to

lie for I to be within 0.1 amp of the value I0 = 5?

V I R

_

+

T

When Is a Number L Not the Limit of ƒ(x) as x u c?

Showing L is not a limit We can prove that limxSc ƒ(x) _ L by

providing an P 7 0 such that no possible d 7 0 satisfies the condition

for all x, 0 6 0 x – c 0 6 d 1 0 ƒ(x) – L 0 6 P.

We accomplish this for our candidate P by showing that for each

d 7 0 there exists a value of x such that

0 6 0 x – c 0 6 d and 0 ƒ(x) – L 0 Ú P.

y

x

0 c _ d c c + d

L

L _ P

L + P

y _ f (x)

a value of x for which

0 < 0 x _ c 0 < d and 0 f (x) _ L 0 _ P

f (x)

- Let ƒ(x) = e

x, x 6 1

x + 1, x 7 1.

x

y

y _ x + 1

y _ x

y _ f (x)

1

1

2

- Let P = 1>2. Show that no possible d 7 0 satisfies the following

condition:

For all x, 0 6 0 x – 1 0 6 d 1 0 ƒ(x) – 2 0 6 1>2.

That is, for each d 7 0 show that there is a value of x such

that

0 6 0 x – 1 0 6 d and 0 ƒ(x) – 2 0 Ú 1>2.

This will show that limxS1 ƒ(x) _ 2.

- Show that limxS1 ƒ(x) _ 1.
- Show that limxS1 ƒ(x) _ 1.5.
- Let h(x) = c

x2, x 6 2

3, x = 2

2, x 7 2.

x

y

0 2

1

2

3

4 y _ h(x)

y _ x2

y _ 2

Show that

- lim

xS2

h(x) _ 4

- lim

xS2

h(x) _ 3

- lim

xS2

h(x) _ 2

- For the function graphed here, explain why
- lim

xS3

ƒ(x) _ 4

- lim

xS3

ƒ(x) _ 4.8

- lim

xS3

ƒ(x) _ 3

x

y

0 3

3

4

4.8

y _ f (x)

- a. For the function graphed here, show that limxS -1 g(x) _ 2.
- Does limxS -1 g(x) appear to exist? If so, what is the value of

the limit? If not, why not?

y

x

y _ g(x)

_1 0

1

2

COMPUTER EXPLORATIONS

In Exercises 61–66, you will further explore finding deltas graphically.

Use a CAS to perform the following steps:

- Plot the function y = ƒ(x) near the point c being approached.
- Guess the value of the limit L and then evaluate the limit symbolically

to see if you guessed correctly.

- Using the value P = 0.2, graph the banding lines y1 = L – P

and y2 = L + P together with the function ƒ near c.

- From your graph in part (c), estimate a d 7 0 such that for all x

0 6 0 x – c 0 6 d 1 0 ƒ(x) – L 0 6 P.

Test your estimate by plotting ƒ, y1, and y2 over the interval

0 6 0 x – c 0 6 d. For your viewing window use c – 2d …

x … c + 2d and L – 2P … y … L + 2P. If any function values

lie outside the interval 3L – P, L + P], your choice of d

was too large. Try again with a smaller estimate.

- Repeat parts (c) and (d) successively for P = 0.1, 0.05, and 0.001.
- ƒ(x) = x4 – 81

x – 3

, c = 3 62. ƒ(x) = 5×3 + 9×2

2×5 + 3×2 , c = 0

- ƒ(x) = sin 2x

3x

, c = 0 64. ƒ(x) =

x(1 – cos x)

x – sin x

, c = 0

- ƒ(x) =

23

x – 1

x – 1

, c = 1

- ƒ(x) =

3×2 – (7x + 1)2x + 5

x – 1

, c = 1

Exercises 2.4

Finding Limits Graphically

- Which of the following statements about the function y = ƒ(x)

graphed here are true, and which are false?

x

y

_1 1 2

1

0

y _ f (x)

- lim

xS -1+

ƒ(x) = 1 b. lim

xS0-

ƒ(x) = 0

- lim

xS0-

ƒ(x) = 1 d. lim

xS0-

ƒ(x) = lim

xS0+

ƒ(x)

- lim

xS0

ƒ(x) exists. f. lim

xS0

ƒ(x) = 0

- lim

xS0

ƒ(x) = 1 h. lim

xS1

ƒ(x) = 1

- lim

xS1

ƒ(x) = 0 j. lim

xS2-

ƒ(x) = 2

- lim

xS -1-

ƒ(x) does not exist. l. lim

xS2+

ƒ(x) = 0

- Which of the following statements about the function y = ƒ(x)

graphed here are true, and which are false?

x

y

0

1

2

_1 1 2 3

y _ f (x)

- lim

xS -1+

ƒ(x) = 1 b. lim

xS2

ƒ(x) does not exist.

- lim

xS2

ƒ(x) = 2 d. lim

xS1-

ƒ(x) = 2

- lim

xS1+

ƒ(x) = 1 f. lim

xS1

ƒ(x) does not exist.

- lim

xS0+

ƒ(x) = lim

xS0-

ƒ(x)

- lim

xSc

ƒ(x) exists at every c in the open interval (-1, 1).

- lim

xSc

ƒ(x) exists at every c in the open interval (1, 3).

- lim

xS -1-

ƒ(x) = 0 k. lim

xS3+

ƒ(x) does not exist.

- Let ƒ(x) = c

3 – x, x 6 2

x

2 + 1, x 7 2.

x

y

3

0 2 4

y _ 3 _ x

y _ x + 1

2

- Find limxS2+ ƒ(x) and limxS2- ƒ(x).
- Does limxS2 ƒ(x) exist? If so, what is it? If not, why not?
- Find limxS4- ƒ(x) and limxS4+ ƒ(x).
- Does limxS4 ƒ(x) exist? If so, what is it? If not, why not?
- Let ƒ(x) = d

3 – x, x 6 2

2, x = 2

x

2

, x 7 2.

x

y

y _ 3 _ x

0

3

_2 2

y _

2 x

- Find limxS2+ ƒ(x), limxS2- ƒ(x), and ƒ(2).
- Does limxS2 ƒ(x) exist? If so, what is it? If not, why not?
- Find limxS -1- ƒ(x) and limxS -1+ ƒ(x).
- Does limxS -1 ƒ(x) exist? If so, what is it? If not, why not?
- Let ƒ(x) = c

0, x … 0

sin

1x

, x 7 0.

x

y

0

_1

1

1x

sin ,

y _

0, x _ 0

x > 0

- Does limxS0+ ƒ(x) exist? If so, what is it? If not, why not?
- Does limxS0- ƒ(x) exist? If so, what is it? If not, why not?
- Does limxS0 ƒ(x) exist? If so, what is it? If not, why not?
- Let g(x) = 2x sin(1>x).

x

0

_1

1

y

y _ “x

y _ _”x

1 1p

1

2p

2p

y _ “x sin 1x

- Does limxS0+ g(x) exist? If so, what is it? If not, why not?
- Does limxS0- g(x) exist? If so, what is it? If not, why not?
- Does limxS0 g(x) exist? If so, what is it? If not, why not?
- a. Graph ƒ(x) = e

x3, x _ 1

0, x = 1.

- Find limxS1- ƒ(x) and limxS1+ ƒ(x).
- Does limxS1 ƒ(x) exist? If so, what is it? If not, why not?
- a. Graph ƒ(x) = e

1 – x2, x _ 1

2, x = 1.

- Find limxS1+ ƒ(x) and limxS1- ƒ(x).
- Does limxS1 ƒ(x) exist? If so, what is it? If not, why not?

Graph the functions in Exercises 9 and 10. Then answer these questions.

- What are the domain and range of ƒ?
- At what points c, if any, does limxSc ƒ(x) exist?
- At what points does only the left-hand limit exist?
- At what points does only the right-hand limit exist?
- ƒ(x) = c

21 – x2, 0 … x 6 1

1, 1 … x 6 2

2, x = 2

- ƒ(x) = c

x, -1 … x 6 0, or 0 6 x … 1

1, x = 0

0, x 6 -1 or x 7 1

Finding One-Sided Limits Algebraically

Find the limits in Exercises 11–18.

- lim

xS -0.5-A

x + 2

x + 1

- lim

xS1+ A

x – 1

x + 2

- lim

xS -2+

a x

x + 1b a2x + 5

x2 + x

b

- lim

xS1-

a 1

x + 1b ax + 6

x b a3 – x

7 b

- lim

hS0+

2h2 + 6h + 13 – 213

- lim

hS0-

26 – 25h2 + 11h + 6

h

- a. lim

xS -2+

(x + 3)

0 x + 2 0

x + 2

- lim

xS -2-

(x + 3)

0 x + 2 0

x + 2

- a. lim

xS1+

22x (x – 1)

0 x – 1 0

- lim

xS1-

22x (x – 1)

0 x – 1 0

Use the graph of the greatest integer function y = :x;, Figure 1.10 in

Section 1.1, to help you find the limits in Exercises 19 and 20.

- a. lim

uS3+

:u;

u

- lim

uS3-

:u;

u

- a. lim

tS4+

(t – :t;) b. lim

tS4-

(t – :t;)

Using lim

Uu0

sin U

U

_ 1

Find the limits in Exercises 21–42.

- lim

uS0

2sin 29u

29u

- lim

tS0

sin kt

t (k constant)

- lim

yS0

sin 19y

7y

- lim

hS0-

h

sin 3h

- lim

xS0

tan 3x

x 26. lim

tS0

2t

tan t

- lim

xS0

x csc 5x

cos 12x

- lim

xS0

6×2(cot x)(csc 2x)

- lim

xS0

11x + 11x cos 11x

8sin 11x cos 11x

- lim

xS0

x2 – x + sin x

2x

- lim

uS0

1 – cos u

sin 2u

- lim

xS0

x – x cos x

sin2 3x

- lim

tS0

sin (1 – cos t)

1 – cos t

- lim

hS0

sin (sin h)

sin h

- lim

uS0

sin u

sin 2u

- lim

xS0

sin 5x

sin 4x

- lim

uS0

u cos u 38. lim

uS0

sin u cot 2u

- lim

xS0

tan 3x

sin 8x

- lim

yS0

sin 3y cot 5y

y cot 4y

- lim

uS0

tan u

u2 cot 3u

- lim

uS0

u cot 4u

sin2 u cot2 2u

Theory and Examples

- Once you know limxSa+ ƒ(x) and limxSa- ƒ(x) at an interior point

of the domain of ƒ, do you then know limxSa ƒ(x)? Give reasons

for your answer.

- If you know that limxSc ƒ(x) exists, can you find its value by calculating

limxSc+ ƒ(x)? Give reasons for your answer.

- Suppose that ƒ is an odd function of x. Does knowing that

limxS0+ ƒ(x) = 3 tell you anything about limxS0- ƒ(x)? Give reasons

for your answer.

- Suppose that ƒ is an even function of x. Does knowing that

limxS2- ƒ(x) = 7 tell you anything about either limxS -2- ƒ(x) or

limxS -2+ ƒ(x)? Give reasons for your answer.

Formal Definitions of One-Sided Limits

- Given P 7 0, find an interval I = (5, 5 + d), d 7 0, such that if

x lies in I, then 2x – 5 6 P. What limit is being verified and

what is its value?

- Given P 7 0, find an interval I = (4 – d, 4), d 7 0, such that if

x lies in I, then 24 – x 6 P. What limit is being verified and

what is its value?

Use the definitions of right-hand and left-hand limits to prove the

limit statements in Exercises 49 and 50.

- lim

xS0-

x

0 x 0 = -1 50. lim

xS2+

x – 2

0 x – 2 0 = 1

- Greatest integer function Find (a) limxS400+ :x; and (b)

limxS400- :x;; then use limit definitions to verify your findings.

(c) Based on your conclusions in parts (a) and (b), can you say

anything about limxS400 :x;? Give reasons for your answer.

- One-sided limits Let ƒ(x) = e

x2 sin (1>x), x 6 0

2x, x 7 0.

Find (a) limxS0+ ƒ(x) and (b) limxS0- ƒ(x); then use limit definitions

to verify your findings. (c) Based on your conclusions in

parts (a) and (b), can you say anything about limxS0 ƒ(x)? Give

reasons for your answer.

Exercises 2.5

Continuity from Graphs

In Exercises 1–4, say whether the function graphed is continuous on

3-1, 34. If not, where does it fail to be continuous and why?

- 2.

x

y

_1 0 1 3

1

2

2

y _ f (x)

x

y

_1 0 1 3

1

2

2

y _ g(x)

- 4.

x

y

0 1 3

2

_1 2

1

y _ h(x)

x

y

_1 0 1 3

1

2

2

y _ k(x)

Exercises 5–10 refer to the function

ƒ(x) = e

x2 – 1, -1 … x 6 0

2x, 0 6 x 6 1

1, x = 1

-2x + 4, 1 6 x 6 2

0, 2 6 x 6 3

graphed in the accompanying figure.

2

x

y

0 3

(1, 2)

_1 1 2

(1, 1)

y _ f (x)

y _ _2x + 4

y _ x2 _ 1 _1

y _ 2x

The graph for Exercises 5–10.

- a. Does ƒ(-1) exist?
- Does limxS -1+ ƒ(x) exist?
- Does limxS -1+ ƒ(x) = ƒ(-1)?
- Is ƒ continuous at x = -1?
- a. Does ƒ(1) exist?
- Does limxS1 ƒ(x) exist?
- Does limxS1 ƒ(x) = ƒ(1)?
- Is ƒ continuous at x = 1?
- a. Is ƒ defined at x = 2? (Look at the definition of ƒ.)
- Is ƒ continuous at x = 2?
- At what values of x is ƒ continuous?
- What value should be assigned to ƒ(2) to make the extended

function continuous at x = 2?

- To what new value should ƒ(1) be changed to remove the discontinuity?

Applying the Continuity Test

At which points do the functions in Exercises 11 and 12 fail to be continuous?

At which points, if any, are the discontinuities removable?

Not removable? Give reasons for your answers.

- Exercise 1, Section 2.4 12. Exercise 2, Section 2.4

At what points are the functions in Exercises 13–30 continuous?

- y = 1

x – 2 – 3x 14. y = 1

(x + 2)2 + 4

- y = x + 1

x2 – 4x + 3

- y = x + 3

x2 – 3x – 10

- y = 0 x – 1 0 + sin x 18. y = 1

0 x 0 + 1

– x2

2

- y = cos x

x 20. y = x + 2

cos x

- y = csc 2x 22. y = tan px

2

- y = x tan x

x2 + 1

- y =

2×4 + 1

1 + sin2 x

- y = 22x + 3 26. y = 24

3x – 1

- y = (2x – 1)1>3 28. y = (2 – x)1>5
- g(x) = c

x2 – x – 6

x – 3

, x _ 3

5, x = 3

- ƒ(x) = d

x3 – 8

x2 – 4

, x _ 2, x _ -2

3, x = 2

4, x = -2

Limits Involving Trigonometric Functions

Find the limits in Exercises 31–38. Are the functions continuous at the

point being approached?

- lim

xSp

sin (x – sin x) 32. lim

tS0

sinap

2

cos (tan t)b

- lim

yS1

sec (y sec2 y – tan2 y – 1)

- lim

xS0

tanap

4

cos (sin x1>3)b

- lim

tS0

cos a p

219 – 3 sec 2t

b 36. lim

xSp/6

2csc2 x + 513 tan x

- lim

xS0+

sin ap

2

e2xb 38. lim

xS1

cos-1 1ln 2×2

Continuous Extensions

- Define g(3) in a way that extends g(x) = (x2 – 9)>(x – 3) to be

continuous at x = 3.

- Define h(2) in a way that extends h(t) = (t2 + 3t – 10)>(t – 2)

to be continuous at t = 2.

- Define ƒ(1) in a way that extends ƒ(s) = (s3 – 1)>(s2 – 1) to be

continuous at s = 1.

- Define g(4) in a way that extends

g(x) = (x2 – 16)> (x2 – 3x – 4)

to be continuous at x = 4.

- For what value of a is

ƒ(x) = e

x2 – 1, x 6 3

2ax, x Ú 3

continuous at every x?

- For what value of b is

g(x) = e

x, x 6 -2

bx2, x Ú -2

continuous at every x?

- For what values of a is

ƒ(x) = b

a2x – 2a, x Ú 2

12, x 6 2

continuous at every x?

- For what value of b is

g(x) = c

x – b

b + 1

, x 6 0

x2 + b, x 7 0

continuous at every x?

- For what values of a and b is

ƒ(x) = c

-2, x … -1

ax – b, -1 6 x 6 1

3, x Ú 1

continuous at every x?

- For what values of a and b is

g(x) = c

ax + 2b, x … 0

x2 + 3a – b, 0 6 x … 2

3x – 5, x 7 2

continuous at every x?

In Exercises 49–52, graph the function ƒ to see whether it appears to

have a continuous extension to the origin. If it does, use Trace and Zoom

to find a good candidate for the extended function’s value at x = 0. If

the function does not appear to have a continuous extension, can it be

extended to be continuous at the origin from the right or from the left? If

so, what do you think the extended function’s value(s) should be?

- ƒ(x) = 10 x – 1

x 50. ƒ(x) = 10 0 x 0 – 1

x

- ƒ(x) = sin x

0 x 0 52. ƒ(x) = (1 + 2x)1>x

Theory and Examples

- A continuous function y = ƒ(x) is known to be negative at

x = 0 and positive at x = 1. Why does the equation ƒ(x) = 0

have at least one solution between x = 0 and x = 1? Illustrate

with a sketch.

- Explain why the equation cos x = x has at least one solution.
- Roots of a cubic Show that the equation x3 – 15x + 1 = 0

has three solutions in the interval 3-4, 4].

- A function value Show that the function F(x) = (x – a)2 #

(x – b)2 + x takes on the value (a + b)>2 for some value of x.

- Solving an equation If ƒ(x) = x3 – 8x + 10, show that there

are values c for which ƒ(c) equals (a) p; (b) -23; (c) 5,000,000.

- Explain why the following five statements ask for the same information.
- Find the roots of ƒ(x) = x3 – 3x – 1.
- Find the x-coordinates of the points where the curve y = x3

crosses the line y = 3x + 1.

- Find all the values of x for which x3 – 3x = 1.
- Find the x-coordinates of the points where the cubic curve

y = x3 – 3x crosses the line y = 1.

- Solve the equation x3 – 3x – 1 = 0.
- Removable discontinuity Give an example of a function ƒ(x)

that is continuous for all values of x except x = 2, where it has

a removable discontinuity. Explain how you know that ƒ is discontinuous

at x = 2, and how you know the discontinuity is

removable.

- Nonremovable discontinuity Give an example of a function

g(x) that is continuous for all values of x except x = -1, where it

has a nonremovable discontinuity. Explain how you know that g

is discontinuous there and why the discontinuity is not removable.

- A function discontinuous at every point
- Use the fact that every nonempty interval of real numbers

contains both rational and irrational numbers to show that the

function

ƒ(x) = e

1, if x is rational

0, if x is irrational

is discontinuous at every point.

- Is ƒ right-continuous or left-continuous at any point?
- If functions ƒ(x) and g(x) are continuous for 0 … x … 1, could

ƒ(x)>g(x) possibly be discontinuous at a point of 30, 14? Give

reasons for your answer.

- If the product function h(x) = ƒ(x) # g(x) is continuous at x = 0,

must ƒ(x) and g(x) be continuous at x = 0? Give reasons for

your answer.

- Discontinuous composite of continuous functions Give an

example of functions ƒ and g, both continuous at x = 0, for

which the composite ƒ _ g is discontinuous at x = 0. Does this

contradict Theorem 9? Give reasons for your answer.

- Never-zero continuous functions Is it true that a continuous

function that is never zero on an interval never changes sign on

that interval? Give reasons for your answer.

- Stretching a rubber band Is it true that if you stretch a rubber

band by moving one end to the right and the other to the left,

some point of the band will end up in its original position? Give

reasons for your answer.

- A fixed point theorem Suppose that a function ƒ is continuous

on the closed interval 30, 14 and that 0 … ƒ(x) … 1 for every x

in 30, 14 . Show that there must exist a number c in 30, 14 such

that ƒ(c) = c (c is called a fixed point of ƒ).

- The sign-preserving property of continuous functions Let ƒ

be defined on an interval (a, b) and suppose that ƒ(c) _ 0 at

some c where ƒ is continuous. Show that there is an interval

(c – d, c + d) about c where ƒ has the same sign as ƒ(c).

- Prove that ƒ is continuous at c if and only if

lim

hS0

ƒ(c + h) = ƒ(c).

- Use Exercise 69 together with the identities

sin (h + c) = sin h cos c + cos h sin c,

cos (h + c) = cos h cos c – sin h sin c

to prove that both ƒ(x) = sin x and g(x) = cos x are continuous

at every point x = c.

Solving Equations Graphically

Use the Intermediate Value Theorem in Exercises 71–78 to prove that

each equation has a solution. Then use a graphing calculator or computer

grapher to solve the equations.

- x3 – 3x – 1 = 0
- 2×3 – 2×2 – 2x + 1 = 0
- x(x – 1)2 = 1 (one root)
- xx = 2
- 2x + 21 + x = 4
- x3 – 15x + 1 = 0 (three roots)
- cos x = x (one root). Make sure you are using radian mode.
- 2 sin x = x (three roots). Make sure you are using radian

mode.

Exercises 2.6

Finding Limits

- For the function ƒ whose graph is given, determine the following

limits.

- lim

xS2

ƒ(x) b. lim

xS -3 +

ƒ(x) c. lim

xS -3 –

ƒ(x)

- lim

xS -3

ƒ(x) e. lim

xS0 +

ƒ(x) f. lim

xS0 –

ƒ(x)

- lim

xS0

ƒ(x) h. lim

xSq

ƒ(x) i. lim

xS -q

ƒ(x)

y

x

_2

_1

1

2

3

_3

_6_5_4_3_2_1 1 2 3 4 5 6

f

- For the function ƒ whose graph is given, determine the following

limits.

- lim

xS4

ƒ(x) b. lim

xS2 +

ƒ(x) c. lim

xS2 –

ƒ(x)

- lim

xS2

ƒ(x) e. lim

xS -3 +

ƒ(x) f. lim

xS -3 –

ƒ(x)

- lim

xS -3

ƒ(x) h. lim

xS0 +

ƒ(x) i. lim

xS0 –

ƒ(x)

- lim

xS0

ƒ(x) k. lim

xSq

ƒ(x) l. lim

xS -q

ƒ(x)

y

x

_2

_3

_6_5_4_3_2_1 1 2 3 4 5 6

f

3

2

1

_1

In Exercises 3–8, find the limit of each function (a) as xS q and

(b) as xS -q. (You may wish to visualize your answer with a

graphing calculator or computer.)

- ƒ(x) = 2x

– 3 4. ƒ(x) = p – 2

x2

- g(x) = 1

5 + (1>x)

- g(x) = 1

8 – (5>x2)

- f(x) =

-3 + (4>x)

4 – (3>x2)

- h(x) =

3 – (2>x)

4 + (22>x2)

Find the limits in Exercises 9–12.

- lim

xSq

sin 2x

x 10. lim

uS -q

cos u

3u

- lim

tS -q

2 – t + sin t

t + cos t 12. lim

rSq

r + sin r

2r + 7 – 5 sin r

Limits of Rational Functions

In Exercises 13–22, find the limit of each rational function (a) as

xS q and (b) as xS -q.

- ƒ(x) = 2x + 3

5x + 7

- ƒ(x) = 2×3 + 7

x3 – x2 + x + 7

- ƒ(x) = x + 1

x2 + 3

- ƒ(x) = 3x + 7

x2 – 2

- h(x) = 7×3

x3 – 3×2 + 6x

- h(x) = 9×4 + x

2×4 + 5×2 – x + 6

- g(x) = 10×5 + x4 + 31

x6 20. g(x) = x3 + 7×2 – 2

x2 – x + 1

- f(x) = 3×7 + 5×2 – 1

6×3 – 7x + 3

- h(x) = 5×8 – 2×3 + 9

3 + x – 4×5

Limits as x u H or x u _H

The process by which we determine limits of rational functions applies

equally well to ratios containing noninteger or negative powers of x:

Divide numerator and denominator by the highest power of x in the

denominator and proceed from there. Find the limits in Exercises 23–36.

- lim

xSq

A

8×2 – 3

2×2 + x

- lim

xS -q

¢x2 + x – 1

8×2 – 3

_

1>3

- lim

xS – q

¢ 1 – x3

x2 + 7x

_

5

- lim

xS q

A

x2 – 5x

x3 + x – 2

- lim

xS q

22x + x-1

3x – 7

- lim

xS q

2 + 2x

2 – 2x

- lim

xS – q

23

x – 25

x

23

x + 25

x

- lim

xS q

x-1 + x-4

x-2 – x-3

- lim

xS q

2×5>3 – x1>3 + 7

x8>5 + 3x + 2x

- lim

xS – q

23

x – 5x + 3

2x + x2>3 – 4

- lim

xS q

2×2 + 1

x + 1

- lim

xS – q

2×2 + 1

x + 1

- lim

xS q

x – 3

24×2 + 25

- lim

xS – q

4 – 3×3

2×6 + 9

Infinite Limits

Find the limits in Exercises 37–48.

- lim

xS0+

1

3x

- lim

xS0-

5

2x

- lim

xS2-

3

x – 2

- lim

xS3+

1

x – 3

- lim

xS -8+

2x

x + 8

- lim

xS -5-

3x

2x + 10

- lim

xS7

4

(x – 7)2 44. lim

xS0

-1

x2(x + 1)

- a. lim

xS0+

2

3×1>3 b. lim

xS0-

2

3×1>3

- a. lim

xS0+

2

x1>5 b. lim

xS0-

2

x1>5

- lim

xS0

4

x2>5 48. lim

xS0

1

x2>3

Find the limits in Exercises 49–52.

- lim

xS(p>2)-

tan x 50. lim

xS(-p>2)+

sec x

- lim

uS0-

(1 + csc u) 52. lim

uS0

(2 – cot u)

Find the limits in Exercises 53–58.

- lim

1

x2 – 4

as

- xS 2+ b. xS 2-
- xS -2+ d. xS -2-
- lim

x

x2 – 1

as

- xS 1+ b. xS 1-
- xS -1+ d. xS -1-
- lim ax2

2 – 1x

b as

- xS 0+ b. xS 0-
- xS 23

2 d. xS -1

- lim

x2 – 1

2x + 4

as

- xS -2+ b. xS -2-
- xS 1+ d. xS 0-
- lim

x2 – 3x + 2

x3 – 2×2 as

- xS 0+ b. xS 2+
- xS 2- d. xS 2
- What, if anything, can be said about the limit as xS 0?
- lim

x2 – 3x + 2

x3 – 4x

as

- xS 2+ b. xS -2+
- xS 0- d. xS 1+
- What, if anything, can be said about the limit as xS 0?

Find the limits in Exercises 59–62.

- lima2 – 3

t1>3b as

- tS 0+ b. tS 0-
- lima 1

t3>5 + 7b as

- tS 0+ b. tS 0-
- lima 1

x2>3 + 2

(x – 1)2>3b as

- xS 0+ b. xS 0-
- xS 1+ d. xS 1-
- lima 1

x1>3 – 1

(x – 1)4>3b as

- xS 0+ b. xS 0-
- xS 1+ d. xS 1-

Graphing Simple Rational Functions

Graph the rational functions in Exercises 63–68. Include the graphs

and equations of the asymptotes and dominant terms.

- y = 1

x – 1

- y = 1

x + 1

- y = 1

2x + 4

- y = -3

x – 3

- y = x + 3

x + 2

- y = 2x

x + 1

Inventing Graphs and Functions

In Exercises 69–72, sketch the graph of a function y = ƒ(x) that satisfies

the given conditions. No formulas are required—just label the

coordinate axes and sketch an appropriate graph. (The answers are not

unique, so your graphs may not be exactly like those in the answer

section.)

- ƒ(0) = 0, ƒ(1) = 2, ƒ(-1) = -2, lim

xS -q

ƒ(x) = -1, and

lim

xSq

ƒ(x) = 1

- ƒ(0) = 0, lim

xS{q

ƒ(x) = 0, lim

xS0+

ƒ(x) = 2, and

lim

xS0-

ƒ(x) = -2

- ƒ(0) = 0, lim

xS{q

ƒ(x) = 0, lim

xS1-

ƒ(x) = lim

xS -1+

ƒ(x) = q,

lim

xS1 +

ƒ(x) = -q, and lim

xS -1-

ƒ(x) = -q

- ƒ(2) = 1, ƒ(-1) = 0, lim

xSq

ƒ(x) = 0, lim

xS0+

ƒ(x) = q,

lim

xS0-

ƒ(x) = -q, and lim

xS -q

ƒ(x) = 1

In Exercises 73–76, find a function that satisfies the given conditions

and sketch its graph. (The answers here are not unique. Any function

that satisfies the conditions is acceptable. Feel free to use formulas

defined in pieces if that will help.)

- lim

xS{q

ƒ(x) = 0, lim

xS2-

ƒ(x) = q, and lim

xS2+

ƒ(x) = q

- lim

xS{q

g(x) = 0, lim

xS3-

g(x) = -q, and lim

xS3+

g(x) = q

- lim

xS – q

h(x) = -1, lim

xS q

h(x) = 1, lim

xS0-

h(x) = -1, and

lim

xS0+

h(x) = 1

- lim

xS{q

k(x) = 1, lim

xS1-

k(x) = q, and lim

xS1+

k(x) = -q

- Suppose that ƒ(x) and g(x) are polynomials in x and that

limxS q (ƒ(x)>g(x)) = 2. Can you conclude anything about

limxS – q (ƒ(x)>g(x))? Give reasons for your answer.

- Suppose that ƒ(x) and g(x) are polynomials in x. Can the graph of

ƒ(x)>g(x) have an asymptote if g(x) is never zero? Give reasons

for your answer.

- How many horizontal asymptotes can the graph of a given rational

function have? Give reasons for your answer.

Finding Limits of Differences When x u tH

Find the limits in Exercises 80–86.

- lim

xS q

(2x + 9 – 2x + 4 )

- lim

xS q

(2×2 + 25 – 2×2 – 1 )

- lim

xS – q

(2×2 + 3 + x )

- lim

xS – q

(2x + 24×2 + 3x – 2 )

- lim

xS q

(29×2 – x – 3x)

- lim

xS q

(2×2 + 3x – 2×2 – 2x )

- lim

xS q

(2×2 + x – 2×2 – x )

Using the Formal Definitions

Use the formal definitions of limits as xS {q to establish the limits

in Exercises 87 and 88.

- If ƒ has the constant value ƒ(x) = k, then lim

xSq

ƒ(x) = k.

- If ƒ has the constant value ƒ(x) = k, then lim

xS -q

ƒ(x) = k.

Use formal definitions to prove the limit statements in Exercises 89–92.

- lim

xS0

-1

x2 = -q 90. lim

xS0

1

0 x 0 = q

- lim

xS3

-2

(x – 3)2 = -q 92. lim

xS -5

1

(x + 5)2 = q

- Here is the definition of infinite right-hand limit.

We say that ƒ(x) approaches infinity as x approaches c from the

right, and write

lim

xSc+

ƒ(x) = q,

if, for every positive real number B, there exists a corresponding

number d 7 0 such that for all x

c 6 x 6 c + d 1 ƒ(x) 7 B.

Modify the definition to cover the following cases.

- lim

xScƒ(

- x) = q
- lim

xSc+

ƒ(x) = -q

- lim

xScƒ(

- x) = -q

Use the formal definitions from Exercise 93 to prove the limit statements

in Exercises 94–98.

- lim

xS0+

1x

= q

- lim

xS0-

1x

= -q

- lim

xS2-

1

x – 2 = -q

- lim

xS2+

1

x – 2 = q

- lim

xS1-

1

1 – x2 = q

Oblique Asymptotes

Graph the rational functions in Exercises 99–104. Include the graphs

and equations of the asymptotes.

- y = x2

x – 1

- y = x2 + 1

x – 1

- y = x2 – 4

x – 1

- y = x2 – 1

2x + 4

- y = x2 – 1

x

- y = x3 + 1

x2

Additional Graphing Exercises

Graph the curves in Exercises 105–108. Explain the relationship

between the curve’s formula and what you see.

- y = x

24 – x2

- y = -1

24 – x2

- y = x2>3 + 1

x1>3

- y = sin a p

x2 + 1

b

Graph the functions in Exercises 109 and 110. Then answer the following

questions.

- How does the graph behave as xS 0+?
- How does the graph behave as xS {q?
- How does the graph behave near x = 1 and x = -1?

Give reasons for your answers.

- y = 3

2

ax – 1x

b

2>3

- y = 3

2

a x

x – 1b

2>3

Chapter 2 Questions to Guide Your Review

- What is the average rate of change of the function g(t) over the

interval from t = a to t = b? How is it related to a secant line?

- What limit must be calculated to find the rate of change of a function

g(t) at t = t0?

- Give an informal or intuitive definition of the limit

lim

xSc

ƒ(x) = L.

Why is the definition “informal”? Give examples.

- Does the existence and value of the limit of a function ƒ(x) as x

approaches c ever depend on what happens at x = c? Explain

and give examples.

- What function behaviors might occur for which the limit may fail

to exist? Give examples.

- What theorems are available for calculating limits? Give examples

of how the theorems are used.

- How are one-sided limits related to limits? How can this relationship

sometimes be used to calculate a limit or prove it does not

exist? Give examples.

- What is the value of limuS0 ((sin u)>u)? Does it matter whether u

is measured in degrees or radians? Explain.

- What exactly does limxSc ƒ(x) = L mean? Give an example in

which you find a d 7 0 for a given ƒ, L, c, and P 7 0 in the precise

definition of limit.

- Give precise definitions of the following statements.
- limxS2- ƒ(x) = 5 b. limxS2+ ƒ(x) = 5
- limxS2 ƒ(x) = q d. limxS2 ƒ(x) = -q
- What conditions must be satisfied by a function if it is to be continuous

at an interior point of its domain? At an endpoint?

- How can looking at the graph of a function help you tell where

the function is continuous?

- What does it mean for a function to be right-continuous at a

point? Left-continuous? How are continuity and one-sided continuity

related?

- What does it mean for a function to be continuous on an interval?

Give examples to illustrate the fact that a function that is not continuous

on its entire domain may still be continuous on selected

intervals within the domain.

- What are the basic types of discontinuity? Give an example of

each. What is a removable discontinuity? Give an example.

- What does it mean for a function to have the Intermediate Value

Property? What conditions guarantee that a function has this

property over an interval? What are the consequences for graphing

and solving the equation ƒ(x) = 0?

- Under what circumstances can you extend a function ƒ(x) to be

continuous at a point x = c? Give an example.

- What exactly do limxSq ƒ(x) = L and limxS -q ƒ(x) = L mean?

Give examples.

- What are limxS{q k (k a constant) and limxS{q (1>x)? How do

you extend these results to other functions? Give examples.

- How do you find the limit of a rational function as xS {q?

Give examples.

- What are horizontal and vertical asymptotes? Give examples.

Chapter 2 Practice Exercises

Limits and Continuity

- Graph the function

ƒ(x) = e

1, x … -1

-x, -1 6 x 6 0

1, x = 0

-x, 0 6 x 6 1

1, x Ú 1.

Then discuss, in detail, limits, one-sided limits, continuity, and

one-sided continuity of ƒ at x = -1, 0, and 1. Are any of the

discontinuities removable? Explain.

- Repeat the instructions of Exercise 1 for

ƒ(x) = d

0, x … -1

1>x, 0 6 0 x 0 6 1

0, x = 1

1, x 7 1.

- Suppose that ƒ(x) and g(x) are defined for all x and that limxSx0

ƒ(x) = -5 and limxSx0 g(x) = 1. Find the limit as xS x0 of the

following functions.

- 5ƒ(x) b. (ƒ(x))3
- ƒ(x) # g(x) d.

ƒ(x)

g(x) – 6

- log g(x) f. 0 g(x) 0
- ƒ(x) – g(x) h. 1>ƒ(x)
- Suppose the functions ƒ(x) and g(x) are defined for all x and that

limxS0 ƒ(x) = 1>2 and limxS0 g(x) = 22. Find the limits as

xS 0 of the following functions.

- -g(x) b. g(x) # ƒ(x)
- ƒ(x) + g(x) d. 1>ƒ(x)
- x + ƒ(x) f.

ƒ(x) # cos x

x – 1

In Exercises 5 and 6, find the value that limxS0 g(x) must have if the

given limit statements hold.

- lim

xS0

a

4 – g(x)

x b = 1 6. lim

xS -4

ax lim

xS0

g(x)b = 2

- On what intervals are the following functions continuous?
- ƒ(x) = x1>3 b. g(x) = x3>4
- h(x) = x-2>3 d. k(x) = x-1>6
- On what intervals are the following functions continuous?
- ƒ(x) = tan x b. g(x) = csc x
- h(x) = cos x

x – p d. k(x) = sin x

x

Finding Limits

In Exercises 9–28, find the limit or explain why it does not exist.

- lim

x2 + 5x + 5

x3 + 3×2 – 11x

- as xS 0 b. as xS 3
- lim

x3 + 1

x7 + 4×6 + x5

- as xS 2 b. as xS -2
- lim

xS0

a21 + x – 1

x b 12. lim

xS2

x5 – 25

x3 – 23

- lim

hS0

(x + h)3 – x3

h

- lim

xS0

(x + h)2 – x2

h

- lim

xS0

1

3 + x – 1

3

x 16. lim

xS0

(3 + x)4 – 81

x

- lim

xS1

x2 – 2x

2x – 1

- lim

xS16

x2>3 – 162>3

2x – 4

- lim

xS0

sin mx

tan nx 20. lim

xSp

sec x = -1

- lim

xSp

cos ax

2 + sin xb 22. lim

xSp

sin2 (x – cot x)

- lim

xS0

7x

8 tan x – x

- lim

xS0

x2 – tan 2x

tan x

- lim

xS0

(1 + x)1>x 26. lim

tS4

t3 ln (3 – 2t )

- lim

uS1

2ue-cos (p>u) 28. lim

xS4

ex – e4

x – 4

In Exercises 29–32, find the limit of g(x) as x approaches the indicated

value.

- lim

xS0+ [9g(x)]1>3 = 3 30. lim

xS27

1

x + g(x) = 5

- lim

xS2

5×4 + 2

g(x) = q 32. lim

xS -3

10 – x2

(g(x))1>3 = 0

Roots

- Let ƒ(x) = x3 – x – 1.
- Use the Intermediate Value Theorem to show that ƒ has a

zero between -1 and 2.

- Solve the equation ƒ(x) = 0 graphically with an error of

magnitude at most 10-8.

T

- It can be shown that the exact value of the solution in part (b) is

a1

2 +

269

18

b

1>3

+ a1

2 –

269

18

b

1>3

.

Evaluate this exact answer and compare it with the value you

found in part (b).

- Let ƒ(u) = u3 – 2u + 2.
- Use the Intermediate Value Theorem to show that ƒ has a

zero between -2 and 0.

- Solve the equation ƒ(u) = 0 graphically with an error of

magnitude at most 10-4.

- It can be shown that the exact value of the solution in part (b) is

a

A

19

27 – 1b

1>3

– a

A

19

27 + 1b

1>3

.

Evaluate this exact answer and compare it with the value you

found in part (b).

Continuous Extension

- Can ƒ(x) = x(x2 – 1)> 0 x2 – 1 0 be extended to be continuous at

x = 1 or -1? Give reasons for your answers. (Graph the function—

you will find the graph interesting.)

- Explain why the function ƒ(x) = sin (1>x) has no continuous

extension to x = 0.

In Exercises 37–40, graph the function to see whether it appears to have

a continuous extension to the given point a. If it does, use Trace and

Zoom to find a good candidate for the extended function’s value at a. If

the function does not appear to have a continuous extension, can it be

extended to be continuous from the right or left? If so, what do you

think the extended function’s value should be?

- ƒ(x) = x – 1

x – 24

x

, a = 1

- g(u) = 5 cos u

4u – 2p

, a = p>2

- h(t) = (1 + 0 t 0 )1>t, a = 0
- k(x) = x

1 – 20 x 0 , a = 0

Limits at Infinity

Find the limits in Exercises 41–54.

- lim

xS q

5x + 8

7x + 4

- lim

xS q

3×4 + 8

5×4 + 9

- lim

xS – q

x2 – 7x + 9

5×3 44. lim

xS q

1

2×2 – 9x + 5

- lim

xS – q

x2 + 11x

x – 1

- lim

xS q

x5 + x4

9×4 + 108

- lim

xS q

sin x

:x;

(If you have a grapher, try graphing the function

for -5 … x … 5.)

- lim

uS q

cos u – 1

u

(If you have a grapher, try graphing

ƒ(x) = x (cos (1>x) – 1) near the origin to

“see” the limit at infinity.)

- lim

xS q

x + tan x + 32x

x + tan x 50. lim

xS q

x5>7 + x-3

x5>7 + cos x

- lim

xS q

e2>x sin

2x

- lim

tS q

ln a1 – 2t

b

- lim

xS – 1

sin-1 x 54. lim

tS – q

e3t sin-1

1t

Horizontal and Vertical Asymptotes

- Use limits to determine the equations for all vertical asymptotes.
- y = x2 + 4

x – 3

- ƒ(x) = x2 – x – 2

x2 – 2x + 1

- y = x2 + x – 6

x2 + 2x – 8

- Use limits to determine the equations for all horizontal asymptotes.
- y = 1 – x2

x2 + 1

- ƒ(x) =

2x + 4

2x + 4

- g(x) =

2×2 + 4

x d. y = B

x2 + 9

9×2 + 1

Chapter 2 Additional and Advanced Exercises

- Assigning a value to 00 The rules of exponents tell us that

a0 = 1 if a is any number different from zero. They also tell us

that 0n = 0 if n is any positive number.

If we tried to extend these rules to include the case 00, we

would get conflicting results. The first rule would say 00 = 1,

whereas the second would say 00 = 0.

We are not dealing with a question of right or wrong here.

Neither rule applies as it stands, so there is no contradiction. We

could, in fact, define 00 to have any value we wanted as long as

we could persuade others to agree.

What value would you like 00 to have? Here is an example

that might help you to decide. (See Exercise 2 below for another

example.)

- Calculate xx for x = 0.1, 0.01, 0.001, and so on as far as

your calculator can go. Record the values you get. What

pattern do you see?

- Graph the function y = xx for 0 6 x … 1. Even though the

function is not defined for x … 0, the graph will approach

the y-axis from the right. Toward what y-value does it seem

to be headed? Zoom in to further support your idea.

- A reason you might want 00 to be something other than 0 or 1

As the number x increases through positive values, the numbers

1>x and 1 > (ln x) both approach zero. What happens to the number

ƒ(x) = a1x

b

1>(ln x)

as x increases? Here are two ways to find out.

- Evaluate ƒ for x = 10, 100, 1000, and so on as far as your

calculator can reasonably go. What pattern do you see?

- Graph ƒ in a variety of graphing windows, including windows

that contain the origin. What do you see? Trace the

y-values along the graph. What do you find?

- Lorentz contraction In relativity theory, the length of an

object, say a rocket, appears to an observer to depend on the

speed at which the object is traveling with respect to the observer.

If the observer measures the rocket’s length as L0 at rest, then at

speed y the length will appear to be

L = L0

B

1 – y2

c2.

This equation is the Lorentz contraction formula. Here, c is the

speed of light in a vacuum, about 3 * 108 m>sec. What happens

to L as y increases? Find limySc- L. Why was the left-hand limit

needed?

T

T

- Controlling the flow from a draining tank Torricelli’s law

says that if you drain a tank like the one in the figure shown, the

rate y at which water runs out is a constant times the square root

of the water’s depth x. The constant depends on the size and

shape of the exit valve.

x

Exit rate y ft3_min

Suppose that y = 2x>2 for a certain tank. You are trying to

maintain a fairly constant exit rate by adding water to the tank

with a hose from time to time. How deep must you keep the water

if you want to maintain the exit rate

- within 0.2 ft3>min of the rate y0 = 1 ft3>min?
- within 0.1 ft3>min of the rate y0 = 1 ft3>min?
- Thermal expansion in precise equipment As you may know,

most metals expand when heated and contract when cooled. The

dimensions of a piece of laboratory equipment are sometimes so

critical that the shop where the equipment is made must be held

at the same temperature as the laboratory where the equipment is

to be used. A typical aluminum bar that is 10 cm wide at 70°F

will be

y = 10 + (t – 70) * 10-4

centimeters wide at a nearby temperature t. Suppose that you are

using a bar like this in a gravity wave detector, where its width

must stay within 0.0005 cm of the ideal 10 cm. How close to

t0 = 70_F must you maintain the temperature to ensure that this

tolerance is not exceeded?

- Stripes on a measuring cup The interior of a typical 1-L measuring

cup is a right circular cylinder of radius 6 cm (see accompanying

figure). The volume of water we put in the cup is therefore

a function of the level h to which the cup is filled, the

formula being

V = p62h = 36ph.

How closely must we measure h to measure out 1 L of water

(1000 cm3) with an error of no more than 1% (10 cm3)?

A 1-L measuring cup (a), modeled as a right circular cylinder (b)

of radius r = 6 cm

Precise Definition of Limit

In Exercises 7–10, use the formal definition of limit to prove that the

function is continuous at c.

- ƒ(x) = x2 – 7, c = 1 8. g(x) = 1>(2x), c = 1>4
- h(x) = 22x – 3, c = 2 10. F(x) = 29 – x, c = 5
- Uniqueness of limits Show that a function cannot have two different

limits at the same point. That is, if limxSc ƒ(x) = L1 and

limxSc ƒ(x) = L2, then L1 = L2.

- Prove the limit Constant Multiple Rule:

lim

xSc

kƒ(x) = k lim

xSc

ƒ(x) for any constant k.

- One-sided limits If limxS0+ ƒ(x) = A and limxS0- ƒ(x) = B,

find

- limxS0+ ƒ(x3 – x) b. limxS0- ƒ(x3 – x)
- limxS0+ ƒ(x2 – x4) d. limxS0- ƒ(x2 – x4)
- Limits and continuity Which of the following statements are

true, and which are false? If true, say why; if false, give a counterexample

(that is, an example confirming the falsehood).

- If limxSc ƒ(x) exists but limxSc g(x) does not exist, then

limxSc(ƒ(x) + g(x)) does not exist.

- If neither limxSc ƒ(x) nor limxSc g(x) exists, then

limxSc (ƒ(x) + g(x)) does not exist.

- If ƒ is continuous at x, then so is 0 ƒ 0 .
- If 0 ƒ 0 is continuous at c, then so is ƒ.

In Exercises 15 and 16, use the formal definition of limit to prove that

the function has a continuous extension to the given value of x.

- ƒ(x) = x2 – 1

x + 1

, x = -1 16. g(x) = x2 – 2x – 3

2x – 6

, x = 3

- A function continuous at only one point Let

ƒ(x) = e

x, if x is rational

0, if x is irrational.

- Show that ƒ is continuous at x = 0.
- Use the fact that every nonempty open interval of real numbers

contains both rational and irrational numbers to show

that ƒ is not continuous at any nonzero value of x.

- The Dirichlet ruler function If x is a rational number, then x

can be written in a unique way as a quotient of integers m>n

where n 7 0 and m and n have no common factors greater than

- (We say that such a fraction is in lowest terms. For example,

6>4 written in lowest terms is 3>2.) Let ƒ(x) be defined for all x

in the interval 30, 14 by

ƒ(x) = e

1>n, if x = m>n is a rational number in lowest terms

0, if x is irrational.

For instance, ƒ(0) = ƒ(1) = 1, ƒ(1>2) = 1>2, ƒ(1>3) = ƒ(2>3) =

1>3, ƒ(1>4) = ƒ(3>4) = 1>4, and so on.

- Show that ƒ is discontinuous at every rational number in 30, 14 .
- Show that ƒ is continuous at every irrational number in 30, 14 .

(Hint: If P is a given positive number, show that there are only

finitely many rational numbers r in 30, 14 such that ƒ(r) Ú P.)

- Sketch the graph of ƒ. Why do you think ƒ is called the

“ruler function”?

- Antipodal points Is there any reason to believe that there is

always a pair of antipodal (diametrically opposite) points on

Earth’s equator where the temperatures are the same? Explain.

- If limxSc (ƒ(x) + g(x)) = 3 and limxSc (ƒ(x) – g(x)) = -1, find

limxSc ƒ(x)g(x).

- Roots of a quadratic equation that is almost linear The equation

ax2 + 2x – 1 = 0, where a is a constant, has two roots if

a 7 -1 and a _ 0, one positive and one negative:

r+(a) = -1 + 21 + a

a , r-(a) = -1 – 21 + a

a ,

- What happens to r+(a) as aS 0? As aS -1+?
- What happens to r-(a) as aS 0? As aS -1+?
- Support your conclusions by graphing r+(a) and r-(a) as

functions of a. Describe what you see.

- For added support, graph ƒ(x) = ax2 + 2x – 1 simultaneously

for a = 1, 0.5, 0.2, 0.1, and 0.05.

- Root of an equation Show that the equation x + 2 cos x = 0

has at least one solution.

- Bounded functions A real-valued function ƒ is bounded from

above on a set D if there exists a number N such that ƒ(x) … N

for all x in D. We call N, when it exists, an upper bound for ƒ on

D and say that ƒ is bounded from above by N. In a similar manner,

we say that ƒ is bounded from below on D if there exists a

number M such that ƒ(x) Ú M for all x in D. We call M, when it

exists, a lower bound for ƒ on D and say that ƒ is bounded from

below by M. We say that ƒ is bounded on D if it is bounded from

both above and below.

- Show that ƒ is bounded on D if and only if there exists a

number B such that 0 ƒ(x) 0 … B for all x in D.

- Suppose that ƒ is bounded from above by N. Show that if

limxSc ƒ(x) = L, then L … N.

- Suppose that ƒ is bounded from below by M. Show that if

limxSc ƒ(x) = L, then L Ú M.

- Max 5a, b6 and min 5a, b6
- Show that the expression

max 5a, b6 = a + b

2 +

0 a – b 0

2

equals a if a Ú b and equals b if b Ú a. In other words,

max 5a, b6 gives the larger of the two numbers a and b.

- Find a similar expression for min 5a, b6, the smaller of a

and b.

Generalized Limits Involving

sin U

U

The formula limuS0 (sin u)>u = 1 can be generalized. If limxSc

ƒ(x) = 0 and ƒ(x) is never zero in an open interval containing the

point x = c, except possibly c itself, then

lim

xSc

sin ƒ(x)

ƒ(x) = 1.

Here are several examples.

- lim

xS0

sin x2

x2 = 1

- lim

xS0

sin x2

x = lim

xS0

sin x2

x2 lim

xS0

x2

x = 1 # 0 = 0

- lim

xS -1

sin (x2 – x – 2)

x + 1 = lim

xS -1

sin (x2 – x – 2)

(x2 – x – 2)

#

lim

xS -1

(x2 – x – 2)

x + 1 = 1 # lim

xS -1

(x + 1)(x – 2)

x + 1 = -3

- lim

xS1

sin 11 – 2×2

x – 1 = lim

xS1

sin 11 – 2×2

1 – 2x

1 – 2x

x – 1 =

1 # lim

xS1

11 – 2x 211 + 2x 2

(x – 1)11 + 2×2 = lim

xS1

1 – x

(x – 1)11 + 2x 2 = –

1

2

Find the limits in Exercises 25–30.

- lim

xS0

sin (1 – cos x)

x 26. lim

xS0+

sin x

sin2x

- lim

xS0

sin (sin x)

x 28. lim

xS0

sin (x2 + x)

x

- lim

xS2

sin (x2 – 4)

x – 2

- lim

xS9

sin 12x – 32

x – 9

Oblique Asymptotes

Find all possible oblique asymptotes in Exercises 31–34.

- y = 2×3>2 + 2x – 3

2x + 1

- y = x + x sin

1x

- y = 2×2 + 1 34. y = 2×2 + 2x

128 Chapter 2: Limits and Continuity

Chapter 3 Derivatives

Exercises 3.1

Slopes and Tangent Lines

In Exercises 1–4, use the grid and a straight edge to make a rough

estimate of the slope of the curve (in y-units per x-unit) at the points

P1 and P2.

- 2.

x

y

1

2

0 1

P1

P2

x

y

0 1 2

2

1

_1

_2

P1

P2

_2 _1

- 4.

x

y

1 2

2

1

0

P1

P2

y

_1 0 1

1

2

3

x

4

_2 2

P1 P2

In Exercises 5–10, find an equation for the tangent to the curve at the

given point. Then sketch the curve and tangent together.

- y = 4 – x2, (-1, 3) 6. y = (x – 1)2 + 1, (1, 1)
- y = 22x, (1, 2) 8. y = 1

x2 , (-1, 1)

- y = x3, (-2, -8) 10. y = 1

x3 , a-2, –

1

8b

In Exercises 11–18, find the slope of the function’s graph at the given

point. Then find an equation for the line tangent to the graph there.

- ƒ(x) = x2 + 1, (2, 5) 12. ƒ(x) = x – 2×2, (1, -1)
- g(x) = x

x – 2

, (3, 3) 14. g(x) = 8

x2 , (2, 2)

- h(t) = t3, (2, 8) 16. h(t) = t3 + 3t, (1, 4)
- ƒ(x) = 2x, (4, 2) 18. ƒ(x) = 2x + 1, (8, 3)

In Exercises 19–22, find the slope of the curve at the point indicated.

- y = 5x – 3×2, x = 1 20. y = x3 – 2x + 7, x = -2
- y = 1

x – 1

, x = 3 22. y = x – 1

x + 1

, x = 0

Interpreting Derivative Values

- Growth of yeast cells In a controlled laboratory experiment,

yeast cells are grown in an automated cell culture system that

counts the number P of cells present at hourly intervals. The number

after t hours is shown in the accompanying figure.

t

p

0

100

1 2 3 4 5 6 7

200

50

150

250

- Explain what is meant by the derivative P_(5). What are its

units?

- Which is larger, P_(2) or P_(3)? Give a reason for your

answer.

- The quadratic curve capturing the trend of the data points

(see Section 1.4) is given by P(t) = 6.10t2 – 9.28t + 16.43.

Find the instantaneous rate of growth when t = 5 hours.

- Effectiveness of a drug On a scale from 0 to 1, the effectiveness

E of a pain-killing drug t hours after entering the bloodstream

is displayed in the accompanying figure.

t

E

0

0.4

1 2 3 4 5

0.8

0.2

0.6

1.0

- At what times does the effectiveness appear to be increasing?

What is true about the derivative at those times?

- At what time would you estimate that the drug reaches its

maximum effectiveness? What is true about the derivative at

that time? What is true about the derivative as time increases

in the 1 hour before your estimated time?

At what points do the graphs of the functions in Exercises 25 and 26

have horizontal tangents?

- ƒ(x) = x2 + 4x – 1 26. g(x) = x3 – 3x
- Find equations of all lines having slope -1 that are tangent to the

curve y = 1>(x – 1).

- Find an equation of the straight line having slope 1>4 that is tangent

to the curve y = 2x.

Rates of Change

- Object dropped from a tower An object is dropped from the

top of a 660-m-high tower. Its height above ground after t sec is

660 – 4.9t2 m. How fast is it falling 2 sec after it is dropped?

- Speed of a rocket At t sec after liftoff, the height of a rocket is

4t2 ft. How fast is the rocket climbing 7 sec after liftoff?

- Circle’s changing area What is the rate of change of the area

of a circle (A = pr2) with respect to the radius when the radius

is r = 5?

- Ball’s changing volume What is the rate of change of the volume

of a ball (V = (4>3)pr3) with respect to the radius when

the radius is r = 1?

- Show that the line y = mx + b is its own tangent line at any

point (x0, mx0 + b).

- Find the slope of the tangent to the curve f(x) = 4>2x at the

point where x = 1/4.

Testing for Tangents

- Does the graph of

ƒ(x) = e

x2 sin (1>x), x _ 0

0, x = 0

have a tangent at the origin? Give reasons for your answer.

- Does the graph of

g(x) = e

x sin (1>x), x _ 0

0, x = 0

have a tangent at the origin? Give reasons for your answer.

Vertical Tangents

We say that a continuous curve y = ƒ(x) has a vertical tangent at the

point where x = x0 if the limit of the difference quotient is q or -q.

For example, y = x1>3 has a vertical tangent at x = 0 (see accompanying

figure):

lim

hS0

ƒ(0 + h) – ƒ(0)

h = lim

hS0

h1>3 – 0

h

= lim

hS0

1

h2>3 = q.

However, y = x2>3 has no vertical tangent at x = 0 (see next figure):

lim

hS0

g(0 + h) – g(0)

h = lim

hS0

h2>3 – 0

h

= lim

hS0

1

h1>3

does not exist, because the limit is q from the right and -q from the

left.

x

y

0

NO VERTICAL TANGENT AT ORIGIN

y _ g(x) _ x2_3

- Does the graph of

ƒ(x) = c

-1, x 6 0

0, x = 0

1, x 7 0

have a vertical tangent at the origin? Give reasons for your answer.

- Does the graph of

U(x) = e

0, x 6 0

1, x Ú 0

have a vertical tangent at the point (0, 1)? Give reasons for your

answer.

Graph the curves in Exercises 39–48.

- Where do the graphs appear to have vertical tangents?

T

- Confirm your findings in part (a) with limit calculations. But

before you do, read the introduction to Exercises 37 and 38.

- y = x2>5 40. y = x4>5
- y = x1>5 42. y = x3>5
- y = 4×2>5 – 2x 44. y = x5>3 – 5×2>3
- y = x2>3 – (x – 1)1>3 46. y = x1>3 + (x – 1)1>3
- y = e-20 x 0 , x … 0

2x, x 7 0

- y = 20 4 – x 0

Computer Explorations

Use a CAS to perform the following steps for the functions in Exercises

49–52:

- Plot y = ƒ(x) over the interval (x0 – 1>2) … x … (x0 + 3).
- Holding x0 fixed, the difference quotient

q(h) =

ƒ(x0 + h) – ƒ(x0)

h

at x0 becomes a function of the step size h. Enter this function

into your CAS workspace.

- Find the limit of q as hS 0.
- Define the secant lines y = ƒ(x0) + q # (x – x0) for h = 3, 2,

and 1. Graph them together with ƒ and the tangent line over

the interval in part (a).

- ƒ(x) = x3 + 2x, x0 = 0 50. ƒ(x) = x + 5x

, x0 = 1

- ƒ(x) = x + sin (2x), x0 = p>2
- ƒ(x) = cos x + 4 sin (2x), x0 = p

Exercises 3.2

Finding Derivative Functions and Values

Using the definition, calculate the derivatives of the functions in

Exercises 1–6. Then find the values of the derivatives as specified.

- ƒ(x) = 2 – x2; ƒ_(-7), ƒ_(0), ƒ_(5)
- F(x) = (x – 1)2 + 1; F_(-1), F_(0), F_(2)
- g(t) = 5

t4 ; g_(-4), g_(2), g_1262

- k(z) =

1 – z

2z

; k_(-1), k_(1), k_1222

- p(u) = 23u ; p_(1), p_(3), p_(2>3)
- r (s) = 22s + 1 ; r_(0), r_(1), r_(1>2)

In Exercises 7–12, find the indicated derivatives.

7.

dy

dx

if y = 4×3 8. dr

ds

if r = s3 – 2s2 + 3

- ds

dt

if s = t

4t + 9

- dy

dt

if y = t – 1t

11.

dy

dx

if y = x3>2 12.

dz

dw

if z = 1

2w2 – 1

Slopes and Tangent Lines

In Exercises 13–16, differentiate the functions and find the slope of

the tangent line at the given value of the independent variable.

- ƒ(x) = 3x + 2x

, x = -1 14. k(x) = 1

2 + x

, x = 2

- s = 9t3 – 8t2, t = 6 16. y = x + 3

1 – x

, x = -2

In Exercises 17–18, differentiate the functions. Then find an equation

of the tangent line at the indicated point on the graph of the function.

- y = ƒ(x) = 8

2x – 2

, (x, y) = (6, 4)

- y = f(x) = 1 + 26 – x, (x, y) = (2, 3)

In Exercises 19–22, find the values of the derivatives.

19.

dy

dx

`

x=-4

if y = 5 – 2×2 20.

dy

dx

`

x=23

if y = 1 – 1x

- dr

du

`

u=0

if r = 2

24 – u

- dw

dz 0 z=4 if w = z + 1z

Using the Alternative Formula for Derivatives

Use the formula

ƒ_(x) = lim

zSx

ƒ(z) – ƒ(x)

z – x

to find the derivative of the functions in Exercises 23–26.

- ƒ(x) = 1

x + 2

- ƒ(x) = x2 – 3x + 4
- g(x) = x

x – 1

- g(x) = 1 + 1x

Graphs

Match the functions graphed in Exercises 27–30 with the derivatives

graphed in the accompanying figures (a)–(d).

y_

0

x

(d)

y_

0

x

(c)

y_

0

x

(a)

y_

0

x

(b)

- 28.

x

y

0

y _ f1(x)

x

y

0

y _ f2(x)

- 30.

y

0

x

y _ f3(x)

y

0

x

y _ f4(x)

- a. The graph in the accompanying figure is made of line segments

joined end to end. At which points of the interval

3-4, 64 is ƒ_ not defined? Give reasons for your answer.

x

y

0 1 6

(0, 2) (6, 2)

(_4, 0)

y _ f (x)

(1, _2) (4, _2)

- Graph the derivative of ƒ.

The graph should show a step function.

- Recovering a function from its derivative
- Use the following information to graph the function ƒ over

the closed interval 3-2, 54.

- i) The graph of ƒ is made of closed line segments joined

end to end.

- ii) The graph starts at the point (-2, 3).

iii) The derivative of ƒ is the step function in the figure

shown here.

x

_2 0 1 3 5

1

y_

y_ _ f _(x)

_2

- Repeat part (a), assuming that the graph starts at (-2, 0)

instead of (-2, 3).

- Growth in the economy The graph in the accompanying figure

shows the average annual percentage change y = ƒ(t) in the U.S.

gross national product (GNP) for the years 2005–2011. Graph

dy>dt (where defined).

2005 2006 2007 2008 2009 2010 2011

10

2

3

4

5

6

7%

- Fruit flies (Continuation of Example 4, Section 2.1.) Populations

starting out in closed environments grow slowly at first,

when there are relatively few members, then more rapidly as the

number of reproducing individuals increases and resources are

still abundant, then slowly again as the population reaches the

carrying capacity of the environment.

- Use the graphical technique of Example 3 to graph the derivative

of the fruit fly population. The graph of the population

is reproduced here.

0 10

50

100

150

200

250

300

350

20 30 40 50

Time (days)

Number of _ies

p

t

- During what days does the population seem to be increasing

fastest? Slowest?

- Temperature The given graph shows the temperature T in °F

at Davis, CA, on April 18, 2008, between 6 a.m. and 6 p.m.

0 3

40

50

60

70

80

6 9 12

6 A.M. 9 A.M. 12 NOON 3 P.M. 6 P.M.

Time (hr)

Temperature (_F)

T

t

- Estimate the rate of temperature change at the times
- i) 7 a.m. ii) 9 a.m. iii) 2 p.m. iv) 4 p.m.
- At what time does the temperature increase most rapidly?

Decrease most rapidly? What is the rate for each of those times?

- Use the graphical technique of Example 3 to graph the derivative

of temperature T versus time t.

- Weight loss Jared Fogle, also known as the “Subway Sandwich

Guy,” weighed 425 lb in 1997 before losing more than 240 lb in

12 months (http://en.wikipedia.org/wiki/Jared_Fogle). A chart

showing his possible dramatic weight loss is given in the accompanying

figure.

0 1 2 3 4 5 7 8 10 11

100

200

300

425

500

6 9 12

Time (months)

Weight (lb)

W

t

- Estimate Jared’s rate of weight loss when
- i) t = 1 ii) t = 4 iii) t = 11
- When does Jared lose weight most rapidly and what is this

rate of weight loss?

- Use the graphical technique of Example 3 to graph the derivative

of weight W.

One-Sided Derivatives

Compute the right-hand and left-hand derivatives as limits to show that

the functions in Exercises 37–40 are not differentiable at the point P.

- 38.

x

y

y _ x2 y _ f (x)

y _ x

P(0, 0)

x

y

y _ f (x)

y _ 2x

y _ 2

1

2

0 1 2

P(1, 2)

- 40.

y

y _ f (x)

y _ 2x _ 1

x

P(1, 1)

0

1

1

y _ “x

y

y _ 1x

y _ f (x)

x

P(1, 1)

y _ x

1

1

In Exercises 41 and 42, determine if the piecewise-defined function is

differentiable at the origin.

- ƒ(x) = e

2x – 1, x Ú 0

x2 + 2x + 7, x 6 0

- g(x) = e

x2>3, x Ú 0

x1>3, x 6 0

Differentiability and Continuity on an Interval

Each figure in Exercises 43–48 shows the graph of a function over a

closed interval D. At what domain points does the function appear to be

- differentiable?
- continuous but not differentiable?
- neither continuous nor differentiable?

In Exercises 19–22, find the values of the derivatives.

19.

dy

dx

`

x=-4

if y = 5 – 2×2 20.

dy

dx

`

x=23

if y = 1 – 1x

- dr

du

`

u=0

if r = 2

24 – u

- dw

dz 0 z=4 if w = z + 1z

Using the Alternative Formula for Derivatives

Use the formula

ƒ_(x) = lim

zSx

ƒ(z) – ƒ(x)

z – x

to find the derivative of the functions in Exercises 23–26.

- ƒ(x) = 1

x + 2

- ƒ(x) = x2 – 3x + 4
- g(x) = x

x – 1

- g(x) = 1 + 1x

Graphs

Match the functions graphed in Exercises 27–30 with the derivatives

graphed in the accompanying figures (a)–(d).

y_

0

x

(d)

y_

0

x

(c)

y_

0

x

(a)

y_

0

x

(b)

- 28.

x

y

0

y _ f1(x)

x

y

0

y _ f2(x)

- 30.

y

0

x

y _ f3(x)

y

0

x

y _ f4(x)

- a. The graph in the accompanying figure is made of line segments

joined end to end. At which points of the interval

3-4, 64 is ƒ_ not defined? Give reasons for your answer.

x

y

0 1 6

(0, 2) (6, 2)

(_4, 0)

y _ f (x)

(1, _2) (4, _2)

- Graph the derivative of ƒ.

The graph should show a step function.

- Recovering a function from its derivative
- Use the following information to graph the function ƒ over

the closed interval 3-2, 54.

- i) The graph of ƒ is made of closed line segments joined

end to end.

- ii) The graph starts at the point (-2, 3).

iii) The derivative of ƒ is the step function in the figure

shown here.

x

_2 0 1 3 5

1

y_

y_ _ f _(x)

_2

- Repeat part (a), assuming that the graph starts at (-2, 0)

instead of (-2, 3).

- Growth in the economy The graph in the accompanying figure

shows the average annual percentage change y = ƒ(t) in the U.S.

gross national product (GNP) for the years 2005–2011. Graph

dy>dt (where defined).

2005 2006 2007 2008 2009 2010 2011

10

2

3

4

5

6

7%

- Fruit flies (Continuation of Example 4, Section 2.1.) Populations

starting out in closed environments grow slowly at first,

when there are relatively few members, then more rapidly as the

number of reproducing individuals increases and resources are

still abundant, then slowly again as the population reaches the

carrying capacity of the environment.

- Use the graphical technique of Example 3 to graph the derivative

of the fruit fly population. The graph of the population

is reproduced here.

0 10

50

100

150

200

250

300

350

20 30 40 50

Time (days)

Number of _ies

p

t

- During what days does the population seem to be increasing

fastest? Slowest?

- Temperature The given graph shows the temperature T in °F

at Davis, CA, on April 18, 2008, between 6 a.m. and 6 p.m.

0 3

40

50

60

70

80

6 9 12

6 A.M. 9 A.M. 12 NOON 3 P.M. 6 P.M.

Time (hr)

Temperature (_F)

T

t

- Estimate the rate of temperature change at the times
- i) 7 a.m. ii) 9 a.m. iii) 2 p.m. iv) 4 p.m.
- At what time does the temperature increase most rapidly?

Decrease most rapidly? What is the rate for each of those times?

- Use the graphical technique of Example 3 to graph the derivative

of temperature T versus time t.

- Weight loss Jared Fogle, also known as the “Subway Sandwich

Guy,” weighed 425 lb in 1997 before losing more than 240 lb in

12 months (http://en.wikipedia.org/wiki/Jared_Fogle). A chart

showing his possible dramatic weight loss is given in the accompanying

figure.

0 1 2 3 4 5 7 8 10 11

100

200

300

425

500

6 9 12

Time (months)

Weight (lb)

W

t

- Estimate Jared’s rate of weight loss when
- i) t = 1 ii) t = 4 iii) t = 11
- When does Jared lose weight most rapidly and what is this

rate of weight loss?

- Use the graphical technique of Example 3 to graph the derivative

of weight W.

One-Sided Derivatives

Compute the right-hand and left-hand derivatives as limits to show that

the functions in Exercises 37–40 are not differentiable at the point P.

- 38.

x

y

y _ x2 y _ f (x)

y _ x

P(0, 0)

x

y

y _ f (x)

y _ 2x

y _ 2

1

2

0 1 2

P(1, 2)

- 40.

y

y _ f (x)

y _ 2x _ 1

x

P(1, 1)

0

1

1

y _ “x

y

y _ 1x

y _ f (x)

x

P(1, 1)

y _ x

1

1

In Exercises 41 and 42, determine if the piecewise-defined function is

differentiable at the origin.

- ƒ(x) = e

2x – 1, x Ú 0

x2 + 2x + 7, x 6 0

- g(x) = e

x2>3, x Ú 0

x1>3, x 6 0

Differentiability and Continuity on an Interval

Each figure in Exercises 43–48 shows the graph of a function over a

closed interval D. At what domain points does the function appear to be

- differentiable?
- continuous but not differentiable?
- neither continuous nor differentiable?

Give reasons for your answers.

- 46.

x

y

y _ f (x)

D: _3 _ x _ 3

_1 0

_1

1

_2

_3 _2 1 2 3

x

y

y _ f (x)

D: _2 _ x _ 3

_2 _1 0 1 2 3

1

2

3

- 48.

x

y

y _ f (x)

D: _1 _ x _ 2

_1 0 1 2

1

y _ f (x)

D: _3 _ x _ 3

x

y

_3_2 _1 0

2

4

1 2 3

Theory and Examples

In Exercises 49–52,

- Find the derivative ƒ_(x) of the given function y = ƒ(x).
- Graph y = ƒ(x) and y = ƒ_(x) side by side using separate

sets of coordinate axes, and answer the following questions.

- For what values of x, if any, is ƒ_ positive? Zero? Negative?
- Over what intervals of x-values, if any, does the function

y = ƒ(x) increase as x increases? Decrease as x increases?

How is this related to what you found in part (c)? (We will

say more about this relationship in Section 4.3.)

- y = -x2 50. y = -1>x
- y = x3>3 52. y = x4>4
- Tangent to a parabola Does the parabola y = 2×2 – 13x + 5

have a tangent whose slope is -1? If so, find an equation for the

line and the point of tangency. If not, why not?

- Tangent to y _ 2x Does any tangent to the curve y = 2x

cross the x-axis at x = -1? If so, find an equation for the line and

the point of tangency. If not, why not?

- Derivative of _ƒ Does knowing that a function ƒ(x) is differentiable

at x = x0 tell you anything about the differentiability of

the function -ƒ at x = x0? Give reasons for your answer.

- Derivative of multiples Does knowing that a function g(t) is

differentiable at t = 7 tell you anything about the differentiability

of the function 3g at t = 7? Give reasons for your answer.

- Limit of a quotient Suppose that functions g(t) and h(t) are

defined for all values of t and g(0) = h(0) = 0. Can

limtS0 (g(t))>(h(t)) exist? If it does exist, must it equal zero?

Give reasons for your answers.

- a. Let ƒ(x) be a function satisfying 0 ƒ(x) 0 … x2 for -1 … x … 1.

Show that ƒ is differentiable at x = 0 and find ƒ_(0).

- Show that

ƒ(x) = c

x2 sin

1x

, x _ 0

0, x = 0

is differentiable at x = 0 and find ƒ_(0).

- Graph y = 1>122×2 in a window that has 0 … x … 2. Then, on

the same screen, graph

y =

2x + h – 2x

h

for h = 1, 0.5, 0.1. Then try h = -1, -0.5, -0.1. Explain what

is going on.

- Graph y = 3×2 in a window that has -2 … x … 2, 0 … y … 3.

Then, on the same screen, graph

y =

(x + h)3 – x3

h

for h = 2, 1, 0.2. Then try h = -2, -1, -0.2. Explain what is

going on.

- Derivative of y _ _ x _ Graph the derivative of ƒ(x) = 0 x 0 .

Then graph y = ( 0 x 0 – 0) >(x – 0) = 0 x 0 >x. What can you

conclude?

- Weierstrass’s nowhere differentiable continuous function

The sum of the first eight terms of the Weierstrass function

ƒ(x) = gqn

=0 (2>3)n cos (9npx) is

g(x) = cos (px) + (2>3)1 cos (9px) + (2>3)2 cos (92px)

+ (2>3)3 cos (93px) + g + (2>3)7 cos (97px).

Graph this sum. Zoom in several times. How wiggly and bumpy

is this graph? Specify a viewing window in which the displayed

portion of the graph is smooth.

COMPUTER EXPLORATIONS

Use a CAS to perform the following steps for the functions in Exercises

63–68.

- Plot y = ƒ(x) to see that function’s global behavior.
- Define the difference quotient q at a general point x, with

general step size h.

- Take the limit as hS 0. What formula does this give?
- Substitute the value x = x0 and plot the function y = ƒ(x)

together with its tangent line at that point.

- Substitute various values for x larger and smaller than x0 into

the formula obtained in part (c). Do the numbers make sense

with your picture?

- Graph the formula obtained in part (c). What does it mean

when its values are negative? Zero? Positive? Does this make

sense with your plot from part (a)? Give reasons for your

answer.

- ƒ(x) = x3 + x2 – x, x0 = 1
- ƒ(x) = x1>3 + x2>3, x0 = 1
- ƒ(x) = 4x

x2 + 1

, x0 = 2

- ƒ(x) = x – 1

3×2 + 1

, x0 = -1

- ƒ(x) = sin 2x, x0 = p>2
- ƒ(x) = x2 cos x, x0 = p>4

Exercises 3.3

Derivative Calculations

In Exercises 1–12, find the first and second derivatives.

- y = -3×9 – 1 2. y = x2 + x + 8
- s = 7t4 – 4t7 4. w = 3z7 – 7z3 + 21z2
- y = 4×3

3 – x + 2ex 6. y = x3

3 + x2

2 + e-x

- y = 8x-7 – 6x
- s = -2t-1 + 4

t2

- y = 5×2 – 15x – 4x-2 10. y = 4 – 2x – x-3
- r = 1

3s2 – 5

2s

- r = 12

u – 4

u3 + 1

u4

In Exercises 13–16, find y_ (a) by applying the Product Rule and

(b) by multiplying the factors to produce a sum of simpler terms to

differentiate.

- y = (3 – x2) (x3 – x + 1) 14. y = (2x + 3) (5×2 – 4x)
- y = (x2 + 1) ax + 5 + 1x

b 16. y = (1 + x2)(x3>4 – x-3)

Find the derivatives of the functions in Exercises 17–40.

- y = 3x – 1

7x + 2

- z = 4 – 3x

3×2 + x

- y = 9×2 + 2

x2 + 3

- ƒ(t) = t2 – 1

t2 + t – 2

- s = (5 – t) (1 + t2)-1 22. w = (2x – 7)-1(x + 5)
- ƒ(s) =

1s – 3

1s + 3

- u = 5x + 1

21x

- y =

1 + x – 42x

x 26. r = 2a 1

2u

+ 2ub

- y = 1

(x2 – 1) (x2 + x + 1) 28. y =

(x + 1) (x + 2)

(x – 1) (x – 2)

- y = 2e-x + e3x 30. y = x2 + 3ex

2ex – x

- y = x3ex 32. w = re-r
- y = x9>4 + e-2x 34. y = x-3>5 + p3>2
- s = 5t8>5 + 5e4 36. w = 1

z1.4 + p

2z

- y = 27

x2 – xe 38. y = 23

x9.6 + 2e1.3

- r = es

s 40. r = eua 1

u2 + u-p>2b

Find the derivatives of all orders of the functions in Exercises 41–44.

- y = x4

2 – 3

2

x2 – x 42. y = x5

120

- y = (x – 1) (x + 2)(x + 3) 44. y = (4×2 + 3)(2 – x) x

Find the first and second derivatives of the functions in Exercises

45–52.

- y = x3 + 7

x 46. s = t2 + 5t – 1

t2

- r =

(u – 1)(u2 + u + 1)

u3 48. u =

(x2 + x)(x2 – x + 1)

x4

- w = a1 + 3z

3z b(3 – z) 50. p =

q2 + 3

(q – 1)3 + (q + 1)3

- w = 3z2e2z 52. w = ez(z – 1)(z2 + 1)
- Suppose u and y are functions of x that are differentiable at

x = 0 and that

u(0) = 5, u_(0) = -3, y(0) = -1, y_(0) = 2.

Find the values of the following derivatives at x = 0.

- d

dx

(uy) b. d

dx

au

yb c. d

dx

ayu

b d. d

dx

(7y – 2u)

- Suppose u and y are differentiable functions of x and that

u(1) = 2, u_(1) = 0, y(1) = 5, y_(1) = -1.

Find the values of the following derivatives at x = 1.

- d

dx

(uy) b. d

dx

au

yb c. d

dx

ayu

b d. d

dx

(7y – 2u)

Slopes and Tangents

- a. Normal to a curve Find an equation for the line perpendicular

to the tangent to the curve y = x3 – 4x + 1 at the point (2, 1).

- Smallest slope What is the smallest slope on the curve? At

what point on the curve does the curve have this slope?

- Tangents having specified slope Find equations for the tangents

to the curve at the points where the slope of the curve is 8.

- a. Horizontal tangents Find equations for the horizontal tangents

to the curve y = x3 – 3x – 2. Also find equations for the lines

that are perpendicular to these tangents at the points of tangency.

- Smallest slope What is the smallest slope on the curve? At

what point on the curve does the curve have this slope? Find

an equation for the line that is perpendicular to the curve’s

tangent at this point.

- Find the tangents to Newton’s serpentine (graphed here) at the

origin and the point (1, 2).

x

y

0

1

1 2

2

(1, 2)

3 4

y _ 4x

x2 + 1

- Find the tangent to the Witch of Agnesi (graphed here) at the point

(2, 1).

x

y

0

1

1 2

2

(2, 1)

3

y _ 8

x2 + 4

- Quadratic tangent to identity function The curve y =

ax2 + bx + c passes through the point (1, 2) and is tangent to the

line y = x at the origin. Find a, b, and c.

- Quadratics having a common tangent The curves y =

x2 + ax + b and y = cx – x2 have a common tangent line at

the point (1, 0). Find a, b, and c.

- Find all points (x, y) on the graph of ƒ(x) = 3×2 – 4x with tangent

lines parallel to the line y = 8x + 5.

- Find all points (x, y) on the graph of g(x) = 13

x3 – 32

x2 + 1 with

tangent lines parallel to the line 8x – 2y = 1.

- Find all points (x, y) on the graph of y = x>(x – 2) with tangent

lines perpendicular to the line y = 2x + 3.

- Find all points (x, y) on the graph of ƒ(x) = x2 with tangent lines

passing through the point (3, 8).

y

x

(3, 8)

_2

2

2 4

6

10

f (x) _ x2

(x, y)

- a. Find an equation for the line that is tangent to the curve

y = x3 – x at the point (-1, 0).

- Graph the curve and tangent line together. The tangent intersects

the curve at another point. Use Zoom and Trace to estimate

the point’s coordinates.

T

- Confirm your estimates of the coordinates of the second

intersection point by solving the equations for the curve and

tangent simultaneously (Solver key).

- a. Find an equation for the line that is tangent to the curve

y = x3 – 6×2 + 5x at the origin.

- Graph the curve and tangent together. The tangent intersects

the curve at another point. Use Zoom and Trace to estimate

the point’s coordinates.

- Confirm your estimates of the coordinates of the second

intersection point by solving the equations for the curve and

tangent simultaneously (Solver key).

Theory and Examples

For Exercises 67 and 68 evaluate each limit by first converting each to

a derivative at a particular x-value.

- lim

xS1

x50 – 1

x – 1

- lim

xS-1

x2>9 – 1

x + 1

- Find the value of a that makes the following function differentiable

for all x-values.

g(x) = e

ax, if x 6 0

x2 – 3x, if x Ú 0

- Find the values of a and b that make the following function differentiable

for all x-values.

ƒ(x) = e

ax + b, x 7 -1

bx2 – 3, x … -1

- The general polynomial of degree n has the form

P(x) = an xn + an-1 xn-1 + g + a2 x2 + a1 x + a0

where an _ 0. Find P_(x).

- The body’s reaction to medicine The reaction of the body to a

dose of medicine can sometimes be represented by an equation of

the form

R = M2 aC

2 – M

3 b,

where C is a positive constant and M is the amount of medicine

absorbed in the blood. If the reaction is a change in blood pressure,

R is measured in millimeters of mercury. If the reaction is a

change in temperature, R is measured in degrees, and so on.

Find dR>dM. This derivative, as a function of M, is called the

sensitivity of the body to the medicine. In Section 4.5, we will see

how to find the amount of medicine to which the body is most

sensitive.

- Suppose that the function y in the Derivative Product Rule has a

constant value c. What does the Derivative Product Rule then say?

What does this say about the Derivative Constant Multiple Rule?

- The Reciprocal Rule
- The Reciprocal Rule says that at any point where the function

y(x) is differentiable and different from zero,

d

dx

a1y

b = –

1

y2

dy

dx

.

Show that the Reciprocal Rule is a special case of the Derivative

Quotient Rule.

T

T

T

- Show that the Reciprocal Rule and the Derivative Product

Rule together imply the Derivative Quotient Rule.

- Generalizing the Product Rule The Derivative Product Rule

gives the formula

d

dx

(uy) = u

dy

dx + y

du

dx

for the derivative of the product uy of two differentiable functions

of x.

- What is the analogous formula for the derivative of the product

uyw of three differentiable functions of x?

- What is the formula for the derivative of the product u1 u2 u3 u4

of four differentiable functions of x?

- What is the formula for the derivative of a product u1 u2 u3gun

of a finite number n of differentiable functions of x?

- Power Rule for negative integers Use the Derivative Quotient

Rule to prove the Power Rule for negative integers, that is,

d

dx

(x-m) = -mx-m-1

where m is a positive integer.

- Cylinder pressure If gas in a cylinder is maintained at a constant

temperature T, the pressure P is related to the volume V by a

formula of the form

P = nRT

V – nb – an2

V2 ,

in which a, b, n, and R are constants. Find dP>dV. (See accompanying

figure.)

- The best quantity to order One of the formulas for inventory

management says that the average weekly cost of ordering, paying

for, and holding merchandise is

A(q) = km

q + cm +

hq

2

,

where q is the quantity you order when things run low (shoes,

TVs, brooms, or whatever the item might be); k is the cost of

placing an order (the same, no matter how often you order); c is

the cost of one item (a constant); m is the number of items sold

each week (a constant); and h is the weekly holding cost per item

(a constant that takes into account things such as space, utilities,

insurance, and security). Find dA>dq and d2A>dq2.

Exercises 3.4

Motion Along a Coordinate Line

Exercises 1–6 give the positions s = ƒ(t) of a body moving on a coordinate

line, with s in meters and t in seconds.

- Find the body’s displacement and average velocity for the

given time interval.

- Find the body’s speed and acceleration at the endpoints of the

interval.

- When, if ever, during the interval does the body change direction?
- s = t2 – 3t + 2, 0 … t … 2
- s = 6t – t2, 0 … t … 6
- s = -t3 + 3t2 – 3t, 0 … t … 3
- s = (t4>4) – t3 + t2, 0 … t … 3
- s = 25

t2 – 5t

, 1 … t … 5

- s = 25

t + 5

, -4 … t … 0

- Particle motion At time t, the position of a body moving along

the s-axis is s = t3 – 6t2 + 9t m.

- Find the body’s acceleration each time the velocity is zero.
- Find the body’s speed each time the acceleration is zero.
- Find the total distance traveled by the body from t = 0 to t = 2.
- Particle motion At time t Ú 0, the velocity of a body moving

along the horizontal s-axis is y = t2 – 4t + 3.

- Find the body’s acceleration each time the velocity is zero.
- When is the body moving forward? Backward?
- When is the body’s velocity increasing? Decreasing?

Free-Fall Applications

- Free fall on Mars and Jupiter The equations for free fall at

the surfaces of Mars and Jupiter (s in meters, t in seconds) are

s = 1.86t2 on Mars and s = 11.44t2 on Jupiter. How long does it

take a rock falling from rest to reach a velocity of 27.8 m> sec

(about 100 km > h) on each planet?

- Lunar projectile motion A rock thrown vertically upward

from the surface of the moon at a velocity of 24 m > sec (about

86 km > h) reaches a height of s = 24t – 0.8t2 m in t sec.

- Find the rock’s velocity and acceleration at time t. (The acceleration

in this case is the acceleration of gravity on the moon.)

- How long does it take the rock to reach its highest point?
- How high does the rock go?
- How long does it take the rock to reach half its maximum

height?

- How long is the rock aloft?
- Finding g on a small airless planet Explorers on a small airless

planet used a spring gun to launch a ball bearing vertically upward

from the surface at a launch velocity of 15 m > sec. Because the acceleration

of gravity at the planet’s surface was gs m>sec2, the explorers

expected the ball bearing to reach a height of s = 15t – (1>2)gs t2 m

t sec later. The ball bearing reached its maximum height 20 sec after

being launched. What was the value of gs?

- Speeding bullet A 45-caliber bullet shot straight up from the

surface of the moon would reach a height of s = 832t – 2.6t2 ft

after t sec. On Earth, in the absence of air, its height would be

s = 832t – 16t2 ft after t sec. How long will the bullet be aloft in

each case? How high will the bullet go?

- Free fall from the Tower of Pisa Had Galileo dropped a cannonball

from the Tower of Pisa, 179 ft above the ground, the

ball’s height above the ground t sec into the fall would have been

s = 179 – 16t2.

- What would have been the ball’s velocity, speed, and acceleration

at time t?

- About how long would it have taken the ball to hit the ground?
- What would have been the ball’s velocity at the moment of

impact?

- Galileo’s free-fall formula Galileo developed a formula for a

body’s velocity during free fall by rolling balls from rest down

increasingly steep inclined planks and looking for a limiting formula

that would predict a ball’s behavior when the plank was

vertical and the ball fell freely; see part (a) of the accompanying

figure. He found that, for any given angle of the plank, the ball’s

velocity t sec into motion was a constant multiple of t. That is, the

velocity was given by a formula of the form y = kt. The value of

the constant k depended on the inclination of the plank.

In modern notation—part (b) of the figure—with distance in

meters and time in seconds, what Galileo determined by experiment

was that, for any given angle u, the ball’s velocity t sec into

the roll was

y = 9.8(sin u)t m>sec.

(a)

?

(b)

Free-fall

position

u

- What is the equation for the ball’s velocity during free fall?
- Building on your work in part (a), what constant acceleration

does a freely falling body experience near the surface of Earth?

Understanding Motion from Graphs

- The accompanying figure shows the velocity y = ds>dt = ƒ(t)

(m> sec) of a body moving along a coordinate line.

0

_3

2 4

3

6 8 10

y (m/sec)

y _ f (t)

t (sec)

- When does the body reverse direction?
- When (approximately) is the body moving at a constant speed?
- Graph the body’s speed for 0 … t … 10.
- Graph the acceleration, where defined.
- A particle P moves on the number line shown in part (a) of the

accompanying figure. Part (b) shows the position of P as a function

of time t.

0

_2

_4

1 2

2

3 4 5 6

(b)

0

(a)

P

s (cm)

s (cm)

s _ f (t)

t (sec)

(6, _4)

- When is P moving to the left? Moving to the right? Standing

still?

- Graph the particle’s velocity and speed (where defined).
- Launching a rocket When a model rocket is launched, the propellant

burns for a few seconds, accelerating the rocket upward.

After burnout, the rocket coasts upward for a while and then

begins to fall. A small explosive charge pops out a parachute

shortly after the rocket starts down. The parachute slows the

rocket to keep it from breaking when it lands.

The figure here shows velocity data from the flight of the

model rocket. Use the data to answer the following.

- How fast was the rocket climbing when the engine stopped?
- For how many seconds did the engine burn?

0 2 4 6 8 10 12

100

50

0

_50

_100

150

200

Time after launch (sec)

Velocity (ft_sec)

- When did the rocket reach its highest point? What was its

velocity then?

- When did the parachute pop out? How fast was the rocket

falling then?

- How long did the rocket fall before the parachute opened?
- When was the rocket’s acceleration greatest?
- When was the acceleration constant? What was its value then

(to the nearest integer)?

- The accompanying figure shows the velocity y = ƒ(t) of a particle

moving on a horizontal coordinate line.

t (sec)

y

0 1 2 3 4 5 6 7 8 9

y _ f(t)

- When does the particle move forward? Move backward?

Speed up? Slow down?

- When is the particle’s acceleration positive? Negative? Zero?
- When does the particle move at its greatest speed?
- When does the particle stand still for more than an instant?
- Two falling balls The multiflash photograph in the accompanying

figure shows two balls falling from rest. The vertical rulers

are marked in centimeters. Use the equation s = 490t2 (the freefall

equation for s in centimeters and t in seconds) to answer the

following questions.

- How long did it take the balls to fall the first 160 cm? What

was their average velocity for the period?

- How fast were the balls falling when they reached the 160-cm

mark? What was their acceleration then?

- About how fast was the light flashing (flashes per second)?
- A traveling truck The accompanying graph shows the position

s of a truck traveling on a highway. The truck starts at t = 0 and

returns 15 h later at t = 15.

- Use the technique described in Section 3.2, Example 3, to

graph the truck’s velocity y = ds>dt for 0 … t … 15. Then

repeat the process, with the velocity curve, to graph the

truck’s acceleration dy>dt.

- Suppose that s = 15t2 – t3. Graph ds>dt and d2s>dt2 and

compare your graphs with those in part (a).

0

100

200

300

400

500

5 10 15

Elapsed time, t (hr)

Position, s (km)

- The graphs in the accompanying figure show the position s,

velocity y = ds>dt, and acceleration a = d2s>dt2 of a body

moving along a coordinate line as functions of time t. Which

graph is which? Give reasons for your answers.

t

y

0

A B

C

- The graphs in the accompanying figure show the position s, the

velocity y = ds>dt, and the acceleration a = d2s>dt2 of a body

moving along a coordinate line as functions of time t. Which

graph is which? Give reasons for your answers.

t

y

0

A

B

C

Economics

- Marginal cost Suppose that the dollar cost of producing x

washing machines is c(x) = 2000 + 100x – 0.1×2.

- Find the average cost per machine of producing the first 100

washing machines.

- Find the marginal cost when 100 washing machines are

produced.

- Show that the marginal cost when 100 washing machines are

produced is approximately the cost of producing one more

washing machine after the first 100 have been made, by calculating

the latter cost directly.

- Marginal revenue Suppose that the revenue from selling x

washing machines is

r(x) = 20,000a1 – 1x

b

dollars.

- Find the marginal revenue when 100 machines are produced.
- Use the function r_(x) to estimate the increase in revenue that

will result from increasing production from 100 machines a

week to 101 machines a week.

- Find the limit of r_(x) as xS q. How would you interpret

this number?

Additional Applications

- Bacterium population When a bactericide was added to a

nutrient broth in which bacteria were growing, the bacterium

population continued to grow for a while, but then stopped growing

and began to decline. The size of the population at time t

(hours) was b = 106 + 104t – 103t2. Find the growth rates at

- t = 0 hours.
- t = 5 hours.
- t = 10 hours.
- Body surface area A typical male’s body surface area S in

square meters is often modeled by the formula S = 1

60 2wh,

where h is the height in cm, and w the weight in kg, of the person.

Find the rate of change of body surface area with respect to

weight for males of constant height h = 180 cm (roughly 5_9_).

Does S increase more rapidly with respect to weight at lower or

higher body weights? Explain.

- Draining a tank It takes 12 hours to drain a storage tank by

opening the valve at the bottom. The depth y of fluid in the tank t

hours after the valve is opened is given by the formula

y = 6a1 – t

12b

2

m.

- Find the rate dy>dt (m > h) at which the tank is draining at

time t.

- When is the fluid level in the tank falling fastest? Slowest?

What are the values of dy>dt at these times?

- Graph y and dy>dt together and discuss the behavior of y in

relation to the signs and values of dy>dt.

- Draining a tank The number of gallons of water in a tank t

minutes after the tank has started to drain is Q(t) = 200(30 – t)2.

How fast is the water running out at the end of 10 min? What is the

average rate at which the water flows out during the first 10 min?

- Vehicular stopping distance Based on data from the U.S.

Bureau of Public Roads, a model for the total stopping distance of

a moving car in terms of its speed is

s = 1.1y + 0.054y2,

where s is measured in ft and y in mph. The linear term 1.1y

models the distance the car travels during the time the driver perceives

a need to stop until the brakes are applied, and the quadratic

term 0.054y2 models the additional braking distance once

they are applied. Find ds>dy at y = 35 and y = 70 mph, and

interpret the meaning of the derivative.

- Inflating a balloon The volume V = (4>3)pr3 of a spherical

balloon changes with the radius.

- At what rate (ft3>ft) does the volume change with respect to

the radius when r = 2 ft?

- By approximately how much does the volume increase when

the radius changes from 2 to 2.2 ft?

- Airplane takeoff Suppose that the distance an aircraft travels

along a runway before takeoff is given by D = (10>9)t2, where D is

measured in meters from the starting point and t is measured in seconds

from the time the brakes are released. The aircraft will become

airborne when its speed reaches 200 km>h. How long will it take to

become airborne, and what distance will it travel in that time?

- Volcanic lava fountains Although the November 1959 Kilauea

Iki eruption on the island of Hawaii began with a line of fountains

along the wall of the crater, activity was later confined to a single

vent in the crater’s floor, which at one point shot lava 1900 ft

straight into the air (a Hawaiian record). What was the lava’s exit

velocity in feet per second? In miles per hour? (Hint: If y0 is the

exit velocity of a particle of lava, its height t sec later will be

s = y0 t – 16t2 ft. Begin by finding the time at which ds>dt = 0.

Neglect air resistance.)

Analyzing Motion Using Graphs

Exercises 33–36 give the position function s = ƒ(t) of an object moving

along the s-axis as a function of time t. Graph ƒ together with the

velocity function y(t) = ds>dt = ƒ_(t) and the acceleration function

a(t) = d2s>dt2 = ƒ_(t). Comment on the object’s behavior in relation

to the signs and values of y and a. Include in your commentary such

topics as the following:

- When is the object momentarily at rest?
- When does it move to the left (down) or to the right (up)?
- When does it change direction?
- When does it speed up and slow down?
- When is it moving fastest (highest speed)? Slowest?
- When is it farthest from the axis origin?
- s = 200t – 16t2, 0 … t … 12.5 (a heavy object fired straight

up from Earth’s surface at 200 ft > sec)

- s = t2 – 3t + 2, 0 … t … 5
- s = t3 – 6t2 + 7t, 0 … t … 4
- s = 4 – 7t + 6t2 – t3, 0 … t … 4

Exercises 3.5

Derivatives

In Exercises 1–18, find dy>dx.

- y = -16x + 9 cos x 2. y = 8x

+ 3 sin x

- y = x5 cos x 4. y = 2x sec x + 3
- y = csc x – 41x + 7

ex 6. y = x2 cot x – 1

x2

- ƒ(x) = sin x tan x 8. g(x) = cos x

sin2 x

- y = xe-x sec x 10. y = (sin x + cos x) sec x
- y = tan x

1 + tan x

- y = 4 cos x

1 + sin x

- y = 4

cos x + 1

tan x 14. y = cos x

x + x

cos x

- y = 3(sec x + tan x) (sec x – tan x)
- y = x2 cos x – 2x sin x – 2 cos x
- ƒ(x) = x3 sin x cos x 18. g(x) = (2 – x) tan2 x

In Exercises 19–22, find ds>dt.

- s = tan t – e-t 20. s = t2 – sec t + 5et
- s = 1 + csc t

1 – csc t

- s = sin t

1 – cos t

In Exercises 23–26, find dr>du.

- r = 4 – u2 sin u 24. r = u sin u + cos u
- r = sec u csc u 26. r = (1 + sec u) sin u

In Exercises 27–32, find dp>dq.

- p = 5 + 1

cot q 28. p = (1 + csc q) cos q

- p =

sin q + cos q

cos q 30. p =

tan q

1 + tan q

- p =

q sin q

q2 – 1

- p =

3q + tan q

q sec q

- Find y_ if
- y = csc x. b. y = sec x.
- Find y(4) = d4 y>dx4 if
- y = -2 sin x. b. y = 9 cos x.

Tangent Lines

In Exercises 35–38, graph the curves over the given intervals, together

with their tangents at the given values of x. Label each curve and tangent

with its equation.

- y = sin x, -3p>2 … x … 2p

x = -p, 0, 3p>2

- y = tan x, -p>2 6 x 6 p>2

x = -p>3, 0, p>3

- y = sec x, -p>2 6 x 6 p>2

x = -p>3, p>4

- y = 1 + cos x, -3p>2 … x … 2p

x = -p>3, 3p>2

Do the graphs of the functions in Exercises 39–42 have any horizontal

tangents in the interval 0 … x … 2p? If so, where? If not, why not?

Visualize your findings by graphing the functions with a grapher.

- y = x + sin x 40. y = 2x + sin x
- y = x – cot x 42. y = x + 2 cos x
- Find all points on the curve y = tan x, -p>2 6 x 6 p>2, where

the tangent line is parallel to the line y = 2x. Sketch the curve

and tangent(s) together, labeling each with its equation.

- Find all points on the curve y = cot x, 0 6 x 6 p, where the

tangent line is parallel to the line y = -x. Sketch the curve and

tangent(s) together, labeling each with its equation.

In Exercises 45 and 46, find an equation for (a) the tangent to the

curve at P and (b) the horizontal tangent to the curve at Q.

Trigonometric Limits

Find the limits in Exercises 47–54.

- lim

xS2

sin a1x

– 1

2b 48. lim

xS -p>6

21 + cos (p csc x)

- lim

uSp>6

sin u – 12

u – p6

- lim

uSp>4

tan u – 1

u – p4

- lim

xS0

secc ex + p tan a p

4 sec xb – 1 d

- lim

xS0

sin a p + tan x

tan x – 2 sec xb

- lim

tS0

tan a1 – sin t

t b 54. lim

uS0

cos a pu

sin u

b

Theory and Examples

The equations in Exercises 55 and 56 give the position s = ƒ(t) of a

body moving on a coordinate line (s in meters, t in seconds). Find the

body’s velocity, speed, acceleration, and jerk at time t = p>4 sec.

- s = 2 – 2 sin t 56. s = sin t + cos t
- Is there a value of c that will make

ƒ(x) = •

sin2 3x

x2 , x _ 0

c, x = 0

continuous at x = 0? Give reasons for your answer.

- Is there a value of b that will make

g(x) = e

x + b, x 6 0

cos x, x Ú 0

continuous at x = 0? Differentiable at x = 0? Give reasons for

your answers.

- By computing the first few derivatives and looking for a pattern,

find d 999>dx999 (cos x).

- Derive the formula for the derivative with respect to x of
- sec x. b. csc x. c. cot x.
- A weight is attached to a spring and reaches its equilibrium position

(x = 0). It is then set in motion resulting in a displacement of

x = 10 cos t,

where x is measured in centimeters and t is measured in seconds.

See the accompanying figure.

x

0

_10

10

Equilibrium

position

at x _ 0

- Find the spring’s displacement when t = 0, t = p>3, and

t = 3p>4.

- Find the spring’s velocity when t = 0, t = p>3, and

t = 3p>4.

- Assume that a particle’s position on the x-axis is given by

x = 3 cos t + 4 sin t,

where x is measured in feet and t is measured in seconds.

- Find the particle’s position when t = 0, t = p>2, and

t = p.

- Find the particle’s velocity when t = 0, t = p>2, and

t = p.

- Graph y = cos x for -p … x … 2p. On the same screen, graph

y =

sin (x + h) – sin x

h

for h = 1, 0.5, 0.3, and 0.1. Then, in a new window, try

h = -1, -0.5, and -0.3. What happens as hS 0+? As hS 0-?

What phenomenon is being illustrated here?

- Graph y = -sin x for -p … x … 2p. On the same screen, graph

y =

cos (x + h) – cos x

h

for h = 1, 0.5, 0.3, and 0.1. Then, in a new window, try

h = -1, -0.5, and -0.3. What happens as hS 0+? As hS 0-?

What phenomenon is being illustrated here?

- Centered difference quotients The centered difference quotient

ƒ(x + h) – ƒ(x – h)

2h

is used to approximate ƒ_(x) in numerical work because (1) its

limit as hS 0 equals ƒ_(x) when ƒ_(x) exists, and (2) it usually

gives a better approximation of ƒ_(x) for a given value of h than

the difference quotient

ƒ(x + h) – ƒ(x)

h

.

- To see how rapidly the centered difference quotient for

ƒ(x) = sin x converges to ƒ_(x) = cos x, graph y = cos x

together with

y =

sin (x + h) – sin (x – h)

2h

over the interval 3-p, 2p4 for h = 1, 0.5, and 0.3. Compare

the results with those obtained in Exercise 63 for the

same values of h.

- To see how rapidly the centered difference quotient for

ƒ(x) = cos x converges to ƒ_(x) = -sin x, graph y = -sin x

together with

y =

cos (x + h) – cos (x – h)

2h

over the interval 3-p, 2p4 for h = 1, 0.5, and 0.3. Compare

the results with those obtained in Exercise 64 for the same

values of h.

- A caution about centered difference quotients (Continuation

of Exercise 65.) The quotient

ƒ(x + h) – ƒ(x – h)

2h

may have a limit as hS 0 when ƒ has no derivative at x. As a

case in point, take ƒ(x) = 0 x 0 and calculate

lim

hS0

0 0 + h 0 – 0 0 – h 0

2h

.

As you will see, the limit exists even though ƒ(x) = 0 x 0 has no

derivative at x = 0. Moral: Before using a centered difference

quotient, be sure the derivative exists.

- Slopes on the graph of the tangent function Graph y = tan x

and its derivative together on (-p>2, p>2). Does the graph of the

tangent function appear to have a smallest slope? A largest slope?

Is the slope ever negative? Give reasons for your answers.

- Exploring (sin kx) ,x Graph y = (sin x)>x, y = (sin 2x)>x, and

y = (sin 4x)>x together over the interval -2 … x … 2. Where

does each graph appear to cross the y-axis? Do the graphs really

intersect the axis? What would you expect the graphs of

y = (sin 5x)>x and y = (sin (-3x))>x to do as xS 0? Why?

What about the graph of y = (sin kx)>x for other values of k?

Give reasons for your answers.

Exercises 3.6

Derivative Calculations

In Exercises 1–8, given y = ƒ(u) and u = g(x), find dy>dx =

ƒ_(g(x))g_(x).

- y = 6u – 9, u = (1>2)x4 2. y = 2u3, u = 8x – 1
- y = sin u, u = 3x + 1 4. y = cos u, u = e-x
- y = 2u, u = sin x 6. y = sin u, u = x – cos x
- y = tan u, u = px2 8. y = -sec u, u = 1x

+ 7x

In Exercises 9–22, write the function in the form y = ƒ(u) and

u = g(x). Then find dy>dx as a function of x.

- y = (2x + 1)5 10. y = (4 – 3x)9
- y = a1 – x

7b

-7

- y = a2x

2 – 1b

-10

- y = ax2

8 + x – 1x

b

4

- y = 23×2 – 4x + 6
- y = sec (tan x) 16. y = cot ap – 1x

b

- y = tan3 x 18. y = 5 cos-4 x
- y = e-5x 20. y = e2x>3
- y = e5-7x 22. y = e142x+x22

Find the derivatives of the functions in Exercises 23–50.

- p = 23 – t 24. q = 23

2r – r2

- s = 4

3p

sin 3t + 4

5p

cos 5t 26. s = sin a3pt

2 b + cos a3pt

2 b

- r = (csc u + cot u)-1 28. r = 6 (sec u – tan u)3>2
- y = x2 sin4 x + x cos-2 x 30. y = 1x

sin-5 x – x

3

cos3 x

- y = 1

18

(3x – 2)6 + a4 – 1

2x2b

-1

- y = (5 – 2x)-3 + 1

8

a2x

+ 1b

4

- y = (4x + 3)4(x + 1)-3 34. y = (2x – 5)-1(x2 – 5x)6
- y = xe-x + ex3 36. y = (1 + 2x)e-2x
- y = (x2 – 2x + 2)e5x>2 38. y = (9×2 – 6x + 2)ex3
- h(x) = x tan 121×2 + 7 40. k(x) = x2 sec a1x

b

- ƒ(x) = 27 + x sec x 42. g(x) = tan 3x

(x + 7)4

- ƒ(u) = a sin u

1 + cos u

b

2

- g(t) = a1 + sin 3t

3 – 2t b

-1

- r = sin (u2) cos (2u) 46. r = sec2u tan a1

u

b

- q = sin a t

2t + 1

b 48. q = cotasin t

t b

- y = cos 1e-u22 50. y = u3e-2u cos 5u

In Exercises 51–70, find dy>dt.

- y = sin2 (pt – 2) 52. y = sec2 pt
- y = (1 + cos 2t)-4 54. y = (1 + cot (t>2))-2
- y = (t tan t)10 56. y = (t-3>4 sin t)4>3
- y = ecos2 (pt-1) 58. y = (esin (t>2))3
- y = a t2

t3 – 4t

b

3

- y = a3t – 4

5t + 2

b

-5

- y = sin (cos (2t – 5)) 62. y = cos a5 sin a t

3b b

- y = a1 + tan4 a t

12b b

3

- y = 1

6

11 + cos2 (7t)23

- y = 21 + cos (t2) 66. y = 4 sin 121 + 1t2
- y = tan2 (sin3 t) 68. y = cos4 (sec2 3t)
- y = 3t (2t2 – 5)4 70. y = 43t + 32 + 21 – t

Second Derivatives

Find y_ in Exercises 71–78.

- y = a1 + 1x

b

3

- y = 11 – 1×2-1
- y = 1

9

cot (3x – 1) 74. y = 9 tan ax

3b

- y = x (2x + 1)4 76. y = x2 (x3 – 1)5
- y = ex2 + 5x 78. y = sin (x2ex)

Finding Derivative Values

In Exercises 79–84, find the value of (ƒ _ g)_ at the given value of x.

- ƒ(u) = u5 + 1, u = g(x) = 1x, x = 1
- ƒ(u) = 1 – 1u

, u = g(x) = 1

1 – x

, x = -1

- ƒ(u) = cot pu

10

, u = g(x) = 51x, x = 1

- ƒ(u) = u + 1

cos2 u

, u = g(x) = px, x = 1>4

- ƒ(u) = 2u

u2 + 1

, u = g(x) = 10×2 + x + 1, x = 0

- ƒ(u) = au – 1

u + 1b

2

, u = g(x) = 1

x2 – 1, x = -1

- Assume that ƒ_(3) = -1, g_(2) = 5, g(2) = 3, and y = ƒ(g(x)).

What is y_ at x = 2?

- If r = sin (ƒ(t)), ƒ(0) = p>3, and ƒ_(0) = 4, then what is dr>dt

at t = 0?

- Suppose that functions ƒ and g and their derivatives with respect

to x have the following values at x = 2 and x = 3.

Find the derivatives with respect to x of the following combinations

at the given value of x.

- 2ƒ(x), x = 2 b. ƒ(x) + g(x), x = 3
- ƒ(x) # g(x), x = 3 d. ƒ(x)>g(x), x = 2
- ƒ(g(x)), x = 2 f. 2ƒ(x), x = 2
- 1>g2(x), x = 3 h. 2ƒ2(x) + g2(x), x = 2
- Suppose that the functions ƒ and g and their derivatives with

respect to x have the following values at x = 0 and x = 1.

Find the derivatives with respect to x of the following combinations

at the given value of x.

- 5ƒ(x) – g(x), x = 1 b. ƒ(x)g3(x), x = 0

c.

ƒ(x)

g(x) + 1

, x = 1 d. ƒ(g(x)), x = 0

- g(ƒ(x)), x = 0 f. (x11 + ƒ(x))-2, x = 1
- ƒ(x + g(x)), x = 0
- Find ds>dt when u = 3p>2 if s = cos u and du>dt = 5.
- Find dy>dt when x = 1 if y = x2 + 7x – 5 and dx>dt = 1>3.

Theory and Examples

What happens if you can write a function as a composite in different

ways? Do you get the same derivative each time? The Chain Rule

says you should. Try it with the functions in Exercises 91 and 92.

- Find dy>dx if y = x by using the Chain Rule with y as a composite

of

- y = (u>5) + 7 and u = 5x – 35
- y = 1 + (1>u) and u = 1>(x – 1).
- Find dy>dx if y = x3>2 by using the Chain Rule with y as a composite

of

- y = u3 and u = 1x
- y = 1u and u = x3.
- Find the tangent to y = ((x – 1)>(x + 1))2 at x = 0.
- Find the tangent to y = 2×2 – x + 7 at x = 2.
- a. Find the tangent to the curve y = 2 tan (px>4) at x = 1.
- Slopes on a tangent curve What is the smallest value the

slope of the curve can ever have on the interval

-2 6 x 6 2? Give reasons for your answer.

- Slopes on sine curves
- Find equations for the tangents to the curves y = sin 2x and

y = -sin (x>2) at the origin. Is there anything special about

how the tangents are related? Give reasons for your answer.

- Can anything be said about the tangents to the curves

y = sin mx and y = -sin (x>m) at the origin

(m a constant _ 0)? Give reasons for your answer.

- For a given m, what are the largest values the slopes of the

curves y = sin mx and y = -sin (x>m) can ever have? Give

reasons for your answer.

- The function y = sin x completes one period on the interval

30, 2p4, the function y = sin 2x completes two periods, the

function y = sin (x>2) completes half a period, and so on. Is

there any relation between the number of periods y = sin mx

completes on 30, 2p4 and the slope of the curve y = sin mx

at the origin? Give reasons for your answer.

- Running machinery too fast Suppose that a piston is moving

straight up and down and that its position at time t sec is

s = A cos (2pbt),

with A and b positive. The value of A is the amplitude of the

motion, and b is the frequency (number of times the piston moves

up and down each second). What effect does doubling the frequency

have on the piston’s velocity, acceleration, and jerk?

(Once you find out, you will know why some machinery breaks

when you run it too fast.)

- Temperatures in Fairbanks, Alaska The graph in the accompanying

figure shows the average Fahrenheit temperature in

Fairbanks, Alaska, during a typical 365-day year. The equation

that approximates the temperature on day x is

y = 37 sin c 2p

365

(x – 101) d + 25

and is graphed in the accompanying figure.

- On what day is the temperature increasing the fastest?
- About how many degrees per day is the temperature increasing

when it is increasing at its fastest?

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Jan

Feb

Mar

0

_20

20

40

60

x

y

………. ……. ……. ….

… .

……………. ……. ….. …………….

……… .

…………….

………. …………………..

….

Temperature (_F)

- Particle motion The position of a particle moving along a

coordinate line is s = 21 + 4t, with s in meters and t in seconds.

Find the particle’s velocity and acceleration at t = 6 sec.

- Constant acceleration Suppose that the velocity of a falling

body is y = k1s m>sec (k a constant) at the instant the body

has fallen s m from its starting point. Show that the body’s

acceleration is constant.

- Falling meteorite The velocity of a heavy meteorite entering

Earth’s atmosphere is inversely proportional to 2s when it is

s km from Earth’s center. Show that the meteorite’s acceleration

is inversely proportional to s2.

- Particle acceleration A particle moves along the x-axis with

velocity dx>dt = ƒ(x). Show that the particle’s acceleration is

ƒ(x)ƒ_(x).

- Temperature and the period of a pendulum For oscillations

of small amplitude (short swings), we may safely model the relationship

between the period T and the length L of a simple pendulum

with the equation

T = 2pA

Lg

,

where g is the constant acceleration of gravity at the pendulum’s

location. If we measure g in centimeters per second squared, we

measure L in centimeters and T in seconds. If the pendulum is

made of metal, its length will vary with temperature, either

increasing or decreasing at a rate that is roughly proportional to

- In symbols, with u being temperature and k the proportionality

constant,

dL

du = kL.

Assuming this to be the case, show that the rate at which the

period changes with respect to temperature is kT>2.

- Chain Rule Suppose that ƒ(x) = x2 and g(x) = _ x _. Then the

composites

(ƒ _ g)(x) = _ x _2 = x2 and (g _ ƒ)(x) = _ x2 _ = x2

are both differentiable at x = 0 even though g itself is not differentiable

at x = 0. Does this contradict the Chain Rule?

Explain.

- The derivative of sin 2x Graph the function y = 2 cos 2x for

-2 … x … 3.5. Then, on the same screen, graph

y =

sin 2(x + h) – sin 2x

h

for h = 1.0, 0.5, and 0.2. Experiment with other values of h,

including negative values. What do you see happening as

hS 0? Explain this behavior.

- The derivative of cos (x2) Graph y = -2x sin (x2) for -2 …

x … 3. Then, on the same screen, graph

y =

cos ((x + h)2) – cos (x2)

h

for h = 1.0, 0.7, and 0.3. Experiment with other values of h.

What do you see happening as hS 0? Explain this behavior.

Using the Chain Rule, show that the Power Rule (d>dx)xn = nxn-1

holds for the functions xn in Exercises 107 and 108.

- x1>4 = 21x 108. x3>4 = 2x1x

COMPUTER EXPLORATIONS

Trigonometric Polynomials

- As the accompanying figure shows, the trigonometric “polynomial”

s = ƒ(t) = 0.78540 – 0.63662 cos 2t – 0.07074 cos 6t

– 0.02546 cos 10t – 0.01299 cos 14t

gives a good approximation of the sawtooth function s = g(t)

on the interval 3-p, p4. How well does the derivative of ƒ

approximate the derivative of g at the points where dg>dt is

defined? To find out, carry out the following steps.

- Graph dg>dt (where defined) over 3-p, p4.
- Find dƒ>dt.
- Graph dƒ>dt. Where does the approximation of dg>dt by

dƒ>dt seem to be best? Least good? Approximations by trigonometric

polynomials are important in the theories of heat

and oscillation, but we must not expect too much of them, as

we see in the next exercise.

t

s

_p 0 p

2 p

s _ g(t)

s _ f (t)

- (Continuation of Exercise 109.) In Exercise 109, the trigonometric

polynomial ƒ(t) that approximated the sawtooth function g(t)

on 3-p, p4 had a derivative that approximated the derivative

of the sawtooth function. It is possible, however, for a trigonometric

polynomial to approximate a function in a reasonable

way without its derivative approximating the function’s derivative

at all well. As a case in point, the trigonometric “polynomial”

s = h(t) = 1.2732 sin 2t + 0.4244 sin 6t + 0.25465 sin 10t

+ 0.18189 sin 14t + 0.14147 sin 18t

graphed in the accompanying figure approximates the step function

s = k(t) shown there. Yet the derivative of h is nothing like

the derivative of k.

1

t

s

0 p2

_p p p

_2

_1

s _ k(t)

s _ h(t)

- Graph dk>dt (where defined) over 3-p, p4.
- Find dh>dt.
- Graph dh>dt to see how badly the graph fits the graph of

dk>dt. Comment on what you see.

Exercises 3.7

Differentiating Implicitly

Use implicit differentiation to find dy>dx in Exercises 1–16.

- x2y + xy2 = 6 2. x3 + y3 = 18xy
- 2xy + y2 = x + y 4. x3 – xy + y3 = 1
- x2(x – y)2 = x2 – y2 6. (3xy + 7)2 = 6y
- y2 = x – 1

x + 1

- x3 =

2x – y

x + 3y

- x = sec y 10. xy = cot (xy)
- x + tan (xy) = 0 12. x4 + sin y = x3y2
- y sin a1y

b = 1 – xy 14. x cos (2x + 3y) = y sin x

- e2x = sin (x + 3y) 16. ex2y = 2x + 2y

Find dr>du in Exercises 17–20.

- u1>2 + r1>2 = 1 18. r – 22u = 3

2

u2>3 + 4

3

u3>4

- sin (r u) = 1

2

- cos r + cot u = er u

Second Derivatives

In Exercises 21–26, use implicit differentiation to find dy>dx and then

d2y>dx2.

- x2 + y2 = 1 22. x2>3 + y2>3 = 1
- y2 = ex2 + 2x 24. y2 – 2x = 1 – 2y
- 21y = x – y 26. xy + y2 = 1
- If x3 + y3 = 16, find the value of d2y>dx2 at the point (2, 2).
- If xy + y2 = 1, find the value of d2y>dx2 at the point (0, -1).

In Exercises 29 and 30, find the slope of the curve at the given points.

- y2 + x2 = y4 – 2x at (-2, 1) and (-2, -1)
- (x2 + y2)2 = (x – y)2 at (1, 0) and (1, -1)

Slopes, Tangents, and Normals

In Exercises 31–40, verify that the given point is on the curve and find

the lines that are (a) tangent and (b) normal to the curve at the given

point.

- x2 + xy – y2 = 1, (2, 3)
- x2 + y2 = 25, (3, -4)
- x2y2 = 9, (-1, 3)
- y2 – 2x – 4y – 1 = 0, (-2, 1)
- 6×2 + 3xy + 2y2 + 17y – 6 = 0, (-1, 0)
- x2 – 23xy + 2y2 = 5, 123, 22
- 2xy + p sin y = 2p, (1, p>2)
- x sin 2y = y cos 2x, (p>4, p>2)
- y = 2 sin (px – y), (1, 0)
- x2 cos2 y – sin y = 0, (0, p)
- Parallel tangents Find the two points where the curve

x2 + xy + y2 = 7 crosses the x-axis, and show that the tangents

to the curve at these points are parallel. What is the common

slope of these tangents?

- Normals parallel to a line Find the normals to the curve

xy + 2x – y = 0 that are parallel to the line 2x + y = 0.

- The eight curve Find the slopes of the curve y4 = y2 – x2 at

the two points shown here.

x

y

0

1

_1

y4 _ y2 _ x2

“3

4

“3

2

,

“3

4

1

2

a , b

a b

- The cissoid of Diocles (from about 200 b.c.) Find equations

for the tangent and normal to the cissoid of Diocles y2(2 – x) = x3

at (1, 1).

x

y

1

1

(1, 1)

0

y2(2 _ x) _ x3

- The devil’s curve (Gabriel Cramer, 1750) Find the slopes of

the devil’s curve y4 – 4y2 = x4 – 9×2 at the four indicated

points.

x

y

_3 3

2

_2

(3, 2)

(3, _2)

(_3, 2)

(_3, _2)

y4 _ 4y2 _ x4 _ 9×2

- The folium of Descartes (See Figure 3.28.)
- Find the slope of the folium of Descartes x3 + y3 – 9xy = 0

at the points (4, 2) and (2, 4).

- At what point other than the origin does the folium have a

horizontal tangent?

- Find the coordinates of the point A in Figure 3.28 where the

folium has a vertical tangent.

Theory and Examples

- Intersecting normal The line that is normal to the curve

x2 + 2xy – 3y2 = 0 at (1, 1) intersects the curve at what other

point?

- Power rule for rational exponents Let p and q be integers

with q 7 0. If y = x p>q, differentiate the equivalent equation

yq = xp implicitly and show that, for y _ 0,

d

dx

x p>q =

p

q x(p>q)-1.

- Normals to a parabola Show that if it is possible to draw three

normals from the point (a, 0) to the parabola x = y2 shown in the

accompanying diagram, then a must be greater than 1>2. One of

the normals is the x-axis. For what value of a are the other two

normals perpendicular?

- Is there anything special about the tangents to the curves y2 = x3

and 2×2 + 3y2 = 5 at the points (1, {1)? Give reasons for your

answer.

x

y

0

(1, 1)

y2 _ x3

2×2 + 3y2 _ 5

(1, _1)

- Verify that the following pairs of curves meet orthogonally.
- x2 + y2 = 4, x2 = 3y2
- x = 1 – y2, x = 1

3

y2

- The graph of y2 = x3 is called a semicubical parabola and is

shown in the accompanying figure. Determine the constant b so

that the line y = -13

x + b meets this graph orthogonally.

x

y

0

y2 _ x3

y _ _ x + b

13

In Exercises 53 and 54, find both dy>dx (treating y as a differentiable

function of x) and dx>dy (treating x as a differentiable function of y).

T

How do dy>dx and dx>dy seem to be related? Explain the relationship

geometrically in terms of the graphs.

- xy3 + x2y = 6
- x3 + y2 = sin2 y
- Derivative of arcsine Assume that y = sin-1 x is a differentiable

function of x. By differentiating the equation x = sin y

implicitly, show that dy>dx = 1>21 – x2 .

- Use the formula in Exercise 55 to find dy>dx if
- y = (sin-1 x)2 b. y = sin-1 a1x

b .

COMPUTER EXPLORATIONS

Use a CAS to perform the following steps in Exercises 57–64.

- Plot the equation with the implicit plotter of a CAS. Check to

see that the given point P satisfies the equation.

- Using implicit differentiation, find a formula for the derivative

dy>dx and evaluate it at the given point P.

- Use the slope found in part (b) to find an equation for the tangent

line to the curve at P. Then plot the implicit curve and

tangent line together on a single graph.

- x3 – xy + y3 = 7, P (2, 1)
- x5 + y3x + yx2 + y4 = 4, P (1, 1)
- y2 + y = 2 + x

1 – x

, P (0, 1)

- y3 + cos xy = x2, P (1, 0)
- x + tan a

y

xb = 2, P a1, p

4 b

- xy3 + tan (x + y) = 1, P ap

4

, 0b

- 2y2 + (xy)1>3 = x2 + 2, P (1, 1)
- x21 + 2y + y = x2, P (1, 0)

Exercises 3.8

Derivatives of Inverse Functions

In Exercises 1–4:

- Find ƒ -1(x).
- Graph ƒ and ƒ -1 together.
- Evaluate dƒ>dx at x = a and dƒ -1>dx at x = ƒ(a) to show that

at these points dƒ -1>dx = 1>(dƒ>dx).

- ƒ(x) = 2x + 3, a = -1 2. ƒ(x) = (1>5)x + 7, a = -1
- ƒ(x) = 5 – 4x, a = 1>2 4. ƒ(x) = 2×2, x Ú 0, a = 5
- a. Show that ƒ(x) = x3 and g(x) = 1

3 x are inverses of one another.

- Graph ƒ and g over an x-interval large enough to show the

graphs intersecting at (1, 1) and (-1, -1). Be sure the picture

shows the required symmetry about the line y = x.

- Find the slopes of the tangents to the graphs of ƒ and g at

(1, 1) and (-1, -1) (four tangents in all).

- What lines are tangent to the curves at the origin?
- a. Show that h(x) = x3>4 and k(x) = (4x)1>3 are inverses of one

another.

- Graph h and k over an x-interval large enough to show the

graphs intersecting at (2, 2) and (-2, -2). Be sure the picture

shows the required symmetry about the line y = x.

- Find the slopes of the tangents to the graphs at h and k at

(2, 2) and (-2, -2).

- What lines are tangent to the curves at the origin?
- Let ƒ(x) = x3 – 3×2 – 1, x Ú 2. Find the value of dƒ -1>dx at

the point x = -1 = ƒ(3).

- Let ƒ(x) = x2 – 4x – 5, x 7 2. Find the value of dƒ -1>dx at

the point x = 0 = ƒ(5).

- Suppose that the differentiable function y = ƒ(x) has an inverse

and that the graph of ƒ passes through the point (2, 4) and has a

slope of 1>3 there. Find the value of dƒ -1>dx at x = 4.

- Suppose that the differentiable function y = g(x) has an inverse

and that the graph of g passes through the origin with slope 2.

Find the slope of the graph of g-1 at the origin.

Derivatives of Logarithms

In Exercises 11–40, find the derivative of y with respect to x, t, or u,

as appropriate.

- y = ln 3x + x 12. y = 1

ln 3x

- y = ln (t2) 14. y = ln (t3>2) + 2t
- y = ln

3x

- y = ln (sin x)
- y = ln (u + 1) – eu 18. y = (cos u) ln (2u + 2)
- y = ln x3 20. y = (ln x)3
- y = t (ln t)2 22. y = t ln 2t
- y = x4

4

ln x – x4

16

- y = (x2 ln x)4
- y = ln t

t 26. y = t

2ln t

- y = ln x

1 + ln x

- y = x ln x

1 + ln x

- y = ln (ln x) 30. y = ln (ln (ln x))
- y = u(sin (ln u) + cos (ln u))
- y = ln (sec u + tan u)
- y = ln

1

x2x + 1

- y = 1

2

ln

1 + x

1 – x

- y = 1 + ln t

1 – ln t

- y = 2ln 1t
- y = ln (sec (ln u)) 38. y = ln a2sin u cos u

1 + 2 ln u

b

- y = ln a

(x2 + 1)5

21 – x

b 40. y = ln C

(x + 1)5

(x + 2)20

Logarithmic Differentiation

In Exercises 41–54, use logarithmic differentiation to find the derivative

of y with respect to the given independent variable.

- y = 2x(x + 1) 42. y = 2(x2 + 1)(x – 1)2
- y = A

t

t + 1

- y = A

1

t(t + 1)

- y = (sin u)2u + 3 46. y = (tan u)22u + 1
- y = t(t + 1)(t + 2) 48. y = 1

t(t + 1)(t + 2)

- y = u + 5

u cos u

- y = u sin u

2sec u

- y =

x2x2 + 1

(x + 1)2>3 52. y = C

(x + 1)10

(2x + 1)5

- y = B

3

x(x – 2)

x2 + 1

- y = B

3

x(x + 1)(x – 2)

(x2 + 1)(2x + 3)

Finding Derivatives

In Exercises 55–62, find the derivative of y with respect to x, t, or u,

as appropriate.

- y = ln (cos2 u) 56. y = ln (3ue-u)
- y = ln (3te-t) 58. y = ln (2e-t sin t)
- y = ln a eu

1 + eub 60. y = ln a 2u

1 + 2u

b

- y = e(cos t + ln t) 62. y = esin t (ln t2 + 1)

In Exercises 63–66, find dy>dx.

- ln y = ey sin x 64. ln xy = ex+y
- xy = yx 66. tan y = ex + ln x

In Exercises 67–88, find the derivative of y with respect to the given

independent variable.

- y = 2x 68. y = 3-x
- y = 52s 70. y = 2(s2)
- y = xp 72. y = t1-e
- y = log2 5u 74. y = log3 (1 + u ln 3)
- y = log4 x + log4 x2 76. y = log25 ex – log51x
- y = log2 r # log4 r 78. y = log3 r # log9 r
- y = log3 a ax + 1

x – 1b

ln 3

b 80. y = log5

B

a 7x

3x + 2b

ln 5

- y = u sin (log7 u) 82. y = log7 asin u cos u

eu 2u b

- y = log5 ex 84. y = log2 a x2e2

22x + 1

b

- y = 3log2 t 86. y = 3 log8 (log2 t)
- y = log2 (8tln 2) 88. y = t log3 1e(sin t)(ln 3)2

Logarithmic Differentiation with Exponentials

In Exercises 89–96, use logarithmic differentiation to find the derivative

of y with respect to the given independent variable.

- y = (x + 1)x 90. y = x(x+1)
- y = (1t)t 92. y = t2t
- y = (sin x)x 94. y = xsin x
- y = xln x 96. y = (ln x)ln x

Theory and Applications

- If we write g(x) for ƒ-1(x), Equation (1) can be written as

g_(ƒ(a)) = 1

ƒ_(a)

, or g_(ƒ(a)) # ƒ_(a) = 1.

If we then write x for a, we get

g_(ƒ(x)) # ƒ_(x) = 1.

The latter equation may remind you of the Chain Rule, and

indeed there is a connection.

Assume that ƒ and g are differentiable functions that are

inverses of one another, so that (g _ ƒ)(x) = x. Differentiate both

sides of this equation with respect to x, using the Chain Rule to

express (g _ ƒ)_(x) as a product of derivatives of g and ƒ.

What do you find? (This is not a proof of Theorem 3 because

we assume here the theorem’s conclusion that g = ƒ -1 is

differentiable.)

- Show that limnSq a1 + xn

b

n

= ex for any x 7 0.

- If ƒ(x) = xn, n Ú 1, show from the definition of the derivative

that ƒ_(0) = 0.

- Using mathematical induction, show that for n 7 1

dn

dxn ln x = (-1)n-1

(n – 1)!

xn .

COMPUTER EXPLORATIONS

In Exercises 101–108, you will explore some functions and their

inverses together with their derivatives and tangent line approxima-

tions at specified points. Perform the following steps using your

CAS:

- Plot the function y = ƒ(x) together with its derivative over the

given interval. Explain why you know that ƒ is one-to-one over

the interval.

- Solve the equation y = ƒ(x) for x as a function of y, and name the

resulting inverse function g.

- Find the equation for the tangent line to ƒ at the specified point

(x0, ƒ(x0)).

- Find the equation for the tangent line to g at the point (ƒ(x0), x0)

located symmetrically across the 45° line y = x (which is the

graph of the identity function). Use Theorem 3 to find the slope of

this tangent line.

- Plot the functions ƒ and g, the identity, the two tangent lines, and

the line segment joining the points (x0, ƒ(x0)) and (ƒ(x0), x0).

Discuss the symmetries you see across the main diagonal.

- y = 23x – 2,

2

3 … x … 4, x0 = 3

- y = 3x + 2

2x – 11

, -2 … x … 2, x0 = 1>2

- y = 4x

x2 + 1

, -1 … x … 1, x0 = 1>2

- y = x3

x2 + 1

, -1 … x … 1, x0 = 1>2

- y = x3 – 3×2 – 1, 2 … x … 5, x0 = 27

10

- y = 2 – x – x3, -2 … x … 2, x0 = 3

2

- y = ex, -3 … x … 5, x0 = 1
- y = sin x, – p

2 … x … p

2

, x0 = 1

In Exercises 109 and 110, repeat the steps above to solve for the functions

y = ƒ(x) and x = ƒ -1(y) defined implicitly by the given equations

over the interval.

- y1>3 – 1 = (x + 2)3, -5 … x … 5, x0 = -3>2
- cos y = x1>5, 0 … x … 1, x0 = 1>2

Exercises 3.9

Common Values

Use reference triangles in an appropriate quadrant, as in Example 1, to

find the angles in Exercises 1–8.

- a. tan-1 1 b. tan-11-232 c. tan-1 a 1

23

b

- a. tan-1(-1) b. tan-123 c. tan-1 a -1

23

b

- a. sin-1 a-1

2 b b. sin-1 a 1

22

b c. sin-1 a-23

2

b

- a. sin-1 a1

2b b. sin-1 a -1

22

b c. sin-1 a23

2

b

- a. cos-1 a1

2b b. cos-1 a -1

22

b c. cos-1 a23

2

B

- a. csc-1 22 b. csc-1 a -2

23

b c. csc-1 2

- a. sec-11-222 b. sec-1 a 2

23

b c. sec-1(-2)

- a. cot-1 (-1) b. cot-1 1232 c. cot-1 a -1

23

b

Evaluations

Find the values in Exercises 9–12.

- sin acos-1 a22

2

b b 10. sec acos-1

1

2b

- tan asin-1 a-

1

2b b 12. cot asin-1 a-

23

2

b b

Limits

Find the limits in Exercises 13–20. (If in doubt, look at the function’s

graph.)

- lim

xS1-

sin-1 x 14. lim

xS-1+

cos-1 x

- lim

xSq

tan-1 x 16. lim

xS-q

tan-1 x

- lim

xSq

sec-1 x 18. lim

xS-q

sec-1 x

- lim

xSq

csc-1 x 20. lim

xS-q

csc-1 x

Finding Derivatives

In Exercises 21–42, find the derivative of y with respect to the appropriate

variable.

- y = cos-1 (x2) 22. y = cos-1 (1>x)
- y = sin-122 t 24. y = sin-1 (1 – t)
- y = sec-1 (2s + 1) 26. y = sec-1 5s
- y = csc-1 (x2 + 1), x 7 0
- y = csc-1

x

2

- y = sec-1

1t

, 0 6 t 6 1 30. y = sin-1

3

t2

- y = cot-1 2t 32. y = cot-1 2t – 1
- y = ln (tan-1 x) 34. y = tan-1 (ln x)
- y = csc-1 (et) 36. y = cos-1 (e-t)
- y = s21 – s2 + cos-1 s 38. y = 2s2 – 1 – sec-1 s
- y = tan-12×2 – 1 + csc-1 x, x 7 1
- y = cot-1

1x

– tan-1 x 41. y = x sin-1 x + 21 – x2

- y = ln (x2 + 4) – x tan-1 ax

2b

Theory and Examples

- You are sitting in a classroom next to the wall looking at the

blackboard at the front of the room. The blackboard is 12 ft long

and starts 3 ft from the wall you are sitting next to. Show that

your viewing angle is

a = cot-1

x

15 – cot-1

x

3

if you are x ft from the front wall.

Blackboard

12_

3_

Wall

You

a

x

- Two derivations of the identity sec_1 (_x) _ P _ sec_1 x
- (Geometric) Here is a pictorial proof that sec-1 (-x) =

p – sec-1 x. See if you can tell what is going on.

x

y

0

p

_x _1 1 x

y _ sec–1x

p2

- (Algebraic) Derive the identity sec-1 (-x) = p – sec-1 x by

combining the following two equations from the text:

cos-1 (-x) = p – cos-1 x Eq. (4), Section 1.6

sec-1 x = cos-1 (1>x) Eq. (1)

Which of the expressions in Exercises 45–48 are defined, and which

are not? Give reasons for your answers.

- a. tan-1 2 b. cos-1 2
- a. csc-1 (1>2) b. csc-1 2
- a. sec-1 0 b. sin-122
- a. cot-1 (-1>2) b. cos-1 (-5)
- Use the identity

csc-1 u = p

2 – sec-1 u

to derive the formula for the derivative of csc-1 u in Table 3.1

from the formula for the derivative of sec-1 u.

- Derive the formula

dy

dx = 1

1 + x2

for the derivative of y = tan-1 x by differentiating both sides of

the equivalent equation tan y = x.

- Use the Derivative Rule in Section 3.8, Theorem 3, to derive

d

dx

sec-1 x = 1

0 x 0 2×2 – 1

, 0 x 0 7 1.

- Use the identity

cot-1 u = p

2 – tan-1 u

to derive the formula for the derivative of cot-1 u in Table 3.1

from the formula for the derivative of tan-1 u.

- What is special about the functions

ƒ(x) = sin-1

x – 1

x + 1

, x Ú 0, and g(x) = 2 tan-1 1x?

Explain.

- What is special about the functions

ƒ(x) = sin-1

1

2×2 + 1

and g(x) = tan-1

1x

?

Explain.

In Exercises 55–57, find the domain and range of each composite

function. Then graph the composites on separate screens. Do the

graphs make sense in each case? Give reasons for your answers. Comment

on any differences you see.

- a. y = tan-1 (tan x) b. y = tan (tan-1 x)
- a. y = sin-1 (sin x) b. y = sin (sin-1 x)
- a. y = cos-1 (cos x) b. y = cos (cos-1 x)

T

Use your graphing utility for Exercises 58–62.

- Graph y = sec (sec-1 x) = sec (cos-1(1>x)). Explain what you

see.

- Newton’s serpentine Graph Newton’s serpentine, y = 4x>(x2 + 1).

Then graph y = 2 sin (2 tan-1 x) in the same graphing window.

What do you see? Explain.

- Graph the rational function y = (2 – x2)>x2. Then graph y =

cos (2 sec-1 x) in the same graphing window. What do you see?

Explain.

- Graph ƒ(x) = sin-1 x together with its first two derivatives. Comment

on the behavior of ƒ and the shape of its graph in relation to

the signs and values of ƒ_ and ƒ_.

- Graph ƒ(x) = tan-1 x together with its first two derivatives. Comment

on the behavior of ƒ and the shape of its graph in relation to

the signs and values of ƒ_ and ƒ_.

Exercises 3.10

- Area Suppose that the radius r and area A = pr2 of a circle are

differentiable functions of t. Write an equation that relates dA>dt

to dr>dt.

- Surface area Suppose that the radius r and surface area S = 4pr2

of a sphere are differentiable functions of t. Write an equation that

relates dS>dt to dr>dt.

- Assume that y = 5x and dx>dt = 2. Find dy>dt.
- Assume that 2x + 3y = 12 and dy>dt = -2. Find dx>dt.
- If y = x2 and dx>dt = 3, then what is dy>dt when x = -1?
- If x = y3 – y and dy>dt = 5, then what is dx>dt when y = 2?
- If x2 + y2 = 25 and dx>dt = -2, then what is dy>dt when

x = 3 and y = -4?

- If x2y3 = 4>27 and dy>dt = 1>2, then what is dx>dt when

x = 2?

- If L = 2×2 + y2, dx>dt = -1, and dy>dt = 3, find dL>dt

when x = 5 and y = 12.

- If r + s2 + y3 = 12, dr>dt = 4, and ds>dt = -3, find dy>dt

when r = 3 and s = 1.

- If the original 24 m edge length x of a cube decreases at the rate

of 5 m>min, when x = 3 m at what rate does the cube’s

- surface area change?
- volume change?
- A cube’s surface area increases at the rate of 72 in2>sec. At what rate

is the cube’s volume changing when the edge length is x = 3 in?

- Volume The radius r and height h of a right circular cylinder

are related to the cylinder’s volume V by the formula V = pr2h.

- How is dV>dt related to dh>dt if r is constant?
- How is dV>dt related to dr>dt if h is constant?
- How is dV>dt related to dr>dt and dh>dt if neither r nor h is

constant?

- Volume The radius r and height h of a right circular cone are

related to the cone’s volume V by the equation V = (1>3)pr2h.

- How is dV>dt related to dh>dt if r is constant?
- How is dV>dt related to dr>dt if h is constant?
- How is dV>dt related to dr>dt and dh>dt if neither r nor h is

constant?

- Changing voltage The voltage V (volts), current I (amperes),

and resistance R (ohms) of an electric circuit like the one shown

here are related by the equation V = IR. Suppose that V is

increasing at the rate of 1 volt>sec while I is decreasing at the

rate of 1>3 amp>sec. Let t denote time in seconds.

V

R

I

+ _

- What is the value of dV>dt?
- What is the value of dI>dt?
- What equation relates dR>dt to dV>dt and dI>dt?
- Find the rate at which R is changing when V = 12 volts and

I = 2 amps. Is R increasing, or decreasing?

- Electrical power The power P (watts) of an electric circuit is

related to the circuit’s resistance R (ohms) and current I (amperes)

by the equation P = RI 2.

- How are dP>dt, dR>dt, and dI>dt related if none of P, R, and

I are constant?

- How is dR>dt related to dI>dt if P is constant?
- Distance Let x and y be differentiable functions of t and let

s = 2×2 + y2 be the distance between the points (x, 0) and

(0, y) in the xy-plane.

- How is ds>dt related to dx>dt if y is constant?
- How is ds>dt related to dx>dt and dy>dt if neither x nor y is

constant?

- How is dx>dt related to dy>dt if s is constant?
- Diagonals If x, y, and z are lengths of the edges of a rectangular

box, the common length of the box’s diagonals is s =

2×2 + y2 + z2.

- Assuming that x, y, and z are differentiable functions of t,

how is ds>dt related to dx>dt, dy>dt, and dz>dt?

- How is ds>dt related to dy>dt and dz>dt if x is constant?
- How are dx>dt, dy>dt, and dz>dt related if s is constant?
- Area The area A of a triangle with sides of lengths a and b

enclosing an angle of measure u is

A = 1

2

ab sin u.

- How is dA>dt related to du>dt if a and b are constant?
- How is dA>dt related to du>dt and da>dt if only b is constant?
- How is dA>dt related to du>dt, da>dt, and db>dt if none of

a, b, and u are constant?

- Heating a plate When a circular plate of metal is heated in an

oven, its radius increases at the rate of 0.01 cm > min. At what rate

is the plate’s area increasing when the radius is 50 cm?

- Changing dimensions in a rectangle The length l of a rectangle

is decreasing at the rate of 2 cm>sec while the width w is

increasing at the rate of 2 cm>sec. When l = 12 cm and

w = 5 cm, find the rates of change of (a) the area, (b) the perimeter,

and (c) the lengths of the diagonals of the rectangle. Which

of these quantities are decreasing, and which are increasing?

- Changing dimensions in a rectangular box Suppose that the

edge lengths x, y, and z of a closed rectangular box are changing

at the following rates:

dx

dt = 1 m>sec,

dy

dt = -2 m>sec,

dz

dt = 1 m>sec.

Find the rates at which the box’s (a) volume, (b) surface area, and

(c) diagonal length s = 2×2 + y2 + z2 are changing at the

instant when x = 4, y = 3, and z = 2.

- A sliding ladder A 13-ft ladder is leaning against a house when

its base starts to slide away (see accompanying figure). By the

time the base is 12 ft from the house, the base is moving at the

rate of 5 ft>sec.

- How fast is the top of the ladder sliding down the wall then?
- At what rate is the area of the triangle formed by the ladder,

wall, and ground changing then?

- At what rate is the angle u between the ladder and the ground

changing then?

x

0

y

13-ft ladder

y(t)

x(t)

u

- Commercial air traffic Two commercial airplanes are flying

at an altitude of 40,000 ft along straight-line courses that intersect

at right angles. Plane A is approaching the intersection point at a

speed of 442 knots (nautical miles per hour; a nautical mile is

2000 yd). Plane B is approaching the intersection at 481 knots. At

what rate is the distance between the planes changing when A is 5

nautical miles from the intersection point and B is 12 nautical

miles from the intersection point?

- Flying a kite A girl flies a kite at a height of 300 ft, the wind carrying

the kite horizontally away from her at a rate of 25 ft>sec. How

fast must she let out the string when the kite is 500 ft away from her?

- Boring a cylinder The mechanics at Lincoln Automotive are

reboring a 6-in.-deep cylinder to fit a new piston. The machine

they are using increases the cylinder’s radius one-thousandth of

an inch every 3 min. How rapidly is the cylinder volume increasing

when the bore (diameter) is 3.800 in.?

- A growing sand pile Sand falls from a conveyor belt at the rate

of 10 m3>min onto the top of a conical pile. The height of the pile

is always three-eighths of the base diameter. How fast are the (a)

height and (b) radius changing when the pile is 4 m high? Answer

in centimeters per minute.

- A draining conical reservoir Water is flowing at the rate of

50 m3>min from a shallow concrete conical reservoir (vertex

down) of base radius 45 m and height 6 m.

- How fast (centimeters per minute) is the water level falling

when the water is 5 m deep?

- How fast is the radius of the water’s surface changing then?

Answer in centimeters per minute.

- A draining hemispherical reservoir Water is flowing at the

rate of 6 m3>min from a reservoir shaped like a hemispherical bowl

of radius 13 m, shown here in profile. Answer the following questions,

given that the volume of water in a hemispherical bowl of

radius R is V = (p>3)y2(3R – y) when the water is y meters deep.

r

y

13

Center of sphere

Water level

- At what rate is the water level changing when the water is

8 m deep?

- What is the radius r of the water’s surface when the water is

y m deep?

- At what rate is the radius r changing when the water is 8 m

deep?

- A growing raindrop Suppose that a drop of mist is a perfect

sphere and that, through condensation, the drop picks up moisture

at a rate proportional to its surface area. Show that under these

circumstances the drop’s radius increases at a constant rate.

- The radius of an inflating balloon A spherical balloon is

inflated with helium at the rate of 100p ft3>min. How fast is the

balloon’s radius increasing at the instant the radius is 5 ft? How

fast is the surface area increasing?

- Hauling in a dinghy A dinghy is pulled toward a dock by a

rope from the bow through a ring on the dock 6 ft above the bow.

The rope is hauled in at the rate of 2 ft>sec.

- How fast is the boat approaching the dock when 10 ft of rope

are out?

- At what rate is the angle u changing at this instant (see the

figure)?

- A balloon and a bicycle A balloon is rising vertically above a

level, straight road at a constant rate of 1 ft>sec. Just when the

balloon is 65 ft above the ground, a bicycle moving at a constant

rate of 17 ft>sec passes under it. How fast is the distance s(t)

between the bicycle and balloon increasing 3 sec later?

y

x

0

y(t)

s(t)

x(t)

- Making coffee Coffee is draining from a conical filter into a

cylindrical coffeepot at the rate of 10 in3>min.

- How fast is the level in the pot rising when the coffee in the

cone is 5 in. deep?

- How fast is the level in the cone falling then?

6_

6_

6_

How fast

is this

level rising?

How fast

is this

level falling?

- Cardiac output In the late 1860s, Adolf Fick, a professor of

physiology in the Faculty of Medicine in Würzberg, Germany,

developed one of the methods we use today for measuring how much

blood your heart pumps in a minute. Your cardiac output as you read

this sentence is probably about 7 L>min. At rest it is likely to be a bit

under 6 L>min. If you are a trained marathon runner running a marathon,

your cardiac output can be as high as 30 L>min.

Your cardiac output can be calculated with the formula

y =

Q

D

,

where Q is the number of milliliters of CO2 you exhale in a minute

and D is the difference between the CO2 concentration (ml>L) in

the blood pumped to the lungs and the CO2 concentration in the

blood returning from the lungs. With Q = 233 ml>min and

D = 97 – 56 = 41 ml>L,

y =

233 ml>min

41 ml>L

_ 5.68 L>min,

fairly close to the 6 L>min that most people have at basal (resting)

conditions. (Data courtesy of J. Kenneth Herd, M.D., Quillan

College of Medicine, East Tennessee State University.)

Suppose that when Q = 233 and D = 41, we also know

that D is decreasing at the rate of 2 units a minute but that Q

remains unchanged. What is happening to the cardiac output?

- Moving along a parabola A particle moves along the parabola

y = x2 in the first quadrant in such a way that its x-coordinate

(measured in meters) increases at a steady 10 m>sec. How fast is

the angle of inclination u of the line joining the particle to the

origin changing when x = 3 m?

- Motion in the plane The coordinates of a particle in the metric

xy-plane are differentiable functions of time t with dx>dt =

-1 m>sec and dy>dt = -5 m>sec. How fast is the particle’s distance

from the origin changing as it passes through the point (5, 12)?

- Videotaping a moving car You are videotaping a race from a

stand 132 ft from the track, following a car that is moving at

180 mi>h (264 ft>sec), as shown in the accompanying figure.

How fast will your camera angle u be changing when the car is

right in front of you? A half second later?

u

Car

Camera

132_

- A moving shadow A light shines from the top of a pole 50 ft

high. A ball is dropped from the same height from a point 30 ft

away from the light. (See accompanying figure.) How fast is the

shadow of the ball moving along the ground 1>2 sec later?

(Assume the ball falls a distance s = 16t2 ft in t sec.)

- A building’s shadow On a morning of a day when the sun will

pass directly overhead, the shadow of an 80-ft building on level

ground is 60 ft long. At the moment in question, the angle u the

sun makes with the ground is increasing at the rate of 0.27_>min.

At what rate is the shadow decreasing? (Remember to use radians.

Express your answer in inches per minute, to the nearest tenth.)

80_

u

- A melting ice layer A spherical iron ball 8 in. in diameter is

coated with a layer of ice of uniform thickness. If the ice melts at

the rate of 10 in3>min, how fast is the thickness of the ice

decreasing when it is 2 in. thick? How fast is the outer surface

area of ice decreasing?

- Highway patrol A highway patrol plane flies 3 mi above a

level, straight road at a steady 120 mi>h. The pilot sees an

oncoming car and with radar determines that at the instant the

line-of-sight distance from plane to car is 5 mi, the line-of-sight

distance is decreasing at the rate of 160 mi>h. Find the car’s

speed along the highway.

- Baseball players A baseball diamond is a square 90 ft on a

side. A player runs from first base to second at a rate of 16 ft>sec.

- At what rate is the player’s distance from third base changing

when the player is 30 ft from first base?

- At what rates are angles u1 and u2 (see the figure) changing

at that time?

- The player slides into second base at the rate of 15 ft>sec. At

what rates are angles u1 and u2 changing as the player

touches base?

90_

Second base

Player

Home

30_ First

base

Third

base

u1

u2

- Ships Two ships are steaming straight away from a point O

along routes that make a 120° angle. Ship A moves at 14 knots

(nautical miles per hour; a nautical mile is 2000 yd). Ship B

moves at 21 knots. How fast are the ships moving apart when

OA = 5 and OB = 3 nautical miles?

- Clock’s moving hands At what rate is the angle between a

clock’s minute and hour hands changing at 4 o’clock in the afternoon?

- Oil spill An explosion at an oil rig located in gulf waters causes

an elliptical oil slick to spread on the surface from the rig. The slick

is a constant 9 in. thick. After several days, when the major axis of

the slick is 2 mi long and the minor axis is 3/4 mi wide, it is determined

that its length is increasing at the rate of 30 ft/hr, and its

width is increasing at the rate of 10 ft/hr. At what rate (in cubic feet

per hour) is oil flowing from the site of the rig at that time?

Exercises 3.11

Finding Linearizations

In Exercises 1–5, find the linearization L(x) of ƒ(x) at x = a.

- ƒ(x) = x3 – 2x + 3, a = 2
- ƒ(x) = 2×2 + 9, a = -4
- ƒ(x) = x + 1x

, a = 1

- ƒ(x) = 23

x, a = -8

- ƒ(x) = tan x, a = p
- Common linear approximations at x _ 0 Find the linearizations

of the following functions at x = 0.

- sin x b. cos x c. tan x d. ex e. ln (1 + x)

Linearization for Approximation

In Exercises 7–14, find a linearization at a suitably chosen integer near

a at which the given function and its derivative are easy to evaluate.

- ƒ(x) = x2 + 2x, a = 0.1
- ƒ(x) = x-1, a = 0.9
- ƒ(x) = 2×2 + 3x – 3, a = -0.9
- ƒ(x) = 1 + x, a = 8.1
- ƒ(x) = 23

x, a = 8.5

- ƒ(x) = x

x + 1

, a = 1.3

- ƒ(x) = e-x, a = -0.1
- ƒ(x) = sin-1 x, a = p>12
- Show that the linearization of ƒ(x) = (1 + x)k at x = 0 is

L(x) = 1 + kx.

- Use the linear approximation (1 + x)k _ 1 + kx to find an

approximation for the function ƒ(x) for values of x near zero.

- ƒ(x) = (1 – x)6 b. ƒ(x) = 2

1 – x

- ƒ(x) = 1

21 + x

- ƒ(x) = 22 + x2
- ƒ(x) = (4 + 3x)1>3 f. ƒ(x) = B

3 a1 – x

2 + xb

2

- Faster than a calculator Use the approximation (1 + x)k _

1 + kx to estimate the following.

- (1.0002)50 b. 23

1.009

- Find the linearization of ƒ(x) = 2x + 1 + sin x at x = 0. How

is it related to the individual linearizations of 2x + 1 and sin x

at x = 0?

Derivatives in Differential Form

In Exercises 19–38, find dy.

- y = x3 – 32x 20. y = x21 – x2
- y = 2x

1 + x2 22. y =

21x

3(1 + 1x)

- 2y3>2 + xy – x = 0 24. xy2 – 4×3>2 – y = 0
- y = sin (51x) 26. y = cos (x2)
- y = 4 tan (x3>3) 28. y = sec (x2 – 1)
- y = 3 csc 11 – 22×2 30. y = 2 cot a 1

1x

b

- y = e2x 32. y = xe-x
- y = ln (1 + x2) 34. y = ln a x + 1

2x – 1

b

- y = tan-1 (ex2) 36. y = cot-1 a 1

x2b + cos-1 2x

- y = sec-1 (e-x) 38. y = etan-1 2×2+1

Approximation Error

In Exercises 39–44, each function ƒ(x) changes value when x changes

from x0 to x0 + dx. Find

- the change _ƒ = ƒ(x0 + dx) – ƒ(x0);
- the value of the estimate dƒ = ƒ_(x0) dx; and
- the approximation error 0 _ƒ – dƒ 0 .

x

y

0

dx

x0 + dx

df _ f _(x0) dx

_f _ f (x0 + dx) _ f (x0)

Tangent

(x0, f (x0))

y _ f (x)

x0

- ƒ(x) = x2 + 2x, x0 = 1, dx = 0.1
- ƒ(x) = 2×2 + 4x – 3, x0 = -1, dx = 0.1
- ƒ(x) = x3 – x, x0 = 1, dx = 0.1
- ƒ(x) = x4, x0 = 1, dx = 0.1
- ƒ(x) = x-1, x0 = 0.5, dx = 0.1
- ƒ(x) = x3 – 2x + 3, x0 = 2, dx = 0.1

Differential Estimates of Change

In Exercises 45–50, write a differential formula that estimates the

given change in volume or surface area.

- The change in the volume V = (4>3)pr3 of a sphere when the

radius changes from r0 to r0 + dr

- The change in the volume V = x3 of a cube when the edge

lengths change from x0 to x0 + dx

- The change in the surface area S = 6×2 of a cube when the edge

lengths change from x0 to x0 + dx

- The change in the lateral surface area S = pr2r2 + h2 of a right

circular cone when the radius changes from r0 to r0 + dr and the

height does not change

- The change in the volume V = pr2h of a right circular cylinder

when the radius changes from r0 to r0 + dr and the height does

not change

- The change in the lateral surface area S = 2prh of a right circular

cylinder when the height changes from h0 to h0 + dh and the

radius does not change

Applications

- The radius of a circle is increased from 2.00 to 2.02 m.
- Estimate the resulting change in area.
- Express the estimate as a percentage of the circle’s original area.
- The diameter of a tree was 10 in. During the following year, the

circumference increased 2 in. About how much did the tree’s

diameter increase? The tree’s cross-sectional area?

- Estimating volume Estimate the volume of material in a cylindrical

shell with length 30 in., radius 6 in., and shell thickness 0.5 in.

6 in.

0.5 in.

30 in.

- Estimating height of a building A surveyor, standing 30 ft

from the base of a building, measures the angle of elevation to the

top of the building to be 75°. How accurately must the angle be

measured for the percentage error in estimating the height of the

building to be less than 4%?

- The radius r of a circle is measured with an error of at most 2%.

What is the maximum corresponding percentage error in computing

the circle’s

- circumference? b. area?
- The edge x of a cube is measured with an error of at most 0.5%.

What is the maximum corresponding percentage error in computing

the cube’s

- surface area? b. volume?
- Tolerance The height and radius of a right circular cylinder are

equal, so the cylinder’s volume is V = ph3. The volume is to be

calculated with an error of no more than 1% of the true value.

Find approximately the greatest error that can be tolerated in the

measurement of h, expressed as a percentage of h.

- Tolerance
- About how accurately must the interior diameter of a

10-m-high cylindrical storage tank be measured to calculate

the tank’s volume to within 1% of its true value?

- About how accurately must the tank’s exterior diameter be

measured to calculate the amount of paint it will take to paint

the side of the tank to within 5% of the true amount?

- The diameter of a sphere is measured as 100 { 1 cm and the

volume is calculated from this measurement. Estimate the percentage

error in the volume calculation.

- Estimate the allowable percentage error in measuring the diameter D

of a sphere if the volume is to be calculated correctly to within 3%.

- The effect of flight maneuvers on the heart The amount of

work done by the heart’s main pumping chamber, the left ventricle,

is given by the equation

W = PV + Vdy2

2g

,

where W is the work per unit time, P is the average blood pressure,

V is the volume of blood pumped out during the unit of

time, d (“delta”) is the weight density of the blood, y is the

average velocity of the exiting blood, and g is the acceleration

of gravity.

When P, V, d, and y remain constant, W becomes a function

of g, and the equation takes the simplified form

W = a + bg

(a, b constant).

As a member of NASA’s medical team, you want to know how

sensitive W is to apparent changes in g caused by flight maneuvers,

and this depends on the initial value of g. As part of your investigation,

you decide to compare the effect on W of a given change dg

on the moon, where g = 5.2 ft>sec2, with the effect the same

change dg would have on Earth, where g = 32 ft>sec2. Use the

simplified equation above to find the ratio of dWmoon to dWEarth.

- Drug concentration The concentration C in milligrams per

milliliter (mg>ml) of a certain drug in a person’s bloodstream t

hrs after a pill is swallowed is modeled by

C (t) = 1 + 4t

1 + t3 – e-0.06t.

Estimate the change in concentration when t changes from 20 to

30 min.

- Unclogging arteries The formula V = kr4, discovered by the

physiologist Jean Poiseuille (1797–1869), allows us to predict how

much the radius of a partially clogged artery has to be expanded in

order to restore normal blood flow. The formula says that the volume

V of blood flowing through the artery in a unit of time at a

fixed pressure is a constant k times the radius of the artery to the

fourth power. How will a 10% increase in r affect V ?

- Measuring acceleration of gravity When the length L of a

clock pendulum is held constant by controlling its temperature,

the pendulum’s period T depends on the acceleration of gravity g.

The period will therefore vary slightly as the clock is moved from

place to place on the earth’s surface, depending on the change in

- By keeping track of _T, we can estimate the variation in g

from the equation T = 2p(L>g)1>2 that relates T, g, and L.

- With L held constant and g as the independent variable, calculate

dT and use it to answer parts (b) and (c).

- If g increases, will T increase or decrease? Will a pendulum

clock speed up or slow down? Explain.

- A clock with a 100-cm pendulum is moved from a location

where g = 980 cm>sec2 to a new location. This increases the

period by dT = 0.001 sec. Find dg and estimate the value of

g at the new location.

- Quadratic approximations
- Let Q(x) = b0 + b1(x – a) + b2(x – a)2 be a quadratic

approximation to ƒ(x) at x = a with the properties:

- i) Q(a) = ƒ(a)
- ii) Q_(a) = ƒ_(a)

iii) Q_(a) = ƒ_(a).

Determine the coefficients b0, b1, and b2.

- Find the quadratic approximation to ƒ(x) = 1>(1 – x) at

x = 0.

- Graph ƒ(x) = 1>(1 – x) and its quadratic approximation at

x = 0. Then zoom in on the two graphs at the point (0, 1).

Comment on what you see.

- Find the quadratic approximation to g(x) = 1>x at x = 1.

Graph g and its quadratic approximation together. Comment

on what you see.

T

T

- Find the quadratic approximation to h(x) = 21 + x at

x = 0. Graph h and its quadratic approximation together.

Comment on what you see.

- What are the linearizations of ƒ, g, and h at the respective

points in parts (b), (d), and (e)?

- The linearization is the best linear approximation Suppose

that y = ƒ(x) is differentiable at x = a and that g(x) =

m(x – a) + c is a linear function in which m and c are constants.

If the error E(x) = ƒ(x) – g(x) were small enough near x = a,

we might think of using g as a linear approximation of ƒ instead

of the linearization L(x) = ƒ(a) + ƒ_(a)(x – a). Show that if we

impose on g the conditions

- E(a) = 0 The approximation error is zero at x = a.
- lim

xSa

E(x)

x – a = 0

The error is negligible when compared

with x – a.

then g(x) = ƒ(a) + ƒ_(a)(x – a). Thus, the linearization L(x)

gives the only linear approximation whose error is both zero at

x = a and negligible in comparison with x – a.

x

a

y _ f (x)

(a, f (a))

The linearization, L(x):

y _ f (a) + f _(a)(x _ a)

Some other linear

approximation, g(x):

y _ m(x _ a) + c

- The linearization of 2x
- Find the linearization of ƒ(x) = 2x at x = 0. Then round its

coefficients to two decimal places.

- Graph the linearization and function together for

-3 … x … 3 and -1 … x … 1.

- The linearization of log3 x
- Find the linearization of ƒ(x) = log3 x at x = 3. Then round

its coefficients to two decimal places.

- Graph the linearization and function together in the window

0 … x … 8 and 2 … x … 4.

COMPUTER EXPLORATIONS

In Exercises 69–74, use a CAS to estimate the magnitude of the error

in using the linearization in place of the function over a specified

interval I. Perform the following steps:

- Plot the function ƒ over I.
- Find the linearization L of the function at the point a.
- Plot ƒ and L together on a single graph.
- Plot the absolute error _ ƒ(x) – L(x) _ over I and find its maximum

value.

- From your graph in part (d), estimate as large a d 7 0 as you

can, satisfying

0 x – a 0 6 d 1 0 ƒ(x) – L(x) 0 6 P

for P = 0.5, 0.1, and 0.01. Then check graphically to see if

your d@estimate holds true.

- ƒ(x) = x3 + x2 – 2x, 3-1, 24, a = 1
- ƒ(x) = x – 1

4×2 + 1

, c –

3

4

, 1 d , a = 1

2

- ƒ(x) = x2>3(x – 2), 3-2, 34, a = 2
- ƒ(x) = 2x – sin x, 30, 2p4, a = 2
- ƒ(x) = x2x, 30, 24, a = 1
- ƒ(x) = 2x sin-1 x, 30, 14, a = 1

2

Chapter 3 Questions to Guide Your Review

- What is the derivative of a function ƒ? How is its domain related

to the domain of ƒ? Give examples.

- What role does the derivative play in defining slopes, tangents,

and rates of change?

- How can you sometimes graph the derivative of a function when

all you have is a table of the function’s values?

- What does it mean for a function to be differentiable on an open

interval? On a closed interval?

- How are derivatives and one-sided derivatives related?
- Describe geometrically when a function typically does not have a

derivative at a point.

- How is a function’s differentiability at a point related to its continuity

there, if at all?

- What rules do you know for calculating derivatives? Give some

examples.

- Explain how the three formulas
- d

dx

(xn) = nxn-1 b. d

dx

(cu) = c

du

dx

- d

dx

(u1 + u2 + g+ un) =

du1

dx +

du2

dx

+ g+

dun

dx

enable us to differentiate any polynomial.

- What formula do we need, in addition to the three listed in Question

9, to differentiate rational functions?

- What is a second derivative? A third derivative? How many

derivatives do the functions you know have? Give examples.

- What is the derivative of the exponential function ex? How does

the domain of the derivative compare with the domain of the

function?

- What is the relationship between a function’s average and instantaneous

rates of change? Give an example.

- How do derivatives arise in the study of motion? What can you

learn about an object’s motion along a line by examining the

derivatives of the object’s position function? Give examples.

- How can derivatives arise in economics?
- Give examples of still other applications of derivatives.
- What do the limits limhS0 ((sin h)>h) and limhS0 ((cos h – 1)>h)

have to do with the derivatives of the sine and cosine functions?

What are the derivatives of these functions?

- Once you know the derivatives of sin x and cos x, how can you

find the derivatives of tan x, cot x, sec x, and csc x? What are the

derivatives of these functions?

- At what points are the six basic trigonometric functions continuous?

How do you know?

- What is the rule for calculating the derivative of a composite of

two differentiable functions? How is such a derivative evaluated?

Give examples.

- If u is a differentiable function of x, how do you find (d>dx)(un) if

n is an integer? If n is a real number? Give examples.

- What is implicit differentiation? When do you need it? Give

examples.

- What is the derivative of the natural logarithm function ln x? How

does the domain of the derivative compare with the domain of the

function?

- What is the derivative of the exponential function ax, a 7 0 and

a _ 1? What is the geometric significance of the limit of

(ah – 1)>h as hS 0? What is the limit when a is the number e?

- What is the derivative of loga x? Are there any restrictions on a?
- What is logarithmic differentiation? Give an example.
- How can you write any real power of x as a power of e? Are there

any restrictions on x? How does this lead to the Power Rule for

differentiating arbitrary real powers?

- What is one way of expressing the special number e as a limit? What

is an approximate numerical value of e correct to 7 decimal places?

- What are the derivatives of the inverse trigonometric functions?

How do the domains of the derivatives compare with the domains

of the functions?

- How do related rates problems arise? Give examples.
- Outline a strategy for solving related rates problems. Illustrate

with an example.

- What is the linearization L (x) of a function ƒ(x) at a point x = a?

What is required of ƒ at a for the linearization to exist? How are

linearizations used? Give examples.

- If x moves from a to a nearby value a + dx, how do you estimate

the corresponding change in the value of a differentiable function

ƒ(x)? How do you estimate the relative change? The percentage

change? Give an example.

Chapter 3 Practice Exercises

Derivatives of Functions

Find the derivatives of the functions in Exercises 1–64.

- y = x3 – 0.375×2 + 0.70x 2. y = 8 + 0.5×5 – 0.7×6
- y = x5 – 5(x2 + 4) 4. y = x5 + 25x – 1

p + 5

- y = (x + 3)2(x3 + 5×2) 6. y = (4x + 7)(2 – x)-2
- y = (x3 + sin x + 3)4 8. y = a-3 – sec u

3 – u3

6 b

3

- r =

1s

5 – 1s

- r = 5

1s – 8

- y = 2 cot2 x + csc2 x 12. y = 1

cos2 x – 3

tan x

- r = tan4 (3 + 4t) 14. r = cot2 a6t

b

- r = (cos t + cot t)6 16. s = sec6 (8 – t2 + 6t3)
- r = 23u tan u 18. r = 3u2tan u
- r = tan 25u 20. r = cos 1u – 2u – 12
- y = 1

3

x3 sec

3x

- y = 52x cos 2x
- y = x-3>2 csc (3x)3 24. y = 2x sec (x – 4)4
- y = 6 tan x3 26. y = x3 cot 8x
- y = x3 cos2 (4×2) 28. y = x-3 cos2 (x5)
- s = a 6t

t + 3b

-3

- s = -8

25(25t + 4)3

- y = a 2x

2 – xb

3

- y = a 32x

32x – 4

b

3

- y = B

x3 + x

x3 34. y = 6x22x3 + 1×3

- r = a cos u

sin u + 1b

3

- r = a2 – cos u

2 + sin u

b

3

- y = (3x + 2)23x + 2 38. y = 30 (5x – 3)1>5 (5x – 3)-1>6
- y = 8

(9×3 + cos 2x)5>2 40. y = (2 + sin2 2x)-1>2

- y = 16e-x>8 42. y = 25e25x
- y = 1

5

xe5x – 1

25

e5x 44. y = x3e-3>x

- y = ln (tan2 u) 46. y = ln (cos2 u)
- y = log3 (x3>3) 48. y = log3 (8x + 5)
- y = 9-x 50. y = 34x
- y = 8×7.2 52. y = 25t-25
- y = (x + 3)x+3 54. y = 5 (ln x)x>5
- y = cos-1(2×21 – x2),

1

12 6 x 6 1

- y = cos-1 a 1

2x

b, x 7 1

- y = ln sin-1 x
- y = x sin-1 x + 21 – x2
- y = x cot-1 x – 1

2

ln x

- y = (1 – x2) tan-1 3x
- y = t csc-1 t + 2t2 – 1, t 7 1
- y = 52t + 1 tan-1 1t
- y = csc-1 (sec u), 0 6 u 6 p>2
- y = 21 – x2 esin-1 x

Implicit Differentiation

In Exercises 65–78, find dy>dx by implicit differentiation.

- xy + 2x + 3y = 1 66. x2 + xy + y2 – 5x = 2
- x3 + 4xy – 3y4>3 = 2x 68. 5×4>5 + 10y6>5 = 15
- 1xy = 1 70. x2y2 = 1
- y2 = x

x + 1

- y2 = A

1 + x

1 – x

- ex+2y = 1 74. y2 = 2e-1>x
- ln (x>y) = 1 76. x sin-1 y = 1 + x2
- yetan-1 x = 2 78. xy = 22

In Exercises 79 and 80, find dp>dq.

- p3 + 4pq – 3q2 = 2 80. q = (5p2 + 2p)-3>2

In Exercises 81 and 82, find dr>ds.

- r cos 2s + sin2 s = p 82. 2rs – r – s + s2 = -3
- Find d2y>dx2 by implicit differentiation:
- x3 + y3 = 1 b. y2 = 1 – 2x
- a. By differentiating x2 – y2 = 1 implicitly, show that

dy>dx = x>y.

- Then show that d2y>dx2 = -1>y3.

Numerical Values of Derivatives

- Suppose that functions ƒ(x) and g(x) and their first derivatives

have the following values at x = 0 and x = 1.

Find the first derivatives of the following combinations at the

given value of x.

- 6ƒ(x) – g(x), x = 1 b. ƒ(x)g2(x), x = 0

c.

ƒ(x)

g(x) + 1

, x = 1 d. ƒ(g(x)), x = 0

- g(ƒ(x)), x = 0 f. (x + ƒ(x))3>2, x = 1
- ƒ(x + g(x)), x = 0
- Suppose that the function ƒ(x) and its first derivative have the following

values at x = 0 and x = 1.

Find the first derivatives of the following combinations at the

given value of x.

- 1x ƒ(x), x = 1 b. 2ƒ(x), x = 0
- ƒ12×2, x = 1 d. ƒ(1 – 5 tan x), x = 0

e.

ƒ(x)

2 + cos x

, x = 0 f. 10 sin apx

2 b ƒ2(x), x = 1

- Find the value of dy>dt at t = 0 if y = 3 sin 2x and x = t2 + p.
- Find the value of ds>du at u = 2 if s = t2 + 5t and t =

(u2 + 2u)1>3.

- Find the value of dw>ds at s = 0 if w = sin 1e1r2 and

r = 3 sin (s + p>6).

- Find the value of dr>dt at t = 0 if r = (u2 + 7)1>3 and

u2t + u = 1.

- If y3 + y = 2 cos x, find the value of d2y>dx2 at the point (0, 1).
- If x1>3 + y1>3 = 4, find d2y>dx2 at the point (8, 8).

Applying the Derivative Definition

In Exercises 93 and 94, find the derivative using the definition.

- ƒ(t) = 1

2t + 1

- g(x) = 2×2 + 1
- a. Graph the function

ƒ(x) = e

x2, -1 … x 6 0

-x2, 0 … x … 1.

- Is ƒ continuous at x = 0?
- Is ƒ differentiable at x = 0?

Give reasons for your answers.

- a. Graph the function

ƒ(x) = e

x, -1 … x 6 0

tan x, 0 … x … p>4.

- Is ƒ continuous at x = 0?
- Is ƒ differentiable at x = 0?

Give reasons for your answers.

- a. Graph the function

ƒ(x) = e

x, 0 … x … 1

2 – x, 1 6 x … 2.

- Is ƒ continuous at x = 1?
- Is ƒ differentiable at x = 1?

Give reasons for your answers.

- For what value or values of the constant m, if any, is

ƒ(x) = e

sin 2x, x … 0

mx, x 7 0

- continuous at x = 0?
- differentiable at x = 0?

Give reasons for your answers.

Slopes, Tangents, and Normals

- Tangents with specified slope Are there any points on the

curve y = (x>2) + 1>(2x – 4) where the slope is -3>2? If so,

find them.

- Tangents with specified slope Are there any points on the

curve y = x – e-x where the slope is 2? If so, find them.

- Horizontal tangents Find the points on the curve y =

2×3 – 3×2 – 12x + 20 where the tangent is parallel to the

x-axis.

- Tangent intercepts Find the x- and y-intercepts of the line

that is tangent to the curve y = x3 at the point (-2, -8).

- Tangents perpendicular or parallel to lines Find the points

on the curve y = 2×3 – 3×2 – 12x + 20 where the tangent is

- perpendicular to the line y = 1 – (x>24).
- parallel to the line y = 22 – 12x.
- Intersecting tangents Show that the tangents to the curve

y = (p sin x)>x at x = p and x = -p intersect at right angles.

- Normals parallel to a line Find the points on the curve

y = tan x, -p>2 6 x 6 p>2, where the normal is parallel to

the line y = -x>2. Sketch the curve and normals together,

labeling each with its equation.

- Tangent and normal lines Find equations for the tangent and

normal to the curve y = 1 + cos x at the point (p>2, 1). Sketch

the curve, tangent, and normal together, labeling each with its

equation.

- Tangent parabola The parabola y = x2 + C is to be tangent

to the line y = x. Find C.

- Slope of tangent Show that the tangent to the curve y = x3 at

any point (a, a3) meets the curve again at a point where the slope

is four times the slope at (a, a3).

- Tangent curve For what value of c is the curve y = c>(x + 1)

tangent to the line through the points (0, 3) and (5, -2)?

- Normal to a circle Show that the normal line at any point of

the circle x2 + y2 = a2 passes through the origin.

In Exercises 111–116, find equations for the lines that are tangent and

normal to the curve at the given point.

- x2 + 2y2 = 9, (1, 2)
- ex + y2 = 2, (0, 1)
- xy + 2x – 5y = 2, (3, 2)
- (y – x)2 = 2x + 4, (6, 2)
- x + 1xy = 6, (4, 1)
- x3>2 + 2y3>2 = 17, (1, 4)
- Find the slope of the curve x3y3 + y2 = x + y at the points

(1, 1) and (1, -1).

- The graph shown suggests that the curve y = sin (x – sin x)

might have horizontal tangents at the x-axis. Does it? Give reasons

for your answer.

x

y

0

_1

1

y _ sin (x _ sin x)

_2p _p p 2p

Analyzing Graphs

Each of the figures in Exercises 119 and 120 shows two graphs, the

graph of a function y = ƒ(x) together with the graph of its derivative

ƒ_(x). Which graph is which? How do you know?

- 120.

x

y

_1 0 1

1

_1

_2

A 2

B

x

y

0 1

1

A

B

2

2

3

4

- Use the following information to graph the function y = ƒ(x)

for -1 … x … 6.

- i) The graph of ƒ is made of line segments joined end to end.
- ii) The graph starts at the point (-1, 2).

iii) The derivative of ƒ, where defined, agrees with the step

function shown here.

x

y

_1 1 2

1

_1

3 4 5 6

_2

y _ f _(x)

- Repeat Exercise 121, supposing that the graph starts at (-1, 0)

instead of (-1, 2).

Trigonometric Limits

Find the limits in Exercises 123–130.

- lim

xS0

sin x

2×2 – x

- lim

xS0

3x – tan 7x

2x

- lim

rS0

sin r

tan 2r

- lim

uS0

sin (sin u)

u

- lim

uS(p>2)-

4 tan2 u + tan u + 1

tan2 u + 5

- lim

uS0+

1 – 2 cot2 u

5 cot2 u – 7 cot u – 8

- lim

xS0

x sin x

2 – 2 cos x

- lim

uS0

1 – cos u

u2

Show how to extend the functions in Exercises 131 and 132 to be continuous

at the origin.

- g(x) =

tan (tan x)

tan x 132. ƒ(x) =

tan (tan x)

sin (sin x)

Logarithmic Differentiation

In Exercises 133–138, use logarithmic differentiation to find the

derivative of y with respect to the appropriate variable.

- y =

2(x2 + 1)

2cos 2x

- y = 10A

3x + 4

2x – 4

- y = a

(t + 1)(t – 1)

(t – 2)(t + 3)b

5

, t 7 2

- y = 2u2u

2u2 + 1

- y = (sin u)2u 138. y = (ln x)1>(ln x)

Related Rates

- Right circular cylinder The total surface area S of a right circular

cylinder is related to the base radius r and height h by the

equation S = 2pr2 + 2prh.

- How is dS>dt related to dr>dt if h is constant?
- How is dS>dt related to dh>dt if r is constant?
- How is dS>dt related to dr>dt and dh>dt if neither r nor h is

constant?

- How is dr>dt related to dh>dt if S is constant?
- Cube’s changing edges The volume of a cube is increasing

at the rate of 1200 cm3>min at the instant its edges are 20 cm

long. At what rate are the lengths of the edges changing at that

instant?

- Resistors connected in parallel If two resistors of R1 and R2

ohms are connected in parallel in an electric circuit to make an

R-ohm resistor, the value of R can be found from the equation

1

R = 1

R1

+ 1

R2

.

+

R

_

R1 R2

If R1 is decreasing at the rate of 1 ohm > sec and R2 is increasing

at the rate of 0.5 ohm > sec, at what rate is R changing when

R1 = 75 ohms and R2 = 50 ohms?

- Impedance in a series circuit The impedance Z (ohms) in a

series circuit is related to the resistance R (ohms) and reactance

X (ohms) by the equation Z = 2R2 + X2. If R is increasing at

3 ohms > sec and X is decreasing at 2 ohms > sec, at what rate is Z

changing when R = 10 ohms and X = 20 ohms?

- Speed of moving particle The coordinates of a particle moving

in the metric xy-plane are differentiable functions of time t with

dx>dt = 10 m>sec and dy>dt = 5 m>sec. How fast is the particle

moving away from the origin as it passes through the point (3, -4)?

- Motion of a particle A particle moves along the curve y = x3>2

in the first quadrant in such a way that its distance from the origin

increases at the rate of 11 units per second. Find dx>dt when x = 3.

- Draining a tank Water drains from the conical tank shown in

the accompanying figure at the rate of 5 ft3>min.

- What is the relation between the variables h and r in the figure?
- How fast is the water level dropping when h = 6 ft?

r

h

Exit rate: 5 ft3_min

10_

4_

- Rotating spool As television cable is pulled from a large spool

to be strung from the telephone poles along a street, it unwinds

from the spool in layers of constant radius (see accompanying

figure). If the truck pulling the cable moves at a steady 6 ft > sec

(a touch over 4 mph), use the equation s = r u to find how fast

(radians per second) the spool is turning when the layer of radius

1.2 ft is being unwound.

1.2_

- Moving searchlight beam The figure shows a boat 1 km offshore,

sweeping the shore with a searchlight. The light turns at a

constant rate, du>dt = -0.6 rad/sec.

- How fast is the light moving along the shore when it reaches

point A?

- How many revolutions per minute is 0.6 rad>sec?

1 km

A

x

u

- Points moving on coordinate axes Points A and B move

along the x- and y-axes, respectively, in such a way that the distance

r (meters) along the perpendicular from the origin to the

line AB remains constant. How fast is OA changing, and is it

increasing, or decreasing, when OB = 2r and B is moving

toward O at the rate of 0.3r m > sec?

Linearization

- Find the linearizations of
- tan x at x = -p>4 b. sec x at x = -p>4.

Graph the curves and linearizations together.

- We can obtain a useful linear approximation of the function

ƒ(x) = 1>(1 + tan x) at x = 0 by combining the approximations

1

1 + x _ 1 – x and tan x _ x

to get

1

1 + tan x _ 1 – x.

Show that this result is the standard linear approximation of

1>(1 + tan x) at x = 0.

- Find the linearization of ƒ(x) = 21 + x + sin x – 0.5 at x = 0.
- Find the linearization of ƒ(x) = 2>(1 – x) + 21 + x – 3.1

at x = 0.

Differential Estimates of Change

- Surface area of a cone Write a formula that estimates the

change that occurs in the lateral surface area of a right circular

cone when the height changes from h0 to h0 + dh and the radius

does not change.

- Controlling error
- How accurately should you measure the edge of a cube to be

reasonably sure of calculating the cube’s surface area with an

error of no more than 2%?

- Suppose that the edge is measured with the accuracy required

in part (a). About how accurately can the cube’s volume be

calculated from the edge measurement? To find out, estimate

the percentage error in the volume calculation that might

result from using the edge measurement.

- Compounding error The circumference of the equator of a

sphere is measured as 10 cm with a possible error of 0.4 cm.

This measurement is used to calculate the radius. The radius is

then used to calculate the surface area and volume of the sphere.

Estimate the percentage errors in the calculated values of

- the radius. b. the surface area. c. the volume.
- Finding height To find the height of a lamppost (see accompanying

figure), you stand a 6 ft pole 20 ft from the lamp and

measure the length a of its shadow, finding it to be 15 ft, give or

take an inch. Calculate the height of the lamppost using the

value a = 15 and estimate the possible error in the result.

h

20 ft

6 ft

a

Chapter 3 A dditional and Advanced Exercises

- An equation like sin2 u + cos2 u = 1 is called an identity

because it holds for all values of u. An equation like sin u = 0.5

is not an identity because it holds only for selected values of u,

not all. If you differentiate both sides of a trigonometric identity

in u with respect to u, the resulting new equation will also be an

identity.

Differentiate the following to show that the resulting equations

hold for all u.

- sin 2u = 2 sin u cos u
- cos 2u = cos2 u – sin2 u
- If the identity sin (x + a) = sin x cos a + cos x sin a is differentiated

with respect to x, is the resulting equation also an identity?

Does this principle apply to the equation x2 – 2x – 8 = 0?

Explain.

- a. Find values for the constants a, b, and c that will make

ƒ(x) = cos x and g(x) = a + bx + cx2

satisfy the conditions

ƒ(0) = g(0), ƒ_(0) = g_(0), and ƒ_(0) = g_(0).

- Find values for b and c that will make

ƒ(x) = sin (x + a) and g(x) = b sin x + c cos x

satisfy the conditions

ƒ(0) = g(0) and ƒ_(0) = g_(0).

- For the determined values of a, b, and c, what happens for the

third and fourth derivatives of ƒ and g in each of parts (a) and (b)?

- An osculating circle Find the values of h, k, and a that make

the circle (x – h)2 + (y – k)2 = a2 tangent to the parabola

y = x2 + 1 at the point (1, 2) and that also make the second

derivatives d2y>dx2 have the same value on both curves there.

Circles like this one that are tangent to a curve and have the same

second derivative as the curve at the point of tangency are called

osculating circles (from the Latin osculari, meaning “to kiss”).

We encounter them again in Chapter 12.

- Industrial production
- Economists often use the expression “rate of growth” in relative

rather than absolute terms. For example, let u = ƒ(t) be

the number of people in the labor force at time t in a given

industry. (We treat this function as though it were differentiable

even though it is an integer-valued step function.)

Let y = g(t) be the average production per person in

the labor force at time t. The total production is then y = uy.

If the labor force is growing at the rate of 4% per year

(du>dt = 0.04u) and the production per worker is growing at

the rate of 5% per year (dy>dt = 0.05y), find the rate of

growth of the total production, y.

- Suppose that the labor force in part (a) is decreasing at the

rate of 2% per year while the production per person is

increasing at the rate of 3% per year. Is the total production

increasing, or is it decreasing, and at what rate?

- Designing a gondola The designer of a 30-ft-diameter spherical

hot air balloon wants to suspend the gondola 8 ft below the

bottom of the balloon with cables tangent to the surface of the

balloon, as shown. Two of the cables are shown running from the

top edges of the gondola to their points of tangency, (-12, -9)

and (12, -9). How wide should the gondola be?

- Pisa by parachute On August 5, 1988, Mike McCarthy of

London jumped from the top of the Tower of Pisa. He then

opened his parachute in what he said was a world record lowlevel

parachute jump of 179 ft. Make a rough sketch to show the

shape of the graph of his speed during the jump. (Source: Boston

Globe, Aug. 6, 1988.)

- Motion of a particle The position at time t Ú 0 of a particle

moving along a coordinate line is

s = 10 cos (t + p>4).

- What is the particle’s starting position (t = 0)?
- What are the points farthest to the left and right of the origin

reached by the particle?

- Find the particle’s velocity and acceleration at the points in

part (b).

- When does the particle first reach the origin? What are its

velocity, speed, and acceleration then?

- Shooting a paper clip On Earth, you can easily shoot a paper

clip 64 ft straight up into the air with a rubber band. In t sec after

firing, the paper clip is s = 64t – 16t2 ft above your hand.

- How long does it take the paper clip to reach its maximum

height? With what velocity does it leave your hand?

- On the moon, the same acceleration will send the paper clip

to a height of s = 64t – 2.6t2 ft in t sec. About how long

will it take the paper clip to reach its maximum height, and

how high will it go?

- Velocities of two particles At time t sec, the positions of two

particles on a coordinate line are s1 = 3t3 – 12t2 + 18t + 5 m

and s2 = -t3 + 9t2 – 12t m. When do the particles have the

same velocities?

- Velocity of a particle A particle of constant mass m moves

along the x-axis. Its velocity y and position x satisfy the equation

1

2

m (y2 – y0

2) = 1

2

k (x0

2 – x2),

where k, y0, and x0 are constants. Show that whenever y _ 0,

m

dy

dt = -kx.

- Average and instantaneous velocity
- Show that if the position x of a moving point is given by a

quadratic function of t, x = At2 + Bt + C, then the average

velocity over any time interval 3t1, t2 4 is equal to the instantaneous

velocity at the midpoint of the time interval.

- What is the geometric significance of the result in part (a)?
- Find all values of the constants m and b for which the function

y = e

sin x, x 6 p

mx + b, x Ú p

is

- continuous at x = p.
- differentiable at x = p.
- Does the function

ƒ(x) = •

1 – cos x

x , x _ 0

0, x = 0

have a derivative at x = 0? Explain.

- a. For what values of a and b will

ƒ(x) = e

ax, x 6 2

ax2 – bx + 3, x Ú 2

be differentiable for all values of x?

- Discuss the geometry of the resulting graph of ƒ.
- a. For what values of a and b will

g(x) = e

ax + b, x … -1

ax3 + x + 2b, x 7 -1

be differentiable for all values of x?

- Discuss the geometry of the resulting graph of g.
- Odd differentiable functions Is there anything special about

the derivative of an odd differentiable function of x? Give reasons

for your answer.

- Even differentiable functions Is there anything special about

the derivative of an even differentiable function of x? Give reasons

for your answer.

- Suppose that the functions ƒ and g are defined throughout an

open interval containing the point x0, that ƒ is differentiable at x0,

that ƒ(x0) = 0, and that g is continuous at x0. Show that the product

ƒg is differentiable at x0. This process shows, for example,

that although 0 x 0 is not differentiable at x = 0, the product x 0 x 0

is differentiable at x = 0.

- (Continuation of Exercise 19.) Use the result of Exercise 19 to

show that the following functions are differentiable at x = 0.

- 0 x 0 sin x b. x2>3 sin x c. 23

x (1 – cos x)

- h(x) = e

x2 sin (1>x), x _ 0

0, x = 0

- Suppose that a function ƒ satisfies the following conditions for all

real values of x and y:

- i) ƒ(x + y) = ƒ(x) # ƒ(y).
- ii) ƒ(x) = 1 + xg(x), where limxS0 g(x) = 1.

Show that the derivative ƒ_(x) exists at every value of x and that

ƒ_(x) = ƒ(x).

- Leibniz’s rule for higher-order derivatives of products Leibniz’s

rule for higher-order derivatives of products of differentiable

functions says that

a.

d2(uy)

dx2 = d2u

dx2 y + 2

du

dx

dy

dx + u

d2y

dx2.

b.

d3(uy)

dx3 = d3u

dx3 y + 3

d2u

dx2

dy

dx + 3

du

dx

d2y

dx2 + u

d3y

dx3.

c.

dn(uy)

dxn = dnu

dxn y + n

dn-1u

dxn-1

dy

dx + g

+

n(n – 1)g(n – k + 1)

k!

dn-ku

dxn-k

dky

dxk

+ g + u

dny

dxn .

The equations in parts (a) and (b) are special cases of the

equation in part (c). Derive the equation in part (c) by mathematical

induction, using

a

m

k

b + a

m

k + 1

b = m!

k!(m – k)! + m!

(k + 1)!(m – k – 1)!

.

- The generalized product rule Use mathematical induction to

prove that if y = u1 u2gun is a finite product of differentiable

functions, then y is differentiable on their common domain and

dy

dx =

du1

dx

u2gun + u1

du2

dx gun + g+ u1 u2gun-1

dun

dx

.

- The period of a clock pendulum The period T of a clock pendulum

(time for one full swing and back) is given by the formula

T2 = 4p2L>g, where T is measured in seconds, g = 32.2 ft>sec2,

and L, the length of the pendulum, is measured in feet. Find

approximately

- the length of a clock pendulum whose period is T = 1 sec.
- the change dT in T if the pendulum in part (a) is lengthened

0.01 ft.

- the amount the clock gains or loses in a day as a result of the

period’s changing by the amount dT found in part (b).

Chapter 4 Applications of Derivatives

Chapter 4 Questions to Guide Your Review

- What can be said about the extreme values of a function that is

continuous on a closed interval?

- What does it mean for a function to have a local extreme value on

its domain? An absolute extreme value? How are local and absolute

extreme values related, if at all? Give examples.

- How do you find the absolute extrema of a continuous function

on a closed interval? Give examples.

- What are the hypotheses and conclusion of Rolle’s Theorem? Are

the hypotheses really necessary? Explain.

- What are the hypotheses and conclusion of the Mean Value Theorem?

What physical interpretations might the theorem have?

- State the Mean Value Theorem’s three corollaries.
- What is the First Derivative Test for Local Extreme Values? Give

examples of how it is applied.

- How do you test a twice-differentiable function to determine

where its graph is concave up or concave down? Give examples.

- What is an inflection point? Give an example. What physical significance

do inflection points sometimes have?

- What is the Second Derivative Test for Local Extreme Values?

Give examples of how it is applied.

- What do the derivatives of a function tell you about the shape of

its graph?

- What is a cusp? Give examples.
- List the steps you would take to graph a rational function. Illustrate

with an example.

- Outline a general strategy for solving max-min problems. Give

examples.

- Describe l’Hôpital’s Rule. How do you know when to use the rule

and when to stop? Give an example.

- How can you sometimes handle limits that lead to indeterminate

forms q>q, q # 0, q – q, 1q, 00, and qq? Give examples.

- Describe Newton’s method for solving equations. Give an example.

What is the theory behind the method? What are some of the

things to watch out for when you use the method?

- Can a function have more than one antiderivative? If so, how are

the antiderivatives related? Explain.

- What is an indefinite integral? How do you evaluate one? What

general formulas do you know for finding indefinite integrals?

- What is an initial value problem? How do you solve one? Give an

example.

Chapter 4 Practice Exercises

Extreme Values

- Does ƒ(x) = x3 + 2x + tan x have any local maximum or minimum

values? Give reasons for your answer.

- Does g(x) = csc x + 2 cot x have any local maximum values?

Give reasons for your answer.

- Does ƒ(x) = (7 + x)(11 – 3x)1>3 have an absolute minimum

value? An absolute maximum? If so, find them or give reasons

why they fail to exist. List all critical points of ƒ.

- Find values of a and b such that the function

ƒ(x) = ax + b

x2 – 1

has a local extreme value of 1 at x = 3. Is this extreme value a

local maximum, or a local minimum? Give reasons for your

answer.

- Does g(x) = ex – x have an absolute minimum value? An absolute

maximum? If so, find them or give reasons why they fail to

exist. List all critical points of g.

- Does ƒ(x) = 2ex>(1 + x2) have an absolute minimum value? An

absolute maximum? If so, find them or give reasons why they fail

to exist. List all critical points of ƒ.

In Exercises 7 and 8, find the absolute maximum and absolute minimum

values of ƒ over the interval.

- ƒ(x) = x – 2 ln x, 1 … x … 3
- ƒ(x) = (4>x) + ln x2, 1 … x … 4
- The greatest integer function ƒ(x) = :x;, defined for all values

of x, assumes a local maximum value of 0 at each point of 30, 1).

Could any of these local maximum values also be local minimum

values of ƒ? Give reasons for your answer.

- a. Give an example of a differentiable function ƒ whose first

derivative is zero at some point c even though ƒ has neither a

local maximum nor a local minimum at c.

- How is this consistent with Theorem 2 in Section 4.1? Give

reasons for your answer.

- The function y = 1>x does not take on either a maximum or a

minimum on the interval 0 6 x 6 1 even though the function is

continuous on this interval. Does this contradict the Extreme

Value Theorem for continuous functions? Why?

- What are the maximum and minimum values of the function

y = 0 x 0 on the interval -1 … x 6 1? Notice that the interval is

not closed. Is this consistent with the Extreme Value Theorem for

continuous functions? Why?

- A graph that is large enough to show a function’s global behavior

may fail to reveal important local features. The graph of ƒ(x) =

(x8>8) – (x6>2) – x5 + 5×3 is a case in point.

- Graph ƒ over the interval -2.5 … x … 2.5. Where does the

graph appear to have local extreme values or points of inflection?

- Now factor ƒ_(x) and show that ƒ has a local maximum at

x =23

5 _ 1.70998 and local minima at x = {23 _

{1.73205.

T

- Zoom in on the graph to find a viewing window that shows

the presence of the extreme values at x = 23

5 and x = 23.

The moral here is that without calculus the existence of two

of the three extreme values would probably have gone unnoticed.

On any normal graph of the function, the values would lie close

enough together to fall within the dimensions of a single pixel on

the screen.

(Source: Uses of Technology in the Mathematics Curriculum,

by Benny Evans and Jerry Johnson, Oklahoma State University,

published in 1990 under a grant from the National Science

Foundation, USE-8950044.)

- (Continuation of Exercise 13.)
- Graph ƒ(x) = (x8>8) – (2>5)x5 – 5x – (5>x2) + 11 over

the interval -2 … x … 2. Where does the graph appear to

have local extreme values or points of inflection?

- Show that ƒ has a local maximum value at x = 27

5 _ 1.2585

and a local minimum value at x = 23

2 _ 1.2599.

- Zoom in to find a viewing window that shows the presence of

the extreme values at x = 27

5 and x = 23

2.

The Mean Value Theorem

- a. Show that g(t) = sin2 t – 3t decreases on every interval in its

domain.

- How many solutions does the equation sin2 t – 3t = 5 have?

Give reasons for your answer.

- a. Show that y = tan u increases on every open interval in its

domain.

- If the conclusion in part (a) is really correct, how do you

explain the fact that tan p = 0 is less than tan (p>4) = 1?

- a. Show that the equation x4 + 2×2 – 2 = 0 has exactly one

solution on 30, 14 .

- Find the solution to as many decimal places as you can.
- a. Show that ƒ(x) = x>(x + 1) increases on every open interval

in its domain.

- Show that ƒ(x) = x3 + 2x has no local maximum or minimum

values.

- Water in a reservoir As a result of a heavy rain, the volume of

water in a reservoir increased by 1400 acre-ft in 24 hours. Show

that at some instant during that period the reservoir’s volume was

increasing at a rate in excess of 225,000 gal>min. (An acre-foot

is 43,560 ft3, the volume that would cover 1 acre to the depth of

1 ft. A cubic foot holds 7.48 gal.)

- The formula F(x) = 3x + C gives a different function for each

value of C. All of these functions, however, have the same derivative

with respect to x, namely F_(x) = 3. Are these the only differentiable

functions whose derivative is 3? Could there be any

others? Give reasons for your answers.

- Show that

d

dx

a x

x + 1b = d

dx

a-

1

x + 1b

even though

x

x + 1 _ –

1

x + 1

.

Doesn’t this contradict Corollary 2 of the Mean Value Theorem?

Give reasons for your answer.

- Calculate the first derivatives of ƒ(x) = x2>(x2 + 1) and g(x) =

-1>(x2 + 1). What can you conclude about the graphs of these

functions?

Analyzing Graphs

In Exercises 23 and 24, use the graph to answer the questions.

- Identify any global extreme values of ƒ and the values of x at

which they occur.

y

x

(1, 1)

2, 1

2

0

y _ f (x)

a b

- Estimate the open intervals on which the function y = ƒ(x) is
- increasing.
- decreasing.
- Use the given graph of ƒ_ to indicate where any local extreme

values of the function occur, and whether each extreme is a

relative maximum or minimum.

y

x

(_3, 1)

(2, 3)

_1

_2

y _ f _(x)

Each of the graphs in Exercises 25 and 26 is the graph of the position

function s = ƒ(t) of an object moving on a coordinate line (t represents

time). At approximately what times (if any) is each object’s (a) velocity

equal to zero? (b) Acceleration equal to zero? During approximately

what time intervals does the object move (c) forward? (d) Backward?

Graphs and Graphing

Graph the curves in Exercises 27– 42.

- y = x2 – (x3>6) 28. y = x3 – 3×2 + 3
- y = -x3 + 6×2 – 9x + 3
- y = (1>8)(x3 + 3×2 – 9x – 27)
- y = x3(8 – x) 32. y = x2(2×2 – 9)
- y = x – 3×2>3 34. y = x1>3(x – 4)
- y = x23 – x 36. y = x24 – x2
- y = (x – 3)2 ex 38. y = xe-x2
- y = ln (x2 – 4x + 3) 40. y = ln (sin x)
- y = sin-1 a1x

b 42. y = tan-1 a1x

b

Each of Exercises 43– 48 gives the first derivative of a function

y = ƒ(x). (a) At what points, if any, does the graph of ƒ have a local

maximum, local minimum, or inflection point? (b) Sketch the general

shape of the graph.

- y_ = 16 – x2 44. y_ = x2 – x – 6
- y_ = 6x(x + 1)(x – 2) 46. y_ = x2(6 – 4x)
- y_ = x4 – 2×2 48. y_ = 4×2 – x4

In Exercises 49–52, graph each function. Then use the function’s first

derivative to explain what you see.

- y = x2>3 + (x – 1)1>3 50. y = x2>3 + (x – 1)2>3
- y = x1>3 + (x – 1)1>3 52. y = x2>3 – (x – 1)1>3

Sketch the graphs of the rational functions in Exercises 53–60.

- y = x + 1

x – 3

- y = 2x

x + 5

- y = x2 + 1

x 56. y = x2 – x + 1

x

- y = x3 + 2

2x

- y = x4 – 1

x2

- y = x2 – 4

x2 – 3

- y = x2

x2 – 4

Using L’Hôpital’s Rule

Use l’Hôpital’s Rule to find the limits in Exercises 61–72.

- lim

xS2

x3 – 5x + 2

x – 2

- lim

xS1

x5 – 1

x7 – 1

- lim

xS0

sin x

x 64. lim

xS0

sin x

2x + tan x

- lim

xS1

1 – x

cos (px

2 )

- lim

xS0

tan mx

tan nx

- lim

xSp>2

sec 9x cos 5x 68. lim

xS0+

2x sec x

- lim

xSp>2

(sec x – tan x) 70. lim

xS0

a 1

x5 – 1

x3b

- lim

xSq

12×4 – x2 + 5 – 2×4 + x22

- lim

xSq

a x4

x3 – 1 – x4

x3 + 1

b

Find the limits in Exercises 73–84.

- lim

xS0

15x – 1

x 74. lim

uS0

5u – 1

u

- lim

xS0

5tan x – 1

ex – 1

- lim

xS0

5-tan x – 1

ex – 1

- lim

xS0

atan 3x – 2x

3x – sin2 x

b 78. lim

xS0

a7 – 7ex

xex b

- lim

tS0+

t – ln (1 + 2t)

t2 80. lim

xS2

x2 – 4

2x + 2 – 23x – 2

- lim

tS0+

e4x

x – 1x

- lim

yS0+

e-1>y ln y

- lim

xSq

a1 + bx

b

kx

- lim

xSq

a6 – 5x

– 10

x3 b

Optimization

- The sum of two nonnegative numbers is 36. Find the numbers if
- the difference of their square roots is to be as large as possible.
- the sum of their square roots is to be as large as possible.
- The sum of two nonnegative numbers is 20. Find the numbers
- if the product of one number and the square root of the other

is to be as large as possible.

- if one number plus the square root of the other is to be as

large as possible.

- An isosceles triangle has its vertex at the origin and its base parallel

to the x-axis with the vertices above the axis on the curve

y = 27 – x2. Find the largest area the triangle can have.

- A customer has asked you to design an open-top rectangular

stainless steel vat. It is to have a square base and a volume of

32 ft3, to be welded from quarter-inch plate, and to weigh no

more than necessary. What dimensions do you recommend?

- Find the height and radius of the largest right circular cylinder

that can be put in a sphere of radius 23.

- The figure here shows two right circular cones, one upside down

inside the other. The two bases are parallel, and the vertex of the

smaller cone lies at the center of the larger cone’s base. What values

of r and h will give the smaller cone the largest possible volume?

r

6_

h

12_

- Manufacturing tires Your company can manufacture x hundred

grade A tires and y hundred grade B tires a day, where

0 … x … 4 and

Your profit on a grade A tire is twice your profit on a grade B tire.

What is the most profitable number of each kind to make?

- Particle motion The positions of two particles on the s-axis are

s1 = cos t and s2 = cos (t + p>4).

- What is the farthest apart the particles ever get?
- When do the particles collide?
- Open-top box An open-top rectangular box is constructed from

a 10-in.-by-16-in. piece of cardboard by cutting squares of equal

side length from the corners and folding up the sides. Find analytically

the dimensions of the box of largest volume and the

maximum volume. Support your answers graphically.

- The ladder problem What is the approximate length (in feet)

of the longest ladder you can carry horizontally around the corner

of the corridor shown here? Round your answer down to the nearest

foot.

x

y

0

6

8

(8, 6)

Newton’s Method

- Let ƒ(x) = 3x – x3. Show that the equation ƒ(x) = -4 has a

solution in the interval 32, 34 and use Newton’s method to find it.

- Let ƒ(x) = x4 – x3. Show that the equation ƒ(x) = 75 has a solution

in the interval 33, 44 and use Newton’s method to find it.

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises

97–120. You may need to try a solution and then adjust your

guess. Check your answers by differentiation.

97.

L

(x5 + 4×2 + 9) dx 98.

L

a6t5 – t3

4 – tb dt

99.

L

at2t + 5

t3b dt 100.

L

a 5

t2t

– 4

t3b dt

101.

L

dr

(r – 3)3 102.

L

5 dr

1r + 2322

103.

L

9u22u3 – 1 du 104.

L

x

25 + x2

dx

105.

L

x4(1 – x5)-1>5 dx 106.

L

(5 – x)5>7 dx

107.

L

sec2

s

10

ds 108.

L

csc2 es ds

109.

L

csc 23u cot 23u du 110.

L

sec

t

5

tan

t

5

Dt

111.

L

cos2

x

6

dx aHint: cos2 x = 1 + cos 2x

2 b

112.

L

sec2

x

2

dx

113.

L

a5x

+ x3b dx

114.

L

a 9

x3 + 4

x2 + 9

b dx

115.

L

a5

2

e2x – e-2xb dx

116.

L

(ax – xa) dx

117.

L

x4+p dx 118.

L

8p+x dx

119.

L

5

4x2x2 – 9

dx 120.

L

dx

225 – x2

Initial Value Problems

Solve the initial value problems in Exercises 121–124.

121.

dy

dx = x2 + 1

x2 , y(1) = -1

122.

dy

dx = ax + 1x

b

2

, y(1) = 1

- d2r

dt2 = 152t + 3

2t

; r_(1) = 8, r (1) = 0

- d3r

dt3 = -cos t; r_(0) = r_(0) = 0, r (0) = -1

Applications and Examples

- Can the integrations in (a) and (b) both be correct? Explain.

a.

L

dx

21 – x2 = sin-1 x + C

b.

L

dx

21 – x2 = –

L

–

dx

21 – x2 = -cos-1 x + C

- Can the integrations in (a) and (b) both be correct? Explain.

a.

L

dx

21 – x2 = –

L

–

dx

21 – x2 = -cos-1 x + C

b.

L

dx

21 – x2 =

L

-du

21 – (-u)2

x = -u

dx = -du

=

L

-du

21 – u2

= cos-1 u + C

= cos-1 (-x) + C u = -x

- The rectangle shown here has one side on the positive y-axis,

one side on the positive x-axis, and its upper right-hand vertex

on the curve y = e-x2. What dimensions give the rectangle its

largest area, and what is that area?

x

y

0

1 y _ e_x2

- The rectangle shown here has one side on the positive y-axis,

one side on the positive x-axis, and its upper right-hand vertex

on the curve y = (ln x)>x2. What dimensions give the rectangle

its largest area, and what is that area?

x

y

0

0.2 y _

1

0.1

x2

ln x

In Exercises 129 and 130, find the absolute maximum and minimum

values of each function on the given interval.

- y = x ln 2x – x, c 1

2e

,

e

2 d

- y = 10x(2 – ln x), (0, e2 4

In Exercises 131 and 132, find the absolute maxima and minima of

the functions and say where they are assumed.

- ƒ(x) = ex>2×4+1 132. g(x) = e23-2x-x2
- Graph the following functions and use what you see to locate

and estimate the extreme values, identify the coordinates of the

inflection points, and identify the intervals on which the graphs

are concave up and concave down. Then confirm your estimates

by working with the functions’ derivatives.

- y = (ln x)>2x b. y = e-x2 c. y = (1 + x)e-x
- Graph ƒ(x) = x ln x. Does the function appear to have an absolute

minimum value? Confirm your answer with calculus.

- Graph ƒ(x) = (sin x)sin x over 30, 3p4. Explain what you see.
- A round underwater transmission cable consists of a core of copper

wires surrounded by nonconducting insulation. If x denotes

the ratio of the radius of the core to the thickness of the insulation,

it is known that the speed of the transmission signal is

given by the equation y = x2 ln (1>x). If the radius of the core is

1 cm, what insulation thickness h will allow the greatest transmission

speed?

Chapter 4 Additional and Advanced Exercises

Functions and Derivatives

- What can you say about a function whose maximum and minimum

values on an interval are equal? Give reasons for your

answer.

- Is it true that a discontinuous function cannot have both an absolute

maximum and an absolute minimum value on a closed interval?

Give reasons for your answer.

- Can you conclude anything about the extreme values of a continuous

function on an open interval? On a half-open interval? Give

reasons for your answer.

- Local extrema Use the sign pattern for the derivative

dƒ

dx = 6(x – 1)(x – 2)2(x – 3)3(x – 4)4

to identify the points where ƒ has local maximum and minimum

values.

- Local extrema
- Suppose that the first derivative of y = ƒ(x) is

y_ = 6(x + 1)(x – 2)2.

At what points, if any, does the graph of ƒ have a local maximum,

local minimum, or point of inflection?

- Suppose that the first derivative of y = ƒ(x) is

y_ = 6x(x + 1)(x – 2).

At what points, if any, does the graph of ƒ have a local maximum,

local minimum, or point of inflection?

- If ƒ_(x) … 2 for all x, what is the most the values of ƒ can

increase on 30, 64 ? Give reasons for your answer.

- Bounding a function Suppose that ƒ is continuous on 3a, b4

and that c is an interior point of the interval. Show that if

ƒ_(x) … 0 on 3a, c) and ƒ_(x) Ú 0 on (c, b4 , then ƒ(x) is never

less than ƒ(c) on 3a, b4 .

- An inequality
- Show that -1>2 … x>(1 + x2) … 1>2 for every value of x.
- Suppose that ƒ is a function whose derivative is ƒ_(x) =

x>(1 + x2). Use the result in part (a) to show that

0 ƒ(b) – ƒ(a) 0 … 1

2

0 b – a 0

for any a and b.

- The derivative of ƒ(x) = x2 is zero at x = 0, but ƒ is not a constant

function. Doesn’t this contradict the corollary of the Mean

Value Theorem that says that functions with zero derivatives are

constant? Give reasons for your answer.

- Extrema and inflection points Let h = ƒg be the product of

two differentiable functions of x.

- If ƒ and g are positive, with local maxima at x = a, and if ƒ_

and g_ change sign at a, does h have a local maximum at a?

- If the graphs of ƒ and g have inflection points at x = a, does

the graph of h have an inflection point at a?

In either case, if the answer is yes, give a proof. If the answer is no,

give a counterexample.

- Finding a function Use the following information to find the

values of a, b, and c in the formula ƒ(x) = (x + a)>

(bx2 + cx + 2).

- i) The values of a, b, and c are either 0 or 1.
- ii) The graph of ƒ passes through the point (-1, 0).

iii) The line y = 1 is an asymptote of the graph of ƒ.

- Horizontal tangent For what value or values of the constant k

will the curve y = x3 + kx2 + 3x – 4 have exactly one horizontal

tangent?

Optimization

- Largest inscribed triangle Points A and B lie at the ends of a

diameter of a unit circle and point C lies on the circumference. Is

it true that the area of triangle ABC is largest when the triangle is

isosceles? How do you know?

- Proving the second derivative test The Second Derivative

Test for Local Maxima and Minima (Section 4.4) says:

- ƒ has a local maximum value at x = c if ƒ_(c) = 0 and

ƒ_(c) 6 0

- ƒ has a local minimum value at x = c if ƒ_(c) = 0 and

ƒ_(c) 7 0.

To prove statement (a), let P = (1>2) 0 ƒ_(c) 0 . Then use the fact that

ƒ_(c) = lim

hS0

ƒ_(c + h) – ƒ_(c)

h = lim

hS0

ƒ_(c + h)

h

to conclude that for some d 7 0,

0 6 0 h 0 6 d 1

ƒ_(c + h)

h 6 ƒ_(c) + P 6 0.

Thus, ƒ_(c + h) is positive for -d 6 h 6 0 and negative for

0 6 h 6 d. Prove statement (b) in a similar way.

- Hole in a water tank You want to bore a hole in the side of the

tank shown here at a height that will make the stream of water

coming out hit the ground as far from the tank as possible. If you

drill the hole near the top, where the pressure is low, the water

will exit slowly but spend a relatively long time in the air. If you

drill the hole near the bottom, the water will exit at a higher

velocity but have only a short time to fall. Where is the best place,

if any, for the hole? (Hint: How long will it take an exiting droplet

of water to fall from height y to the ground?)

- Kicking a field goal An American football player wants to kick

a field goal with the ball being on a right hash mark. Assume that

the goal posts are b feet apart and that the hash mark line is a distance

a 7 0 feet from the right goal post. (See the accompanying

figure.) Find the distance h from the goal post line that gives the

kicker his largest angle b. Assume that the football field is flat.

Goal post line

Football

h

b a

Goal posts

b u

- A max-min problem with a variable answer Sometimes the

solution of a max-min problem depends on the proportions of the

shapes involved. As a case in point, suppose that a right circular

cylinder of radius r and height h is inscribed in a right circular

cone of radius R and height H, as shown here. Find the value of r

(in terms of R and H) that maximizes the total surface area of the

cylinder (including top and bottom). As you will see, the solution

depends on whether H … 2R or H 7 2R.

H

R

r

h

- Minimizing a parameter Find the smallest value of the positive

constant m that will make mx – 1 + (1>x) greater than or

equal to zero for all positive values of x.

Limits

- Evaluate the following limits.
- lim

xS0

2 sin 5x

3x

- lim

xS0

sin 5x cot 3x

- lim

xS0

x csc2 22x d. lim

xSp>2

(sec x – tan x)

- lim

xS0

x – sin x

x – tan x f. lim

xS0

sin x2

x sin x

- lim

xS0

sec x – 1

x2 h. lim

xS2

x3 – 8

x2 – 4

- L’Hôpital’s Rule does not help with the following limits. Find

them some other way.

- lim

xSq

2x + 5

2x + 5

- lim

xSq

2x

x + 72x

Theory and Examples

- Suppose that it costs a company y = a + bx dollars to produce x

units per week. It can sell x units per week at a price of

P = c – ex dollars per unit. Each of a, b, c, and e represents a

positive constant. (a) What production level maximizes the

profit? (b) What is the corresponding price? (c) What is the

weekly profit at this level of production? (d) At what price should

each item be sold to maximize profits if the government imposes

a tax of t dollars per item sold? Comment on the difference

between this price and the price before the tax.

- Estimating reciprocals without division You can estimate the

value of the reciprocal of a number a without ever dividing by a if

you apply Newton’s method to the function ƒ(x) = (1>x) – a.

For example, if a = 3, the function involved is ƒ(x) = (1>x) – 3.

- Graph y = (1>x) – 3. Where does the graph cross the

x-axis?

- Show that the recursion formula in this case is

xn+1 = xn(2 – 3xn),

so there is no need for division.

- To find x = 2q

a, we apply Newton’s method to ƒ(x) = xq – a.

Here we assume that a is a positive real number and q is a positive

integer. Show that x1 is a “weighted average” of x0 and

a>x0

q-1, and find the coefficients m0, m1 such that

x1 = m0 x0 + m1a a

x0

q-1b,

m0 7 0, m1 7 0,

m0 + m1 = 1.

What conclusion would you reach if x0 and a>x0

q-1 were equal?

What would be the value of x1 in that case?

- The family of straight lines y = ax + b (a, b arbitrary constants)

can be characterized by the relation y_ = 0. Find a similar relation

satisfied by the family of all circles

(x – h)2 + (y – h)2 = r2,

where h and r are arbitrary constants. (Hint: Eliminate h and r

from the set of three equations including the given one and two

obtained by successive differentiation.)

- Free fall in the fourteenth century In the middle of the fourteenth

century, Albert of Saxony (1316–1390) proposed a model of

free fall that assumed that the velocity of a falling body was proportional

to the distance fallen. It seemed reasonable to think that a

body that had fallen 20 ft might be moving twice as fast as a body

that had fallen 10 ft. And besides, none of the instruments in use at

the time were accurate enough to prove otherwise. Today we can

see just how far off Albert of Saxony’s model was by solving the

initial value problem implicit in his model. Solve the problem and

compare your solution graphically with the equation s = 16t2.

You will see that it describes a motion that starts too slowly at first

and then becomes too fast too soon to be realistic.

- Group blood testing During World War II it was necessary to

administer blood tests to large numbers of recruits. There are two

standard ways to administer a blood test to N people. In method 1,

each person is tested separately. In method 2, the blood samples

of x people are pooled and tested as one large sample. If the test is

negative, this one test is enough for all x people. If the test is positive,

then each of the x people is tested separately, requiring a

total of x + 1 tests. Using the second method and some probability

theory it can be shown that, on the average, the total number

of tests y will be

y = Na1 – qx + 1x

b.

With q = 0.99 and N = 1000, find the integer value of x that minimizes

- Also find the integer value of x that maximizes y. (This

second result is not important to the real-life situation.) The group

testing method was used in World War II with a savings of 80% over

the individual testing method, but not with the given value of q.

- Assume that the brakes of an automobile produce a constant

deceleration of k ft>sec2. (a) Determine what k must be to bring

an automobile traveling 60 mi>hr (88 ft>sec) to rest in a distance

of 100 ft from the point where the brakes are applied. (b) With

the same k, how far would a car traveling 30 mi>hr go before

being brought to a stop?

- Let ƒ(x), g(x) be two continuously differentiable functions satisfying

the relationships ƒ_(x) = g(x) and ƒ_(x) = -ƒ(x). Let

h(x) = ƒ2(x) + g2(x). If h(0) = 5, find h(10).

- Can there be a curve satisfying the following conditions? d2y>dx2

is everywhere equal to zero and, when x = 0, y = 0 and

dy>dx = 1. Give a reason for your answer.

- Find the equation for the curve in the xy-plane that passes through

the point (1, -1) if its slope at x is always 3×2 + 2.

- A particle moves along the x-axis. Its acceleration is a = -t2. At

t = 0, the particle is at the origin. In the course of its motion, it

reaches the point x = b, where b 7 0, but no point beyond b.

Determine its velocity at t = 0.

- A particle moves with acceleration a = 2t – 11>2t2. Assuming

that the velocity y = 4>3 and the position s = -4>15 when

t = 0, find

- the velocity y in terms of t.
- the position s in terms of t.
- Given ƒ(x) = ax2 + 2bx + c with a 7 0. By considering the

minimum, prove that ƒ(x) Ú 0 for all real x if and only if

b2 – ac … 0.

- Schwarz’s inequality
- In Exercise 33, let

ƒ(x) = (a1 x + b1)2 + (a2 x + b2)2 + g+ (an x + bn)2,

and deduce Schwarz’s inequality:

(a1 b1 + a2 b2 + g+ an bn)2

… 1a1

2 + a2

2 + g+ an

221b1

2 + b2

2 + g+ bn

22.

- Show that equality holds in Schwarz’s inequality only if there

exists a real number x that makes ai x equal -bi for every

value of i from 1 to n.

- The best branching angles for blood vessels and pipes When

a smaller pipe branches off from a larger one in a flow system, we

may want it to run off at an angle that is best from some energysaving

point of view. We might require, for instance, that energy

loss due to friction be minimized along the section AOB shown in

the accompanying figure. In this diagram, B is a given point to be

reached by the smaller pipe, A is a point in the larger pipe

upstream from B, and O is the point where the branching occurs.

A law due to Poiseuille states that the loss of energy due to friction

in nonturbulent flow is proportional to the length of the path

and inversely proportional to the fourth power of the radius.

Thus, the loss along AO is (kd1)>R4 and along OB is (kd2)>r4,

where k is a constant, d1 is the length of AO, d2 is the length of

OB, R is the radius of the larger pipe, and r is the radius of the

smaller pipe. The angle u is to be chosen to minimize the sum of

these two losses:

L = k

d1

R4 + k

d2

r4 .

a

C

B

O

A

d1

d2

d2 cos u

b _ d2 sin u

u

In our model, we assume that AC = a and BC = b are fixed.

Thus we have the relations

d1 + d2 cos u = a d2 sin u = b,

so that

d2 = b csc u,

d1 = a – d2 cos u = a – b cot u.

We can express the total loss L as a function of u:

L = kaa – b cot u

R4 + b csc u

r4 b.

- Show that the critical value of u for which dL>du equals zero

is

uc = cos-1

r4

R4 .

- If the ratio of the pipe radii is r>R = 5>6, estimate to the

nearest degree the optimal branching angle given in part (a).

Chapter 5 Integrals

Chapter 5 Questions to Guide Your Review

- How can you sometimes estimate quantities like distance traveled,

area, and average value with finite sums? Why might you want to

do so?

- What is sigma notation? What advantage does it offer? Give

examples.

- What is a Riemann sum? Why might you want to consider such a

sum?

- What is the norm of a partition of a closed interval?
- What is the definite integral of a function ƒ over a closed interval

3a, b4 ? When can you be sure it exists?

- What is the relation between definite integrals and area? Describe

some other interpretations of definite integrals.

- What is the average value of an integrable function over a closed

interval? Must the function assume its average value? Explain.

- Describe the rules for working with definite integrals (Table 5.6).

Give examples.

- What is the Fundamental Theorem of Calculus? Why is it so

important? Illustrate each part of the theorem with an example.

- What is the Net Change Theorem? What does it say about the

integral of velocity? The integral of marginal cost?

- Discuss how the processes of integration and differentiation can

be considered as “inverses” of each other.

- How does the Fundamental Theorem provide a solution to

the initial value problem dy>dx = ƒ(x), y(x0) = y0 , when ƒ is

continuous?

- How is integration by substitution related to the Chain Rule?
- How can you sometimes evaluate indefinite integrals by substitution?

Give examples.

- How does the method of substitution work for definite integrals?

Give examples.

- How do you define and calculate the area of the region between

the graphs of two continuous functions? Give an example.

Chapter 5 Practice Exercises

Finite Sums and Estimates

- The accompanying figure shows the graph of the velocity (ft > sec)

of a model rocket for the first 8 sec after launch. The rocket accelerated

straight up for the first 2 sec and then coasted to reach its

maximum height at t = 8 sec.

0 2 4 6 8

50

100

150

200

Time after launch (sec)

Velocity (ft/sec)

- Assuming that the rocket was launched from ground level,

about how high did it go? (This is the rocket in Section 3.3,

Exercise 17, but you do not need to do Exercise 17 to do the

exercise here.)

- Sketch a graph of the rocket’s height above ground as a function

of time for 0 … t … 8.

- a. The accompanying figure shows the velocity (m > sec) of a

body moving along the s-axis during the time interval from

t = 0 to t = 10 sec. About how far did the body travel during

those 10 sec?

- Sketch a graph of s as a function of t for 0 … t … 10, assuming

s(0) = 0.

0

1

2 4 6 8 10

2

3

4

5

Time (sec)

Velocity (m/sec)

- Suppose that a

10

k=1

ak = -2 and a

10

k=1

bk = 25. Find the value of

- a

10

k=1

ak

4

- a

10

k=1

(bk – 3ak)

- a

10

k=1

(ak + bk – 1) d. a

10

k=1

a5

2 – bkb

- Suppose that a

20

k=1

ak = 0 and a

20

k=1

bk = 7. Find the values of

- a

20

k=1

3ak b. a

20

k=1

(ak + bk)

- a

20

k=1

a1

2 –

2bk

7 b d. a

20

k=1

(ak – 2)

Definite Integrals

In Exercises 5–8, express each limit as a definite integral. Then evaluate

the integral to find the value of the limit. In each case, P is a

partition of the given interval and the numbers ck are chosen from the

subintervals of P.

- lim

}P}S0

a

n

k=1

(2ck – 1)-1>2 _xk , where P is a partition of 31, 54

- lim

}P}S0

a

n

k=1

ck(ck

2 – 1)1>3 _xk , where P is a partition of 31, 34

- lim

}P}S0

a

n

k=1

acosa

ck

2 b b _xk , where P is a partition of 3-p, 04

- lim

}P}S0

a

n

k=1

(sin ck)(cos ck) _xk , where P is a partition of 30, p>24

- If 1

2

-2 3ƒ(x) dx = 12, 1

5

-2 ƒ(x) dx = 6, and 1

5

-2 g(x) dx = 2, find

the values of the following.

a.

L

2

-2

ƒ(x) dx b.

L

5

2

ƒ(x) dx c.

L

-2

5

g(x) dx

d.

L

5

-2

(-pg(x)) dx e.

L

5

-2

a

ƒ(x) + g(x)

5

b dx

- If 1

2

0 ƒ(x) dx = p, 1

2

0 7g(x) dx = 7, and 1

1

0 g(x) dx = 2, find

the values of the following.

a.

L

2

0

g(x) dx b.

L

2

1

g(x) dx c.

L

0

2

ƒ(x) dx

d.

L

2

0

22 ƒ(x) dx e.

L

2

0

(g(x) – 3ƒ(x)) dx

Area

In Exercises 11–14, find the total area of the region between the graph

of ƒ and the x-axis.

- ƒ(x) = x2 – 4x + 3, 0 … x … 3
- ƒ(x) = 1 – (x2>4), -2 … x … 3
- ƒ(x) = 5 – 5×2>3, -1 … x … 8
- ƒ(x) = 1 – 2x, 0 … x … 4

Find the areas of the regions enclosed by the curves and lines in Exercises

15–26.

- y = x, y = 1>x2, x = 2
- y = x, y = 1>2x, x = 2
- 2x + 2y = 1, x = 0, y = 0

x

y

1

0 1

“x + “y _ 1

- x3 + 2y = 1, x = 0, y = 0, for 0 … x … 1

x

y

0 1

1

x3 + “y _ 1, 0 _ x _ 1

- x = 2y2, x = 0, y = 3 20. x = 4 – y2, x = 0
- y2 = 4x, y = 4x – 2
- y2 = 4x + 4, y = 4x – 16
- y = sin x, y = x, 0 … x … p>4
- y = 0 sin x 0 , y = 1, -p>2 … x … p>2
- y = 2 sin x, y = sin 2x, 0 … x … p
- y = 8 cos x, y = sec2 x, -p>3 … x … p>3
- Find the area of the “triangular” region bounded on the left by

x + y = 2, on the right by y = x2, and above by y = 2.

- Find the area of the “triangular” region bounded on the left by

y = 2x, on the right by y = 6 – x, and below by y = 1.

- Find the extreme values of ƒ(x) = x3 – 3×2 and find the area of

the region enclosed by the graph of ƒ and the x-axis.

- Find the area of the region cut from the first quadrant by the curve

x1>2 + y1>2 = a1>2.

- Find the total area of the region enclosed by the curve x = y2>3 and

the lines x = y and y = -1.

- Find the total area of the region between the curves y = sin x and

y = cos x for 0 … x … 3p>2.

- Area Find the area between the curve y = 2(ln x)>x and the

x-axis from x = 1 to x = e.

- a. Show that the area between the curve y = 1>x and the x-axis

from x = 10 to x = 20 is the same as the area between the

curve and the x-axis from x = 1 to x = 2.

- Show that the area between the curve y = 1>x and the x-axis

from ka to kb is the same as the area between the curve and the

x-axis from x = a to x = b (0 6 a 6 b, k 7 0).

Initial Value Problems

- Show that y = x2 +

L

x

1

1t

dt solves the initial value problem

d2 y

dx2 = 2 – 1

x2 ; y_(1) = 3, y(1) = 1.

- Show that y = 1

x

0 11 + 22sec t2 dt solves the initial value problem

d2y

dx2 = 2sec x tan x; y_(0) = 3, y(0) = 0.

Express the solutions of the initial value problems in Exercises 37

and 38 in terms of integrals.

37.

dy

dx = sin x

x , y(5) = -3

38.

dy

dx = 22 – sin2 x , y(-1) = 2

Solve the initial value problems in Exercises 39–42.

39.

dy

dx = 1

21 – x2

, y(0) = 0

40.

dy

dx = 1

x2 + 1 – 1, y(0) = 1

41.

dy

dx = 1

x2x2 – 1

, x 7 1; y(2) = p

42.

dy

dx = 1

1 + x2 – 2

21 – x2

, y(0) = 2

Evaluating Indefinite Integrals

Evaluate the integrals in Exercises 43–72.

43.

L

8(sin x)-3>2 cos x dx 44.

L

(tan x)-5>2 sec2 x dx

45.

L

(6u – 1 – 4 sin (4u + 1)) du

46.

L

a 2

23u + p

– 3 csc2 (3u + p)b du

47.

L

ax – 5x

b ax + 5x

b dx 48.

L

(t – 2)2 + 2

t5 dt

49.

L

2t cos (6t3>2) dt 50.

L

(csc u cot u) 21 – csc u du

51.

L

ex csc2 (ex + 8) dx 52.

L

ex csc (ex + 3) cot (ex + 3) dx

53.

L

cos x esin x dx 54.

L

sin x ecos x dx

55.

L

2

-2

dx

2x + 5

56.

L

e

3

2ln x

3x

dx

57.

L

3

0

2t

t2 + 16

dt 58.

L

cot (ln x)

x dx

59.

L

(ln x)-5

x dx 60.

L

1x

sec2 (5 + ln x) dx

61.

L

t5t2 dt 62.

L

5cot x csc2 x dx

63.

L

4 dx

21 – 9(x + 1)2

64.

L

8 dx

29 – (x – 2)2

65.

L

dx

3 + (x + 2)2 66.

L

dx

1 + (5x – 2)2

67.

L

dr

(3r – 2)2(3r – 2)2 – 9

68.

L

dx

(x + 5)2(x + 5)2 – 49

69.

L

ecos-12x

22x – x2

dx 70.

L

2cos-1 x

21 – x2

dx

71.

L

dy

2cot-1y (1 + y2)

72.

L

(tan-1 x)5

1 + x2 dx

Evaluating Definite Integrals

Evaluate the integrals in Exercises 73–112.

73.

L

2

-2

(4×3 – 2x + 9) dx 74.

L

3

0

(6s5 – 9s2 + 7) ds

75.

L

3

1

9

x3 dx 76.

L

8

1

x-4>3 dx

77.

L

9

4

dx

x2x

78.

L

9

1

15 – 2×21>3

2x

dx

79.

L

1

0

72 dx

(4x + 3)3 80.

L

2

0

dr

23 (9 + 7r)2

81.

L

1

1>2

x-1>4(1 – x3>4)1>3 dx 82.

L

3>2

0

x4(1 + 5×5)-5>2 dx

83.

L

p

0

cos2 8r dr 84.

L

p>4

0

sin2 a2t + p

4 b dt

85.

L

p>2

0

cos2 u du 86.

L

5p>6

p>3

sin2u du

87.

L

2p

p

tan2

x

6

dx 88.

L

p

-p

cot2 u

3

du

89.

L

0

-p>3

sin x cos x dx 90.

L

5p>6

p>3

sec z tan z dz

91.

L

p>2

0

7(cos x)5>2 sin x dx 92.

L

p>3

-p>3

12 cos2 4x sin 4x dx

93.

L

p>2

0

5 cos x sin x

24 + 5 cos2 x

dx 94.

L

p>4

0

sec2 x

(8 + 19 tan x)4>3 dx

95.

L

5

1

a x

10 + 1

4xb dx 96.

L

9

1

a 3

5x + 9

x3b dx

97.

L

-2

-3

e-(x+2) dx 98.

L

0

-ln 3

e3x dx

99.

L

ln 9

0

ex(2ex + 7)-3>2 dx 100.

L

ln 4

0

eu(eu + 1)3>2 du

101.

L

e

1

1x

(1 + 3 ln x)-1>2 dx 102.

L

5

2

(ln (x + 2))3

x + 2

dx

103.

L

16

1

log8 u

u

du 104.

L

e

1

9 ln 5 log5 u

u

du

105.

L

4>3

-4>3

3 dx

216 – 9×2

dx 106.

L

1>2

-1>2

9 dx

29 – 36×2

107.

L

4

-4

3

16 + 3t2 dt 108.

L

5

25

dx

5 + x2

109.

L

1>22

1>26

dy

y28y2 – 1

110.

L

12

622

36 dy

y2y2 – 36

111.

L

2>5

22>5

dy

0 y 0 225y2 – 1

112.

L

-26>25

-2>25

dy

0 y 0 25y2 – 3

Average Values

- Find the average value of ƒ(x) = mx + b
- over 3-1, 14 b. over 3-k, k4
- Find the average value of
- y = 23x over 30, 34 b. y = 2ax over 30, a4
- Let ƒ be a function that is differentiable on 3a, b4 . In Chapter 2

we defined the average rate of change of ƒ over 3a, b4 to be

ƒ(b) – ƒ(a)

b – a

and the instantaneous rate of change of ƒ at x to be ƒ_(x). In this

chapter we defined the average value of a function. For the new definition

of average to be consistent with the old one, we should have

ƒ(b) – ƒ(a)

b – a = average value of ƒ_ on 3a, b4.

Is this the case? Give reasons for your answer.

- Is it true that the average value of an integrable function over an

interval of length 2 is half the function’s integral over the interval?

Give reasons for your answer.

- a. Verify that 1 ln x dx = x ln x – x + C.
- Find the average value of ln x over 31, e4 .
- Find the average value of ƒ(x) = 1>x on 31, 24 .
- Compute the average value of the temperature function

ƒ(x) = 37 sin a 2p

365

(x – 101)b + 25

for a 365-day year. (See Exercise 98, Section 3.6.) This is one way

to estimate the annual mean air temperature in Fairbanks, Alaska.

The National Weather Service’s official figure, a numerical average

of the daily normal mean air temperatures for the year, is

25.7°F, which is slightly higher than the average value of ƒ(x).

- Specific heat of a gas Specific heat Cy is the amount of heat

required to raise the temperature of one mole (gram molecule) of

a gas with constant volume by 1°C. The specific heat of oxygen

depends on its temperature T and satisfies the formula

Cy = 8.27 + 10-5 (26T – 1.87T2).

T

T

Find the average value of Cy for 20_ … T … 675_C and the

temperature at which it is attained.

Differentiating Integrals

In Exercises 121–128, find dy>dx.

- y =

L

x

2

22 + cos3 t dt 122. y =

L

7×2

2

22 + cos3 t dt

- y =

L

1

x

6

3 + t4 dt 124. y =

L

2

sec x

1

t2 + 1

dt

- y =

L

0

ln x2

ecos t dt 126. y =

L

e2x

1

ln (t2 + 1) dt

- y =

L

sin-1 x

0

dt

21 – 2t2

- y =

L

p>4

tan-1x

e2t dt

Theory and Examples

- a. If

L

1

0

7ƒ(x) dx = 7, does

L

1

0

ƒ(x) dx = 1?

- If

L

1

0

ƒ(x) dx = 4 and ƒ(x) Ú 0, does

L

1

0

2ƒ(x) dx = 24 = 2?

Give reasons for your answers.

- Suppose

L

2

-2

ƒ(x) dx = 4,

L

5

2

ƒ(x) dx = 3,

L

5

-2

g(x) dx = 2.

Which, if any, of the following statements are true?

a.

L

2

5

ƒ(x) dx = -3 b.

L

5

-2

(ƒ(x) + g(x)) = 9

- ƒ(x) … g(x) on the interval -2 … x … 5
- Initial value problem Show that

y = 1

aL

x

0

ƒ(t) sin a(x – t) dt

solves the initial value problem

d2y

dx2 + a2y = ƒ(x),

dy

dx = 0 and y = 0 when x = 0.

(Hint: sin (ax – at) = sin ax cos at – cos ax sin at.)

- Proportionality Suppose that x and y are related by the equation

x =

L

y

0

1

21 + 4t2

dt.

Show that d2y/dx2 is proportional to y and find the constant of

proportionality.

- Find ƒ(4) if

a.

L

x2

0

ƒ(t) dt = x cos px b.

L

ƒ(x)

0

t2 dt = x cos px.

- Find ƒ(p/2) from the following information.
- i) ƒ is positive and continuous.
- ii) The area under the curve y = ƒ(x) from x = 0 to x = a is

a2

2 + a

2

sin a + p

2

cos a.

- The area of the region in the xy-plane enclosed by the x-axis, the

curve y = ƒ(x), ƒ(x) Ú 0, and the lines x = 1 and x = b is

equal to 2b2 + 1 – 22 for all b 7 1. Find ƒ(x).

- Prove that

L

x

0

a

L

u

0

ƒ(t) dtb du =

L

x

0

ƒ(u)(x – u) du.

(Hint: Express the integral on the right-hand side as the difference

of two integrals. Then show that both sides of the equation

have the same derivative with respect to x.)

- Finding a curve Find the equation for the curve in the xy-plane

that passes through the point (1, -1) if its slope at x is always

3×2 + 2.

- Shoveling dirt You sling a shovelful of dirt up from the bottom

of a hole with an initial velocity of 32 ft > sec. The dirt must rise

17 ft above the release point to clear the edge of the hole. Is that

enough speed to get the dirt out, or had you better duck?

Piecewise Continuous Functions

Although we are mainly interested in continuous functions, many

functions in applications are piecewise continuous. A function ƒ(x) is

piecewise continuous on a closed interval I if ƒ has only finitely

many discontinuities in I, the limits

lim

xSc- ƒ(x) and lim

xSc +

ƒ(x)

exist and are finite at every interior point of I, and the appropriate onesided

limits exist and are finite at the endpoints of I. All piecewise

continuous functions are integrable. The points of discontinuity subdivide

I into open and half-open subintervals on which ƒ is continuous,

and the limit criteria above guarantee that ƒ has a continuous extension

to the closure of each subinterval. To integrate a piecewise continuous

function, we integrate the individual extensions and add the

results. The integral of

ƒ(x) = c

1 – x, -1 … x 6 0

x2, 0 … x 6 2

-1, 2 … x … 3

(Figure 5.32) over 3-1, 34 is

L

3

-1

ƒ(x) dx =

L

0

-1

(1 – x) dx +

L

2

0

x2 dx +

L

3

2

(-1) dx

= c x – x2

2 d

-1

0

+ c x3

3 d

0

2

+ c -x d

2

3

= 3

2 + 8

3 – 1 = 19

6

.

Figure 5.32 Piecewise continuous

functions like this are integrated piece by

piece.

x

y

2

_1 0 1 2 3

1

3

4

_1

y _ x2

y _ 1 _ x

y _ _1

The Fundamental Theorem applies to piecewise continuous functions

with the restriction that (d>dx)1

x

a ƒ(t) dt is expected to equal ƒ(x)

only at values of x at which ƒ is continuous. There is a similar restriction

on Leibniz’s Rule (see Exercises 31–38).

Graph the functions in Exercises 11–16 and integrate them over

their domains.

- ƒ(x) = e

x2>3, -8 … x 6 0

-4, 0 … x … 3

- ƒ(x) = e

2-x, -4 … x 6 0

x2 – 4, 0 … x … 3

- g(t) = e

t, 0 … t 6 1

sin pt, 1 … t … 2

- h(z) = e

21 – z, 0 … z 6 1

(7z – 6)-1>3, 1 … z … 2

- ƒ(x) = c

1, -2 … x 6 -1

1 – x2, -1 … x 6 1

2, 1 … x … 2

- h(r) = c

r, -1 … r 6 0

1 – r2, 0 … r 6 1

1, 1 … r … 2

- Find the average value of the function graphed in the accompanying

figure.

x

y

0 1 2

1

- Find the average value of the function graphed in the accompanying

figure.

x

y

1

0 1 2 3

Limits

Find the limits in Exercises 19–22.

- lim

bS1-L

b

0

dx

21 – x2

- lim

xSq

1x

L

x

0

tan-1 t dt

- lim

nSq

a 1

n + 1 + 1

n + 2 + g + 1

2nb

- lim

nSq

1n 1e1>n

+

e2>n

+

g

+

e(n-1)>n

+

en>n2

Defining Functions Using the Fundamental Theorem

- A function defined by an integral The graph of a function ƒ

consists of a semicircle and two line segments as shown. Let

g(x) = 1

x

1 ƒ(t) dt.

y

_3 1 3

y _ f(x)

_1

_1

1

3

x

- Find g(1). b. Find g(3). c. Find g(-1).
- Find all values of x on the open interval (-3, 4) at which g has

a relative maximum.

- Write an equation for the line tangent to the graph of g at x = -1.
- Find the x-coordinate of each point of inflection of the graph

of g on the open interval (-3, 4).

- Find the range of g.
- A differential equation Show that both of the following conditions

are satisfied by y = sin x + 1

p

x cos 2t dt + 1:

- i) y_ = -sin x + 2 sin 2x
- ii) y = 1 and y_ = -2 when x = p.

Leibniz’s Rule In applications, we sometimes encounter functions

defined by integrals that have variable upper limits of integration and

variable lower limits of integration at the same time. We can find the

derivative of such an integral by a formula called Leibniz’s Rule.

- Find ƒ_(2) if ƒ(x) = eg(x) and g(x) =

L

x

2

t

1 + t4 dt.

- Use the accompanying figure to show that

L

p>2

0

sin x dx = p

2 –

L

1

0

sin-1 x dx.

0 1

1

p2

p2

y _ sin x

y _ sin_1 x

x

y

- Napier’s inequality Here are two pictorial proofs that

b 7 a 7 0 1

1

b 6 ln b – ln a

b – a 6 1a

.

Explain what is going on in each case.

a.

x

y

0 a b

L1

L2

L3

y _ ln x

b.

x

y

0 a b

y _ 1x

(Source: Roger B. Nelson, College Mathematics Journal, Vol. 24,

No. 2, March 1993, p. 165.)

- Bound on an integral Let ƒ be a continuously differentiable

function on 3a, b4 satisfying 1

b

a ƒ(x) dx = 0.

- If c = (a + b)>2, show that

L

b

a

xƒ(x) dx =

L

c

a

(x – c)ƒ(x) dx +

L

b

c

(x – c)ƒ(x) dx.

- Let t = 0 x – c 0 and / = (b – a)>2. Show that

L

b

a

xƒ(x) dx =

L

/

0

t(ƒ(c + t) – ƒ(c – t)) dt.

- Apply the Mean Value Theorem from Section 4.2 to part (b)

to prove that

2

L

b

a

xƒ(x) dx 2 …

(b – a)3

12

M,

where M is the absolute maximum of ƒ_ on 3a, b4 .

Leibniz’s Rule

If ƒ is continuous on 3a, b4 and if u(x) and y(x) are differentiable

functions of x whose values lie in 3a, b4 , then

d

dx

L

y(x)

u(x)

ƒ(t) dt = ƒ(y(x)) dy

dx – ƒ(u(x)) du

dx

.

To prove the rule, let F be an antiderivative of ƒ on 3a, b4 . Then

L

y(x)

u(x)

ƒ(t) dt = F(y(x)) – F(u(x)).

Differentiating both sides of this equation with respect to x gives the

equation we want:

d

dxL

y(x)

u(x)

ƒ(t) dt = d

dx

3F(y(x)) – F(u(x)) 4

= F_(y(x))

dy

dx – F_(u(x))

du

dx

Chain Rule

= ƒ(y(x))

dy

dx – ƒ(u(x))

du

dx

.

Use Leibniz’s Rule to find the derivatives of the functions in

Exercises 31–38.

- ƒ(x) =

L

x

1>x

1t

dt 26. ƒ(x) =

L

sin x

cos x

1

1 – t2 dt

- g(y) =

L

22y

2y

sin t2 dt 28. g(y) =

L

y2

2y

e

t

t dt

- y =

L

x2

x2>2

ln 2t dt 30. y =

L

23

x

2x

ln t dt

- y =

L

ln x

0

sin e

t dt 32. y =

L

e2x

e42x

ln t dt

Theory and Examples

- Use Leibniz’s Rule to find the value of x that maximizes the value

of the integral

L

x+3

x

t(5 – t) dt.

- For what x 7 0 does x(xx) = (xx)x? Give reasons for your answer.
- Find the areas between the curves y = 2(log2 x)>x and y =

2(log4 x)>x and the x-axis from x = 1 to x = e. What is the ratio

of the larger area to the smaller?

- a. Find df > dx if

ƒ(x) =

L

ex

1

2 ln t

t dt.

- Find ƒ(0).
- What can you conclude about the graph of ƒ? Give reasons for

your answer.

Chapter 6 Applications of Definite Integrals

Chapter 6 Questions to Guide Your Review

- How do you define and calculate the volumes of solids by the

method of slicing? Give an example.

- How are the disk and washer methods for calculating volumes

derived from the method of slicing? Give examples of volume

calculations by these methods.

- Describe the method of cylindrical shells. Give an example.
- How do you find the length of the graph of a smooth function

over a closed interval? Give an example. What about functions

that do not have continuous first derivatives?

- How do you define and calculate the area of the surface swept out

by revolving the graph of a smooth function y = ƒ(x), a … x … b,

about the x-axis? Give an example.

- How do you define and calculate the work done by a variable

force directed along a portion of the x-axis? How do you calculate

the work it takes to pump a liquid from a tank? Give examples.

- What is a center of mass? a centroid?
- How do you locate the center of mass of a thin flat plate of material?

Give an example.

- How do you locate the center of mass of a thin plate bounded by

two curves y = ƒ(x) and y = g(x) over a … x … b?

Chapter 6 Practice Exercises

Volumes

Find the volumes of the solids in Exercises 1–16.

- The solid lies between planes perpendicular to the x-axis at x = 0

and x = 1. The cross-sections perpendicular to the x-axis

between these planes are circular disks whose diameters run from

the parabola y = x2 to the parabola y = 2x.

- The base of the solid is the region in the first quadrant between

the line y = x and the parabola y = 22x. The cross-sections of

the solid perpendicular to the x-axis are equilateral triangles

whose bases stretch from the line to the curve.

- The solid lies between planes perpendicular to the x-axis at

x = p>4 and x = 5p>4. The cross-sections between these

planes are circular disks whose diameters run from the curve

y = 2 cos x to the curve y = 2 sin x.

- The solid lies between planes perpendicular to the x-axis at

x = 0 and x = 6. The cross-sections between these planes

are squares whose bases run from the x-axis up to the curve

x1>2 + y1>2 = 26.

x

y

6

6

x1_2 + y1_2 _ “6

- The solid lies between planes perpendicular to the x-axis at x = 0

and x = 4. The cross-sections of the solid perpendicular to the

x-axis between these planes are circular disks whose diameters

run from the curve x2 = 4y to the curve y2 = 4x.

- The base of the solid is the region bounded by the parabola

y2 = 4x and the line x = 1 in the xy-plane. Each cross-section

perpendicular to the x-axis is an equilateral triangle with one edge

in the plane. (The triangles all lie on the same side of the plane.)

- Find the volume of the solid generated by revolving the region

bounded by the x-axis, the curve y = 3×4, and the lines x = 1

and x = -1 about (a) the x-axis; (b) the y-axis; (c) the line

x = 1; (d) the line y = 3.

- Find the volume of the solid generated by revolving the “triangular”

region bounded by the curve y = 4>x3 and the lines x = 1

and y = 1>2 about (a) the x-axis; (b) the y-axis; (c) the line

x = 2; (d) the line y = 4.

- Find the volume of the solid generated by revolving the region

bounded on the left by the parabola x = y2 + 1 and on the right by

the line x = 5 about (a) the x-axis; (b) the y-axis; (c) the line x = 5.

- Find the volume of the solid generated by revolving the region

bounded by the parabola y2 = 4x and the line y = x about (a)

the x-axis; (b) the y-axis; (c) the line x = 4; (d) the line y = 4.

- Find the volume of the solid generated by revolving the “triangular”

region bounded by the x-axis, the line x = p>3, and the

curve y = tan x in the first quadrant about the x-axis.

- Find the volume of the solid generated by revolving the region

bounded by the curve y = sin x and the lines x = 0, x = p, and

y = 2 about the line y = 2.

- Find the volume of the solid generated by revolving the region

bounded by the curve x = ey2 and the lines y = 0, x = 0, and

y = 1 about the x-axis.

- Find the volume of the solid generated by revolving about the

x-axis the region bounded by y = 2 tan x, y = 0, x = -p>4, and

x = p>4. (The region lies in the first and third quadrants and

resembles a skewed bowtie.)

- Volume of a solid sphere hole A round hole of radius 23 ft is

bored through the center of a solid sphere of a radius 2 ft. Find the

volume of material removed from the sphere.

- Volume of a football The profile of a football resembles the

ellipse shown here. Find the football’s volume to the nearest

cubic inch.

x

y

_ 0

+ _ 1

4×2

121

y2

12

2

11

2

11

Lengths of Curves

Find the lengths of the curves in Exercises 17–20.

- y = x1>2 – (1>3)x3>2, 1 … x … 4
- x = y2>3, 1 … y … 8
- y = x2 – (ln x)>8, 1 … x … 2
- x = (y3>12) + (1>y), 1 … y … 2

Areas of Surfaces of Revolution

In Exercises 21–24, find the areas of the surfaces generated by revolving

the curves about the given axes.

- y = 22x + 1, 0 … x … 3; x@axis
- y = x3>3, 0 … x … 1; x@axis
- x = 24y – y2, 1 … y … 2; y@axis
- x = 2y, 2 … y … 6; y@axis

Work

- Lifting equipment A rock climber is about to haul up 100 N

(about 22.5 lb) of equipment that has been hanging beneath her on 40

m of rope that weighs 0.8 newton per meter. How much work will it

take? (Hint: Solve for the rope and equipment separately, then add.)

- Leaky tank truck You drove an 800-gal tank truck of water

from the base of Mt. Washington to the summit and discovered

on arrival that the tank was only half full. You started with a full

tank, climbed at a steady rate, and accomplished the 4750-ft

elevation change in 50 min. Assuming that the water leaked out at

a steady rate, how much work was spent in carrying water to the

top? Do not count the work done in getting yourself and the truck

there. Water weighs 8 lb > U.S. gal.

- Earth’s attraction The force of attraction on an object below

Earth’s surface is directly proportional to its distance from Earth’s

center. Find the work done in moving a weight of w lb located a

mi below Earth’s surface up to the surface itself. Assume Earth’s

radius is a constant r mi.

- Garage door spring A force of 200 N will stretch a garage

door spring 0.8 m beyond its unstressed length. How far will a

300-N force stretch the spring? How much work does it take to

stretch the spring this far from its unstressed length?

- Pumping a reservoir A reservoir shaped like a right-circular cone,

point down, 20 ft across the top and 8 ft deep, is full of water. How

much work does it take to pump the water to a level 6 ft above the top?

- Pumping a reservoir (Continuation of Exercise 29.) The reservoir

is filled to a depth of 5 ft, and the water is to be pumped to

the same level as the top. How much work does it take?

- Pumping a conical tank A right-circular conical tank, point

down, with top radius 5 ft and height 10 ft is filled with a liquid

whose weight-density is 60 lb>ft3. How much work does it take

to pump the liquid to a point 2 ft above the tank? If the pump is

driven by a motor rated at 275 ft-lb > sec (1 > 2 hp), how long will

it take to empty the tank?

- Pumping a cylindrical tank A storage tank is a right-circular

cylinder 20 ft long and 8 ft in diameter with its axis horizontal. If

the tank is half full of olive oil weighing 57 lb>ft3, find the work

done in emptying it through a pipe that runs from the bottom of

the tank to an outlet that is 6 ft above the top of the tank.

Centers of Mass and Centroids

- Find the centroid of a thin, flat plate covering the region enclosed

by the parabolas y = 2×2 and y = 3 – x2.

- Find the centroid of a thin, flat plate covering the region enclosed by

the x-axis, the lines x = 2 and x = -2, and the parabola y = x2.

- Find the centroid of a thin, flat plate covering the “triangular”

region in the first quadrant bounded by the y-axis, the parabola

y = x2>4, and the line y = 4.

- Find the centroid of a thin, flat plate covering the region enclosed

by the parabola y2 = x and the line x = 2y.

- Find the center of mass of a thin, flat plate covering the region

enclosed by the parabola y2 = x and the line x = 2y if the density

function is d(y) = 1 + y. (Use horizontal strips.)

- a. Find the center of mass of a thin plate of constant density covering

the region between the curve y = 3>x3>2 and the x-axis

from x = 1 to x = 9.

- Find the plate’s center of mass if, instead of being constant,

the density is d(x) = x. (Use vertical strips.)

Chapter 6 Additional and Advanced Exercises

Volume and Length

- A solid is generated by revolving about the x-axis the region

bounded by the graph of the positive continuous function

y = ƒ(x), the x-axis, the fixed line x = a, and the variable line

x = b, b 7 a. Its volume, for all b, is b2 – ab. Find ƒ(x).

- A solid is generated by revolving about the x-axis the region

bounded by the graph of the positive continuous function

y = ƒ(x), the x-axis, and the lines x = 0 and x = a. Its volume,

for all a 7 0, is a2 + a. Find ƒ(x).

- Suppose that the increasing function ƒ(x) is smooth for x Ú 0

and that ƒ(0) = a. Let s(x) denote the length of the graph of ƒ

from (0, a) to (x, ƒ(x)), x 7 0. Find ƒ(x) if s(x) = Cx for some

constant C. What are the allowable values for C?

- a. Show that for 0 6 a … p>2,

L

a

0

21 + cos2 u du 7 2a2 + sin2 a.

- Generalize the result in part (a).
- Find the volume of the solid formed by revolving the region

bounded by the graphs of y = x and y = x2 about the line y = x.

- Consider a right-circular cylinder of diameter 1. Form a wedge by

making one slice parallel to the base of the cylinder completely

through the cylinder, and another slice at an angle of 45_ to the first

slice and intersecting the first slice at the opposite edge of the cylinder

(see accompanying diagram). Find the volume of the wedge.

45_ wedge

r _ 1

2

Surface Area

- At points on the curve y = 22x, line segments of length h = y

are drawn perpendicular to the xy-plane. (See accompanying figure.)

Find the area of the surface formed by these perpendiculars

from (0, 0) to 13, 2232.

x

0

3

x

y _ 2″x

2″x

2″3

(3, 2″3)

y

- At points on a circle of radius a, line segments are drawn perpendicular

to the plane of the circle, the perpendicular at each point P

being of length ks, where s is the length of the arc of the circle

measured counterclockwise from (a, 0) to P and k is a positive

constant, as shown here. Find the area of the surface formed by

the perpendiculars along the arc beginning at (a, 0) and extending

once around the circle.

0

a

a

x

y

Work

- A particle of mass m starts from rest at time t = 0 and is moved

along the x-axis with constant acceleration a from x = 0 to

x = h against a variable force of magnitude F(t) = t2. Find the

work done.

- Work and kinetic energy Suppose a 1.6-oz golf ball is placed

on a vertical spring with force constant k = 2 lb>in. The spring is

compressed 6 in. and released. About how high does the ball go

(measured from the spring’s rest position)?

Centers of Mass

- Find the centroid of the region bounded below by the x-axis and

above by the curve y = 1 – xn, n an even positive integer. What

is the limiting position of the centroid as nS q?

- If you haul a telephone pole on a two-wheeled carriage behind a

truck, you want the wheels to be 3 ft or so behind the pole’s center

of mass to provide an adequate “tongue” weight. The 40-ft

wooden telephone poles used by Verizon have a 27-in. circumference

at the top and a 43.5-in. circumference at the base. About

how far from the top is the center of mass?

- Suppose that a thin metal plate of area A and constant density d

occupies a region R in the xy-plane, and let My be the plate’s

moment about the y-axis. Show that the plate’s moment about the

line x = b is

- My – bdA if the plate lies to the right of the line, and
- bdA – My if the plate lies to the left of the line.
- Find the center of mass of a thin plate covering the region bounded

by the curve y2 = 4ax and the line x = a, a = positive constant,

if the density at (x, y) is directly proportional to (a) x, (b) 0 y 0 .

- a. Find the centroid of the region in the first quadrant bounded

by two concentric circles and the coordinate axes, if the circles

have radii a and b, 0 6 a 6 b, and their centers are at

the origin.

- Find the limits of the coordinates of the centroid as a approaches

b and discuss the meaning of the result.

- A triangular corner is cut from a square 1 ft on a side. The area of

the triangle removed is 36 in2. If the centroid of the remaining

region is 7 in. from one side of the original square, how far is it

from the remaining sides?

Chapter 7 Integrals and Transcendental Functions

Chapter 7 Questions to Guide Your Review

- How is the natural logarithm function defined as an integral?

What are its domain, range, and derivative? What arithmetic

properties does it have? Comment on its graph.

- What integrals lead to logarithms? Give examples.
- What are the integrals of tan x and cot x? sec x and csc x?
- How is the exponential function ex defined? What are its domain,

range, and derivative? What laws of exponents does it obey?

Comment on its graph.

- How are the functions ax and loga x defined? Are there any

restrictions on a? How is the graph of loga x related to the

graph of ln x? What truth is there in the statement that there is

really only one exponential function and one logarithmic

function?

- How do you solve separable first-order differential equations?
- What is the law of exponential change? How can it be derived

from an initial value problem? What are some of the applications

of the law?

- What are the six basic hyperbolic functions? Comment on their

domains, ranges, and graphs. What are some of the identities

relating them?

- What are the derivatives of the six basic hyperbolic functions?

What are the corresponding integral formulas? What similarities

do you see here with the six basic trigonometric functions?

- How are the inverse hyperbolic functions defined? Comment on

their domains, ranges, and graphs. How can you find values of

sech-1 x, csch-1 x, and coth-1 x using a calculator’s keys for

cosh-1 x, sinh-1 x, and tanh-1 x?

- What integrals lead naturally to inverse hyperbolic functions?

Chapter 7 Practice Exercises

Integration

Evaluate the integrals in Exercises 1–12.

1.

L

ex cos (ex) dx 2.

L

ex sin (5ex – 7) dx

3.

L

p

0

tan

x

6

dx 4.

L

1>6

1>3

3 tan px dx

5.

L

p>2

-p>3

sin t

1 – cos t

dt 6.

L

ex tan ex dx

7.

L

ln (x + 9)

x + 9

dx 8.

L

sin (2 + ln x)

x dx

9.

L

9

1

5x dx 10.

L

81

1

1

4x

dx

11.

L

e3

e2

5

x2ln x

dx 12.

L

9

3

(1 + ln t)t ln t dt

Solving Equations with Logarithmic or Exponential Terms

In Exercises 13–18, solve for y.

- 3y = 2y+1 14. 4-y = 3y+2
- 9e2y = x2 16. 3y = 3 ln x
- ln ( y – 1) = x + ln y 18. ln (10 ln y) = ln 5x

Theory and Applications

- The function ƒ(x) = ex + x, being differentiable and one-to-one,

has a differentiable inverse ƒ

-1(x). Find the value of dƒ-1>dx at

the point ƒ(ln 2).

- Find the inverse of the function ƒ(x) = 1 + (1>x), x _ 0. Then

show that ƒ

-1( ƒ(x)) = ƒ( ƒ

-1(x)) = x and that

dƒ

-1

dx

`

ƒ(x)

= 1

ƒ_(x)

.

- A particle is traveling upward and to the right along the curve

y = ln x. Its x-coordinate is increasing at the rate (dx>dt) =

2x m>sec. At what rate is the y-coordinate changing at the point

(e2, 2)?

- A girl is sliding down a slide shaped like the curve y = 9e-x>3.

Her y-coordinate is changing at the rate dy>dt = (-1>4)29 – y

ft>sec. At approximately what rate is her x-coordinate changing

when she reaches the bottom of the slide at x = 9 ft? (Take e3 to

be 20 and round your answer to the nearest ft > sec.)

- The functions ƒ(x) = ln 5x and g(x) = ln 3x differ by a constant.

What constant? Give reasons for your answer.

- a. If (ln x)>x = (ln 2)>2, must x = 2?
- If (ln x)>x = -2 ln 2, must x = 1>2?

Give reasons for your answers.

- The quotient (log4 x)>(log2 x) has a constant value. What value?

Give reasons for your answer.

- logx (2) vs. log2 (x) How does ƒ(x) = logx (2) compare with

g(x) = log2 (x)? Here is one way to find out.

- Use the equation loga b = (ln b)>(ln a) to express ƒ(x) and

g(x) in terms of natural logarithms.

T

- Graph ƒ and g together. Comment on the behavior of ƒ in

relation to the signs and values of g.

In Exercises 27–30, solve the differential equation.

27.

dy

dx = 2y cos2 2y 28. y_ =

3y(x + 1)2

y – 1

- yy_ = sec y2 sec2 x 30. y cos2 x dy + sin x dx = 0

In Exercises 31–34, solve the initial value problem.

31.

dy

dx = e-x-y-2, y(0) = -2

32.

dy

dx =

y ln y

1 + x2 , y(0) = e2

- x dy – 1y + 2y2 dx = 0, y(1) = 1
- y-2

dx

dy = ex

e2x + 1

, y(0) = 1

- What is the age of a sample of charcoal in which 90% of the

carbon-

14 originally present has decayed?

- Cooling a pie A deep-dish apple pie, whose internal temperature

was 220°F when removed from the oven, was set out on a

breezy 40°F porch to cool. Fifteen minutes later, the pie’s internal

temperature was 180°F. How long did it take the pie to cool from

there to 70°F?

- Let A(t) be the area of the region in the first quadrant enclosed by

the coordinate axes, the curve y = e-x, and the vertical line

x = t, t 7 0. Let V(t) be the volume of the solid generated by

revolving the region about the x-axis. Find the following limits.

- lim

tSq

A(t) b. lim

tSq

V(t)>A(t) c. lim

tS0 +

V(t)>A(t)

- Varying a logarithm’s base
- Find lim loga 2 as aS 0+, 1-, 1+, and q.
- Graph y = loga 2 as a function of a over the interval

0 6 a … 4.

- Graph ƒ(x) = tan-1 x + tan-1(1>x) for -5 … x … 5. Then use

calculus to explain what you see. How would you expect ƒ to behave

beyond the interval 3-5, 54? Give reasons for your answer.

- Graph ƒ(x) = (sin x)sin x over 30, 3p4. Explain what you see.
- Even-odd decompositions
- Suppose that g is an even function of x and h is an odd function

of x. Show that if g(x) + h(x) = 0 for all x then

g(x) = 0 for all x and h(x) = 0 for all x.

- Use the result in part (a) to show that if ƒ(x) =

ƒE (x) + ƒO(x) is the sum of an even function ƒE (x) and an

odd function ƒO(x), then

ƒE (x) = (ƒ(x) + ƒ(-x))>2 and ƒO(x) = (ƒ(x) – ƒ(-x))>2.

- What is the significance of the result in part (b)?

T

T

T

- Let g be a function that is differentiable throughout an open interval

containing the origin. Suppose g has the following properties:

- g(x + y) =

g(x) + g(y)

1 – g(x)g(y)

for all real numbers x, y, and

x + y in the domain of g.

- lim

hS0

g(h) = 0

iii. lim

hS0

g(h)

h = 1

- Show that g(0) = 0.
- Show that g_(x) = 1 + 3g(x) 42.
- Find g(x) by solving the differential equation in part (b).
- Center of mass Find the center of mass of a thin plate of constant

density covering the region in the first and fourth quadrants

enclosed by the curves y = 1>(1 + x2) and y = -1>(1 + x2)

and by the lines x = 0 and x = 1.

- Solid of revolution The region between the curve y = 1>121×2

and the x-axis from x = 1>4 to x = 4 is revolved about the x-axis

to generate a solid.

- Find the volume of the solid.
- Find the centroid of the region.

twice the area of the sector AOP pictured in the accompanying

figure. To see why this is so, carry out the following steps.

- Show that the area A(u) of sector AOP is

A(u) = 1

2

cosh u sinh u –

L

cosh u

1

2×2 – 1 dx.

- Differentiate both sides of the equation in part (a) with respect

to u to show that

A_(u) = 1

2

.

- Solve this last equation for A(u). What is the value of A(0)?

What is the value of the constant of integration C in your solution?

With C determined, what does your solution say about

the relationship of u to A(u)?

One of the analogies between hyperbolic and circular functions

is revealed by these two diagrams (Exercise 86).

x

y

O

Asymptote

Asymptote

A

x

y

O A

x2 _ y2 _ 1

x2 + y2 _ 1 P(cos u, sin u)

u is twice the area

of sector AOP.

u _ 0

u _ 0

u is twice the area

of sector AOP.

P(cosh u, sinh u)

Chapter 7 Questions to Guide Your Review

- How is the natural logarithm function defined as an integral?

What are its domain, range, and derivative? What arithmetic

properties does it have? Comment on its graph.

- What integrals lead to logarithms? Give examples.
- What are the integrals of tan x and cot x? sec x and csc x?
- How is the exponential function ex defined? What are its domain,

range, and derivative? What laws of exponents does it obey?

Comment on its graph.

- How are the functions ax and loga x defined? Are there any

restrictions on a? How is the graph of loga x related to the

graph of ln x? What truth is there in the statement that there is

really only one exponential function and one logarithmic

function?

- How do you solve separable first-order differential equations?
- What is the law of exponential change? How can it be derived

from an initial value problem? What are some of the applications

of the law?

- What are the six basic hyperbolic functions? Comment on their

domains, ranges, and graphs. What are some of the identities

relating them?

- What are the derivatives of the six basic hyperbolic functions?

What are the corresponding integral formulas? What similarities

do you see here with the six basic trigonometric functions?

- How are the inverse hyperbolic functions defined? Comment on

their domains, ranges, and graphs. How can you find values of

sech-1 x, csch-1 x, and coth-1 x using a calculator’s keys for

cosh-1 x, sinh-1 x, and tanh-1 x?

- What integrals lead naturally to inverse hyperbolic functions?

Chapter 7 Practice Exercises

Integration

Evaluate the integrals in Exercises 1–12.

1.

L

ex cos (ex) dx 2.

L

ex sin (5ex – 7) dx

3.

L

p

0

tan

x

6

dx 4.

L

1>6

1>3

3 tan px dx

5.

L

p>2

-p>3

sin t

1 – cos t

dt 6.

L

ex tan ex dx

7.

L

ln (x + 9)

x + 9

dx 8.

L

sin (2 + ln x)

x dx

9.

L

9

1

5x dx 10.

L

81

1

1

4x

dx

11.

L

e3

e2

5

x2ln x

dx 12.

L

9

3

(1 + ln t)t ln t dt

Solving Equations with Logarithmic or Exponential Terms

In Exercises 13–18, solve for y.

- 3y = 2y+1 14. 4-y = 3y+2
- 9e2y = x2 16. 3y = 3 ln x
- ln ( y – 1) = x + ln y 18. ln (10 ln y) = ln 5x

Theory and Applications

- The function ƒ(x) = ex + x, being differentiable and one-to-one,

has a differentiable inverse ƒ

-1(x). Find the value of dƒ-1>dx at

the point ƒ(ln 2).

- Find the inverse of the function ƒ(x) = 1 + (1>x), x _ 0. Then

show that ƒ

-1( ƒ(x)) = ƒ( ƒ

-1(x)) = x and that

dƒ

-1

dx

`

ƒ(x)

= 1

ƒ_(x)

.

- A particle is traveling upward and to the right along the curve

y = ln x. Its x-coordinate is increasing at the rate (dx>dt) =

2x m>sec. At what rate is the y-coordinate changing at the point

(e2, 2)?

- A girl is sliding down a slide shaped like the curve y = 9e-x>3.

Her y-coordinate is changing at the rate dy>dt = (-1>4)29 – y

ft>sec. At approximately what rate is her x-coordinate changing

when she reaches the bottom of the slide at x = 9 ft? (Take e3 to

be 20 and round your answer to the nearest ft > sec.)

- The functions ƒ(x) = ln 5x and g(x) = ln 3x differ by a constant.

What constant? Give reasons for your answer.

- a. If (ln x)>x = (ln 2)>2, must x = 2?
- If (ln x)>x = -2 ln 2, must x = 1>2?

Give reasons for your answers.

- The quotient (log4 x)>(log2 x) has a constant value. What value?

Give reasons for your answer.

- logx (2) vs. log2 (x) How does ƒ(x) = logx (2) compare with

g(x) = log2 (x)? Here is one way to find out.

- Use the equation loga b = (ln b)>(ln a) to express ƒ(x) and

g(x) in terms of natural logarithms.

T

- Graph ƒ and g together. Comment on the behavior of ƒ in

relation to the signs and values of g.

In Exercises 27–30, solve the differential equation.

27.

dy

dx = 2y cos2 2y 28. y_ =

3y(x + 1)2

y – 1

- yy_ = sec y2 sec2 x 30. y cos2 x dy + sin x dx = 0

In Exercises 31–34, solve the initial value problem.

31.

dy

dx = e-x-y-2, y(0) = -2

32.

dy

dx =

y ln y

1 + x2 , y(0) = e2

- x dy – 1y + 2y2 dx = 0, y(1) = 1
- y-2

dx

dy = ex

e2x + 1

, y(0) = 1

- What is the age of a sample of charcoal in which 90% of the

carbon-

14 originally present has decayed?

- Cooling a pie A deep-dish apple pie, whose internal temperature

was 220°F when removed from the oven, was set out on a

breezy 40°F porch to cool. Fifteen minutes later, the pie’s internal

temperature was 180°F. How long did it take the pie to cool from

there to 70°F?

- Let A(t) be the area of the region in the first quadrant enclosed by

the coordinate axes, the curve y = e-x, and the vertical line

x = t, t 7 0. Let V(t) be the volume of the solid generated by

revolving the region about the x-axis. Find the following limits.

- lim

tSq

A(t) b. lim

tSq

V(t)>A(t) c. lim

tS0 +

V(t)>A(t)

- Varying a logarithm’s base
- Find lim loga 2 as aS 0+, 1-, 1+, and q.
- Graph y = loga 2 as a function of a over the interval

0 6 a … 4.

- Graph ƒ(x) = tan-1 x + tan-1(1>x) for -5 … x … 5. Then use

calculus to explain what you see. How would you expect ƒ to behave

beyond the interval 3-5, 54? Give reasons for your answer.

- Graph ƒ(x) = (sin x)sin x over 30, 3p4. Explain what you see.
- Even-odd decompositions
- Suppose that g is an even function of x and h is an odd function

of x. Show that if g(x) + h(x) = 0 for all x then

g(x) = 0 for all x and h(x) = 0 for all x.

- Use the result in part (a) to show that if ƒ(x) =

ƒE (x) + ƒO(x) is the sum of an even function ƒE (x) and an

odd function ƒO(x), then

ƒE (x) = (ƒ(x) + ƒ(-x))>2 and ƒO(x) = (ƒ(x) – ƒ(-x))>2.

- What is the significance of the result in part (b)?

T

T

T

- Let g be a function that is differentiable throughout an open interval

containing the origin. Suppose g has the following properties:

- g(x + y) =

g(x) + g(y)

1 – g(x)g(y)

for all real numbers x, y, and

x + y in the domain of g.

- lim

hS0

g(h) = 0

iii. lim

hS0

g(h)

h = 1

- Show that g(0) = 0.
- Show that g_(x) = 1 + 3g(x) 42.
- Find g(x) by solving the differential equation in part (b).
- Center of mass Find the center of mass of a thin plate of constant

density covering the region in the first and fourth quadrants

enclosed by the curves y = 1>(1 + x2) and y = -1>(1 + x2)

and by the lines x = 0 and x = 1.

- Solid of revolution The region between the curve y = 1>121×2

and the x-axis from x = 1>4 to x = 4 is revolved about the x-axis

to generate a solid.

- Find the volume of the solid.
- Find the centroid of the region.

Chapter 8 Techniques of Integration

Chapter 8 Questions to Guide Your Review

- What is the formula for integration by parts? Where does it come

from? Why might you want to use it?

- When applying the formula for integration by parts, how do you

choose the u and dy? How can you apply integration by parts to

an integral of the form 1ƒ(x) dx?

- If an integrand is a product of the form sinn x cosm x, where m and

n are nonnegative integers, how do you evaluate the integral?

Give a specific example of each case.

- What substitutions are made to evaluate integrals of sin mx sin nx,

sin mx cos nx, and cos mx cos nx? Give an example of each case.

- What substitutions are sometimes used to transform integrals

involving 2a2 – x2, 2a2 + x2, and 2×2 – a2 into integrals

that can be evaluated directly? Give an example of each case.

- What restrictions can you place on the variables involved in the

three basic trigonometric substitutions to make sure the substitutions

are reversible (have inverses)?

- What is the goal of the method of partial fractions?
- When the degree of a polynomial ƒ(x) is less than the degree of a

polynomial g(x), how do you write ƒ(x)>g(x) as a sum of partial

fractions if g(x)

- is a product of distinct linear factors?
- consists of a repeated linear factor?
- contains an irreducible quadratic factor?

What do you do if the degree of ƒ is not less than the degree of g?

- How are integral tables typically used? What do you do if a particular

integral you want to evaluate is not listed in the table?

- What is a reduction formula? How are reduction formulas used?

Give an example.

- How would you compare the relative merits of Simpson’s Rule

and the Trapezoidal Rule?

- What is an improper integral of Type I? Type II? How are the

values of various types of improper integrals defined? Give

examples.

- What tests are available for determining the convergence and

divergence of improper integrals that cannot be evaluated

directly? Give examples of their use.

Chapter 8 Practice Exercises

Integration by Parts

Evaluate the integrals in Exercises 1–8 using integration by parts.

1.

L

x ln x dx 2.

L

x2 cos x dx

3.

L

sin-1 2x dx 4.

L

tan-1 ax

3b dx

5.

L

(1 + x2)ex dx 6.

L

x2 cos (2 – x) dx

7.

L

ex sin 3x dx 8.

L

x sin 2x cos 2x dx

Partial Fractions

Evaluate the integrals in Exercises 9–28. It may be necessary to use a

substitution first.

9.

L

x

x2 – 5x + 6

dx 10.

L

x

x2 + 4x – 5

dx

11.

L

dx

x(x – 2)2 12.

L

x + 3

x2(x – 2)

dx

13.

L

cos x

(1 – sin x)(2 – sin x)

dx 14.

L

sin x

cos2 x + 3 cos x + 2

dx

15.

L

5×2 + 7x + 9

x3 – x

dx 16.

L

9x dx

x3 + 9x

17.

L

x – 5

3×3 – 12x

dx 18.

L

5x + 9

(x – 2)(x + 3)(x + 4)

dx

19.

L

du

u4 + 5u2 + 4

20.

L

t dt

t4 + t2 – 2

21.

L

x3 + x2

x2 + x – 2

dx 22.

L

x3 + 1

x3 – x

dx

23.

L

x3 + 4×2

x2 + 4x + 3

dx 24.

L

2×3 + x2 – 21x + 24

x2 + 2x – 8

dx

25.

L

dx

x(32x + 1)

26.

L

dx

x11 + 23

x2

27.

L

ds

es – 1

28.

L

ds

2es + 1

Trigonometric Substitutions

Evaluate the integrals in Exercises 29–32 (a) without using a trigonometric

substitution, (b) using a trigonometric substitution.

29.

L

y dy

216 – y2

30.

L

x dx

24 + x2

31.

L

x dx

4 – x2 32.

L

t dt

24t2 – 1

Evaluate the integrals in Exercises 33–36.

33.

L

x dx

9 – x2 34.

L

dx

x(9 – x2)

35.

L

dx

9 – x2 36.

L

dx

29 – x2

Trigonometric Integrals

Evaluate the integrals in Exercises 37–44.

37.

L

sin5 x cos6 x dx 38.

L

cos3 x sin3 x dx

39.

L

tan6 x sec2 x dx 40.

L

tan3 x sec3 x dx

41.

L

sin 7u cos 5u du 42.

L

csc2 u cot3 u du

43.

L

21 – cos (t>2) dt 44.

L

eu21 – cos2 eu du

Numerical Integration

- According to the error-bound formula for Simpson’s Rule, how many

subintervals should you use to be sure of estimating the value of

ln 3 =

L

3

1

1x

dx

by Simpson’s Rule with an error of no more than 10-4 in absolute

value? (Remember that for Simpson’s Rule, the number of subintervals

has to be even.)

- A brief calculation shows that if 0 … x … 1, then the second

derivative of ƒ(x) = 21 + x4 lies between 0 and 8. Based on

this, about how many subdivisions would you need to estimate

the integral of ƒ from 0 to 1 with an error no greater than 10-3 in

absolute value using the Trapezoidal Rule?

- A direct calculation shows that

L

p

0

2 sin2 x dx = p.

How close do you come to this value by using the Trapezoidal Rule

with n = 6? Simpson’s Rule with n = 6? Try them and find out.

- You are planning to use Simpson’s Rule to estimate the value of

the integral

L

2

1

ƒ(x) dx

with an error magnitude less than 10-5. You have determined that

0 ƒ(4)(x) 0 … 3 throughout the interval of integration. How many

subintervals should you use to ensure the required accuracy?

(Remember that for Simpson’s Rule the number has to be even.)

- Mean temperature Use Simpson’s Rule to approximate the

average value of the temperature function

ƒ(x) = 37 sin a 2p

365

(x – 101)b + 25

for a 365-day year. This is one way to estimate the annual mean air

temperature in Fairbanks, Alaska. The National Weather Service’s

official figure, a numerical average of the daily normal mean air

temperatures for the year, is 25.7_F, which is slightly higher than

the average value of ƒ(x).

- Heat capacity of a gas Heat capacity Cy is the amount of heat

required to raise the temperature of a given mass of gas with constant

volume by 1_C, measured in units of cal > deg-mol (calories

per degree gram molecular weight). The heat capacity of oxygen

depends on its temperature T and satisfies the formula

Cy = 8.27 + 10-5 (26T – 1.87T2).

Use Simpson’s Rule to find the average value of Cy and the temperature

at which it is attained for 20_ … T … 675_C.

- Fuel efficiency An automobile computer gives a digital readout

of fuel consumption in gallons per hour. During a trip, a passenger

recorded the fuel consumption every 5 min for a full hour of travel.

Time Gal , h Time Gal , h

0 2.5 35 2.5

5 2.4 40 2.4

10 2.3 45 2.3

15 2.4 50 2.4

20 2.4 55 2.4

25 2.5 60 2.3

- Use the Trapezoidal Rule to approximate the total fuel consumption

during the hour.

- If the automobile covered 60 mi in the hour, what was its

fuel efficiency (in miles per gallon) for that portion of the

trip?

- A new parking lot To meet the demand for parking, your town

has allocated the area shown here. As the town engineer, you have

been asked by the town council to find out if the lot can be built

for $11,000. The cost to clear the land will be $0.10 a square foot,

and the lot will cost $2.00 a square foot to pave. Use Simpson’s

Rule to find out if the job can be done for $11,000.

67.5 ft

54 ft

Ignored

51 ft

54 ft

49.5 ft

64.4 ft

36 ft

42 ft

0 ft

Vertical spacing _ 15 ft

Improper Integrals

Evaluate the improper integrals in Exercises 53–62.

53.

L

3

0

dx

29 – x2

54.

L

1

0

ln x dx

55.

L

2

0

dy

(y – 1)2>3 56.

L

0

-2

du

(u + 1)3>5

57.

L

q

3

2 du

u2 – 2u

58.

L

q

1

3y – 1

4y3 – y2 dy

59.

L

q

0

x2e-x dx 60.

L

0

-q

xe3x dx

61.

L

q

-q

dx

4×2 + 9

62.

L

q

-q

4 dx

x2 + 16

Which of the improper integrals in Exercises 63–68 converge and

which diverge?

63.

L

q

6

du

2u2 + 1

64.

L

q

0

e-u cos u du

65.

L

q

1

ln z

z dz 66.

L

q

1

e-t

2t

dt

67.

L

q

-q

2 dx

ex + e-x 68.

L

q

-q

dx

x2(1 + ex)

Assorted Integrations

Evaluate the integrals in Exercises 69–116. The integrals are listed in

random order so you need to decide which integration technique to use.

69.

L

x dx

1 + 2x

70.

L

x3 + 2

4 – x2 dx

71.

L

22x – x2 dx 72.

L

dx

2-2x – x2

73.

L

2 – cos x + sin x

sin2 x

dx 74.

L

sin2 u cos5 u du

75.

L

9 dy

81 – y4 76.

L

q

2

dx

(x – 1)2

77.

L

u cos (2u + 1) du 78.

L

x3 dx

x2 – 2x + 1

79.

L

sin 2u du

(1 + cos 2u)2 80.

L

p>2

p>4

21 + cos 4x dx

81.

L

x dx

22 – x

82.

L

21 – y2

y2 dy

83.

L

dy

y2 – 2y + 2

84.

L

x dx

28 – 2×2 – x4

85.

L

z + 1

z2(z2 + 4)

dz 86.

L

x2(x – 1)1>3 dx

87.

L

t dt

29 – 4t2

88.

L

tan-1 x

x2 dx

89.

L

et dt

e2t + 3et + 2

90.

L

tan3 t dt

91.

L

q

1

ln y

y3 dy 92.

L

y3>2(ln y)2 dy

93.

L

eln2x dx 94.

L

eu23 + 4eu du

95.

L

sin 5t dt

1 + (cos 5t)2 96.

L

dy

2e2y – 1

97.

L

dr

1 + 2r

98.

L

4×3 – 20x

x4 – 10×2 + 9

dx

99.

L

x3

1 + x2 dx 100.

L

x2

1 + x3 dx

101.

L

1 + x2

1 + x3 dx 102.

L

1 + x2

(1 + x)3 dx

103.

L

2x # 31 + 2x dx 104.

L

31 + 21 + x dx

105.

L

1

2x # 21 + x

dx 106.

L

1>2

0

31 + 21 – x2 dx

107.

L

ln x

x + x ln x

dx 108.

L

1

x # ln x # ln (ln x)

dx

109.

L

xln x ln x

x dx 110.

L

(ln x)ln x c1x

+

ln (ln x)

x d dx

111.

L

1

x21 – x4

dx 112.

L

21 – x

x dx

- a. Show that 1

a

0 ƒ(x) dx = 1

a

0 ƒ(a – x) dx.

- Use part (a) to evaluate

L

p>2

0

sin x

sin x + cos x

dx.

114.

L

sin x

sin x + cos x

dx

115.

L

sin2 x

1 + sin2 x

dx

116.

L

1 – cos x

1 + cos x

Dx

Chapter 8 Additional and Advanced Exercises

Evaluating Integrals

Evaluate the integrals in Exercises 1–6.

1.

L

(sin-1 x)2 dx

2.

L

dx

x(x + 1)(x + 2)g(x + m)

3.

L

x sin-1 x dx 4.

L

sin-1 2y dy

5.

L

dt

t – 21 – t2

6.

L

dx

x4 + 4

Evaluate the limits in Exercises 7 and 8.

- lim

xSqL

x

-x

sin t dt 8. lim

xS0 +

x

L

1

x

cos t

t2 dt

Evaluate the limits in Exercises 9 and 10 by identifying them with

definite integrals and evaluating the integrals.

- lim

nSq

a

n

k=1

ln

A

n 1 + kn

- lim

nSq

a

n-1

k=0

1

2n2 – k2

Applications

- Finding arc length Find the length of the curve

y =

L

x

0

2cos 2t dt, 0 … x … p>4.

- Finding arc length Find the length of the graph of the function

y = ln (1 – x2), 0 … x … 1>2.

- Finding volume The region in the first quadrant that is enclosed

by the x-axis and the curve y = 3×21 – x is revolved about the

y-axis to generate a solid. Find the volume of the solid.

- Finding volume The region in the first quadrant that is enclosed

by the x-axis, the curve y = 5>1×25 – x2, and the lines x = 1

and x = 4 is revolved about the x-axis to generate a solid. Find

the volume of the solid.

- Finding volume The region in the first quadrant enclosed by the

coordinate axes, the curve y = ex, and the line x = 1 is revolved

about the y-axis to generate a solid. Find the volume of the solid.

- Finding volume Let R be the “triangular” region in the first

quadrant that is bounded above by the line y = 1, below by the

curve y = ln x, and on the left by the line x = 1. Find the volume

of the solid generated by revolving R about

- the x-axis. b. the line y = 1.
- Finding volume The region between the x-axis and the curve

y = ƒ(x) = e

0, x = 0

x ln x, 0 6 x … 2

is revolved about the x-axis to generate the solid shown here.

- Show that ƒ is continuous at x = 0.
- Find the volume of the solid.

y

0

y _ x ln x

x

1 2

- Finding volume The infinite region bounded by the coordinate

axes and the curve y = -ln x in the first quadrant is

revolved about the x-axis to generate a solid. Find the volume of

the solid.

- Centroid of a region Find the centroid of the region in the first

quadrant that is bounded below by the x-axis, above by the curve

y = ln x, and on the right by the line x = e.

- Length of a curve Find the length of the curve y = ln x from

x = 1 to x = e.

- The surface generated by an astroid The graph of the equation

x2>3 + y2>3 = 1 is an astroid (see accompanying figure).

Find the area of the surface generated by revolving the curve

about the x-axis.

- Length of a curve Find the length of the curve

y =

L

x

1

32t – 1 dt , 1 … x … 16.

- For what value or values of a does

L

q

1

a ax

x2 + 1 – 1

2xb dx

converge? Evaluate the corresponding integral(s).

- For each x 7 0, let G(x) = 1

q

0 e-xt dt. Prove that xG(x) = 1 for

each x 7 0.

- Infinite area and finite volume What values of p have the following

property: The area of the region between the curve

y = x-p, 1 … x 6 q, and the x-axis is infinite but the volume of

the solid generated by revolving the region about the x-axis is

finite.

- Integrating the square of the derivative If ƒ is continuously

differentiable on 30, 14 and ƒ(1) = ƒ(0) = -1>6, prove that

L

1

0

(ƒ_(x))2 dx Ú 2

L

1

0

ƒ(x) dx + 1

4

.

Hint: Consider the inequality 0 …

L

1

0

aƒ_(x) + x – 1

2b

2

dx.

Source: Mathematics Magazine, vol. 84, no. 4, Oct. 2011.

The Gamma Function and Stirling’s Formula

Euler’s gamma function _(x) (“gamma of x”; _ is a Greek capital g)

uses an integral to extend the factorial function from the nonnegative

integers to other real values. The formula is

_(x) =

L

q

0

tx-1e-t dt, x 7 0.

For each positive x, the number _(x) is the integral of tx-1e-t with

respect to t from 0 to q. Figure 8.21 shows the graph of _ near the

origin. You will see how to calculate _(1>2) if you do Additional

Exercise 23 in Chapter 14.

- If n is a nonnegative integer, _(n + 1) _ n!
- Show that _(1) = 1.
- Then apply integration by parts to the integral for _(x + 1) to

show that _(x + 1) = x_(x). This gives

_(2) = 1_(1) = 1

_(3) = 2_(2) = 2

_(4) = 3_(3) = 6

f

_(n + 1) = n _(n) = n! (1)

- Use mathematical induction to verify Equation (5) for every

nonnegative integer n.

- Stirling’s formula Scottish mathematician James Stirling

(1692–1770) showed that

lim

xSq

aexb

x

A

x

2p

_(x) = 1,

so, for large x,

_(x) = axe

b

x

A

2p

x (1 + P(x)), P(x) S 0 as xS q. (2)

Dropping P(x) leads to the approximation

_(x) _ axe

b

x

A

2p

x (Stirling>s formula). (3)

- Stirling’s approximation for n! Use Equation (3) and the

fact that n! = n_(n) to show that

n! _ ane

b

n

22np (Stirling>s approximation). (4)

As you will see if you do Exercise 104 in Section 9.1, Equation

(8) leads to the approximation

2n

n! _ ne

. (5)

- Compare your calculator’s value for n! with the value given

by Stirling’s approximation for n = 10, 20, 30,c, as far as

your calculator can go.

Chapter 9 Infinite Sequences and Series

Chapter 9 Questions to Guide Your Review

- What is an infinite sequence? What does it mean for such a

sequence to converge? To diverge? Give examples.

- What is a monotonic sequence? Under what circumstances does

such a sequence have a limit? Give examples.

- What theorems are available for calculating limits of sequences?

Give examples.

- What theorem sometimes enables us to use l’Hôpital’s Rule to

calculate the limit of a sequence? Give an example.

- What are the six commonly occurring limits in Theorem 5 that

arise frequently when you work with sequences and series?

- What is an infinite series? What does it mean for such a series to

converge? To diverge? Give examples.

- What is a geometric series? When does such a series converge?

Diverge? When it does converge, what is its sum? Give examples.

- Besides geometric series, what other convergent and divergent

series do you know?

- What is the nth-Term Test for Divergence? What is the idea

behind the test?

- What can be said about term-by-term sums and differences of

convergent series? About constant multiples of convergent and

divergent series?

- What happens if you add a finite number of terms to a convergent

series? A divergent series? What happens if you delete a finite

number of terms from a convergent series? A divergent series?

- How do you reindex a series? Why might you want to do this?
- Under what circumstances will an infinite series of nonnegative

terms converge? Diverge? Why study series of nonnegative terms?

- What is the Integral Test? What is the reasoning behind it? Give

an example of its use.

- When do p-series converge? Diverge? How do you know? Give

examples of convergent and divergent p-series.

- What are the Direct Comparison Test and the Limit Comparison

Test? What is the reasoning behind these tests? Give examples of

their use.

- What are the Ratio and Root Tests? Do they always give you the

information you need to determine convergence or divergence?

Give examples.

- What is absolute convergence? Conditional convergence? How

are the two related?

- What is an alternating series? What theorem is available for

determining the convergence of such a series?

- How can you estimate the error involved in approximating the

sum of an alternating series with one of the series’ partial sums?

What is the reasoning behind the estimate?

- What do you know about rearranging the terms of an absolutely

convergent series? Of a conditionally convergent series?

- What is a power series? How do you test a power series for convergence?

What are the possible outcomes?

- What are the basic facts about
- sums, differences, and products of power series?
- substitution of a function for x in a power series?
- term-by-term differentiation of power series?
- term-by-term integration of power series?

Give examples.

- What is the Taylor series generated by a function ƒ(x) at a point

x = a? What information do you need about ƒ to construct the

series? Give an example.

- What is a Maclaurin series?
- Does a Taylor series always converge to its generating function?

Explain.

- What are Taylor polynomials? Of what use are they?
- What is Taylor’s formula? What does it say about the errors

involved in using Taylor polynomials to approximate functions?

In particular, what does Taylor’s formula say about the error in a

linearization? A quadratic approximation?

- What is the binomial series? On what interval does it converge?

How is it used?

- How can you sometimes use power series to estimate the values

of nonelementary definite integrals? To find limits?

- What are the Taylor series for 1>(1 – x), 1>(1 + x), ex, sin x,

cos x, ln (1 + x), and tan-1 x? How do you estimate the errors

involved in replacing these series with their partial sums?

Chapter 9 Practice Exercises

Determining Convergence of Sequences

Which of the sequences whose nth terms appear in Exercises 1–18

converge, and which diverge? Find the limit of each convergent

sequence.

- an = 1 +

(-1)n

n 2. an =

1 – (-1)n

2n

- an = 1 – 2n

2n 4. an = 1 + (0.9)n

- an = sin

np

2

- an = sin np
- an =

ln (n2)

n 8. an =

ln (2n + 1)

n

- an = n + ln n

n 10. an =

ln (2n3 + 1)

n

- an = an – 5

n b

n

- an = a1 + 1n

b

-n

- an = A

n 3n

n 14. an = a3n

b

1>n

- an = n(21>n – 1) 16. an = 2n 2n + 1
- an =

(n + 1)!

n!

- an =

(-4)n

n!

Convergent Series

Find the sums of the series in Exercises 19–24.

- a

q

n=3

1

(2n – 3)(2n – 1)

- a

q

n=2

-2

n(n + 1)

- a

q

n=1

9

(3n – 1)(3n + 2)

- a

q

n=3

-8

(4n – 3)(4n + 1)

- a

q

n=0

e-n 24. a

q

n=1

(-1)n

3

4n

Determining Convergence of Series

Which of the series in Exercises 25–40 converge absolutely, which

converge conditionally, and which diverge? Give reasons for your

answers.

- a

q

n=1

1

2n

- a

q

n=1

-5

n

- a

q

n=1

(-1)n

2n

- a

q

n=1

1

2n3

- a

q

n=1

(-1)n

ln (n + 1)

- a

q

n=2

1

n (ln n)2

- a

q

n=1

ln n

n3 32. a

q

n=3

ln n

ln (ln n)

- a

q

n=1

(-1)n

n2n2 + 1

- a

q

n=1

(-1)n 3n2

n3 + 1

- a

q

n=1

n + 1

n!

- a

q

n=1

(-1)n(n2 + 1)

2n2 + n – 1

- a

q

n=1

(-3)n

n!

- a

q

n=1

2n 3n

nn

- a

q

n=1

1

2n(n + 1)(n + 2)

- a

q

n=2

1

n2n2 – 1

Power Series

In Exercises 41–50, (a) find the series’ radius and interval of convergence.

Then identify the values of x for which the series converges

(b) absolutely and (c) conditionally.

- a

q

n=1

(x + 4)n

n3n 42. a

q

n=1

(x – 1)2n-2

(2n – 1)!

- a

q

n=1

(-1)n-1(3x – 1)n

n2 44. a

q

n=0

(n + 1)(2x + 1)n

(2n + 1)2n

- a

q

n=1

xn

nn 46. a

q

n=1

xn

2n

- a

q

n=0

(n + 1)x2n-1

3n 48. a

q

n=0

(-1)n(x – 1)2n+1

2n + 1

- a

q

n=1

(csch n)xn 50. a

q

n=1

(coth n)xn

Maclaurin Series

Each of the series in Exercises 51–56 is the value of the Taylor series

at x = 0 of a function ƒ(x) at a particular point. What function and

what point? What is the sum of the series?

- 1 – 1

4 + 1

16 – g + (-1)n

1

4n + g

- 2

3 – 4

18 + 8

81 – g + (-1)n-1

2n

n3n + g

- p – p3

3! + p5

5! – g + (-1)n p2n+1

(2n + 1)! + g

- 1 – p2

9 # 2! + p4

81 # 4! – g + (-1)n p2n

32n(2n)!

+ g

- 1 + ln 2 +

(ln 2)2

2! + g +

(ln 2)n

n! + g

- 1

23

– 1

923

+ 1

4523

– g

+ (-1)n-1

1

(2n – 1)12322n-1 + g

Find Taylor series at x = 0 for the functions in Exercises 57–64.

- 1

1 – 2x

- 1

1 + x3

- sin px 60. sin

2x

3

- cos (x5>3) 62. cos

x3

25

- e(px>2) 64. e-x2

Taylor Series

In Exercises 65–68, find the first four nonzero terms of the Taylor

series generated by ƒ at x = a.

- ƒ(x) = 23 + x2 at x = -1
- ƒ(x) = 1>(1 – x) at x = 2
- ƒ(x) = 1>(x + 1) at x = 3
- ƒ(x) = 1>x at x = a 7 0

Nonelementary Integrals

Use series to approximate the values of the integrals in Exercises

69–72 with an error of magnitude less than 10-8. (The answer section

gives the integrals’ values rounded to 10 decimal places.)

69.

L

1>2

0

e-x3 dx 70.

L

1

0

x sin (x3) dx

71.

L

1>2

0

tan-1 x

x dx 72.

L

1>64

0

tan-1 x

2x

dx

Using Series to Find Limits

In Exercises 73–78:

- Use power series to evaluate the limit.
- Then use a grapher to support your calculation.
- lim

xS0

7 sin x

e2x – 1

- lim

uS0

eu – e-u – 2u

u – sin u

- lim

tS0

a 1

2 – 2 cos t – 1

t2b 76. lim

hS0

(sin h)>h – cos h

h2

- lim

zS0

1 – cos2 z

ln (1 – z) + sin z

- lim

yS0

y2

cos y – cosh y

Theory and Examples

- Use a series representation of sin 3x to find values of r and s for

which

lim

xS0

asin 3x

x3 + r

x2 + sb = 0.

- Compare the accuracies of the approximations sin x _ x and

sin x _ 6x>(6 + x2) by comparing the graphs of ƒ(x) = sin x – x

and g(x) = sin x – (6x>(6 + x2)). Describe what you find.

- Find the radius of convergence of the series

a

q

n=1

2 # 5 # 8 # g# (3n – 1)

2 # 4 # 6 # g# (2n)

xn.

- Find the radius of convergence of the series

a

q

n=1

3 # 5 # 7 # g# (2n + 1)

4 # 9 # 14 # g# (5n – 1)

(x – 1)n.

- Find a closed-form formula for the nth partial sum of the series

gqn

=2 ln (1 – (1>n2) ) and use it to determine the convergence or

divergence of the series.

- Evaluate gqk

=2 (1> (k2 – 1) ) by finding the limits as nS q of

the series’ nth partial sum.

- a. Find the interval of convergence of the series

y = 1 + 1

6

x3 + 1

180

x6 + g

+

1 # 4 # 7 #g# (3n – 2)

(3n)!

x3n + g.

- Show that the function defined by the series satisfies a differential

equation of the form

d2y

dx2 = xa y + b

and find the values of the constants a and b.

- a. Find the Maclaurin series for the function x2>(1 + x).
- Does the series converge at x = 1? Explain.
- If gqn

=1 an and gqn

=1 bn are convergent series of nonnegative

numbers, can anything be said about gqn

=1 an bn? Give reasons for

your answer.

- If gqn

=1 an and gqn

=1 bn are divergent series of nonnegative numbers,

can anything be said about gqn

=1 an bn? Give reasons for

your answer.

- Prove that the sequence 5xn6 and the series gqk

=1 (xk+1 – xk)

both converge or both diverge.

- Prove that gqn

=1 (an>(1 + an)) converges if an 7 0 for all n and

gqn

=1 an converges.

- Suppose that a1, a2, a3,c, an are positive numbers satisfying

the following conditions:

- i) a1 Ú a2 Ú a3 Ú g;
- ii) the series a2 + a4 + a8 + a16 + g diverges.

Show that the series

a1

1 +

a2

2 +

a3

3 + g

diverges.

- Use the result in Exercise 91 to show that

1 + a

q

n=2

1

n ln n

Chapter 9 A dditional and Advanced Exercises

Determining Convergence of Series

Which of the series gqn

=1 an defined by the formulas in Exercises 1–4

converge, and which diverge? Give reasons for your answers.

- a

q

n=1

1

(3n – 2)n+(1>2) 2. a

q

n=1

(tan-1 n)2

n2 + 1

- a

q

n=1

(-1)n tanh n 4. a

q

n=2

logn (n!)

n3

Which of the series gqn

=1 an defined by the formulas in Exercises 5–8

converge, and which diverge? Give reasons for your answers.

- a1 = 1, an+1 =

n(n + 1)

(n + 2)(n + 3)

an

(Hint: Write out several terms, see which factors cancel, and then

generalize.)

- a1 = a2 = 7, an+1 = n

(n – 1)(n + 1)

an if n Ú 2

- a1 = a2 = 1, an+1 = 1

1 + an

if n Ú 2

- an = 1>3n if n is odd, an = n>3n if n is even

Choosing Centers for Taylor Series

Taylor’s formula

ƒ(x) = ƒ(a) + ƒ_(a)(x – a) +

ƒ_(a)

2!

(x – a)2 + g

+

ƒ(n)(a)

n!

(x – a)n +

ƒ(n+1)(c)

(n + 1)!

(x – a)n+1

expresses the value of ƒ at x in terms of the values of ƒ and its derivatives

at x = a. In numerical computations, we therefore need a to be a

point where we know the values of ƒ and its derivatives. We also need

a to be close enough to the values of ƒ we are interested in to make

(x – a)n+1 so small we can neglect the remainder.

In Exercises 9–14, what Taylor series would you choose to represent

the function near the given value of x? (There may be more than one

good answer.) Write out the first four nonzero terms of the series you

choose.

- cos x near x = 1 10. sin x near x = 6.3
- ex near x = 0.4 12. ln x near x = 1.3
- cos x near x = 69 14. tan-1 x near x = 2

Theory and Examples

- Let a and b be constants with 0 6 a 6 b. Does the sequence

5(an + bn)1>n6 converge? If it does converge, what is the limit?

- Find the sum of the infinite series

1 + 2

10 + 3

102 + 7

103 + 2

104 + 3

105 + 7

106 + 2

107

+ 3

108 + 7

109 + g.

- Evaluate

a

q

n=0

L

n+1

n

1

1 + x2 dx.

- Find all values of x for which

a

q

n=1

nxn

(n + 1)(2x + 1)n

converges absolutely.

- a. Does the value of

lim

nSq

a1 –

cos (a>n)

n b

n

, a constant,

appear to depend on the value of a? If so, how?

- Does the value of

lim

nSq

a1 –

cos (a>n)

bn

b

n

, a and b constant, b _ 0,

appear to depend on the value of b? If so, how?

- Use calculus to confirm your findings in parts (a) and (b).
- Show that if gqn

=1 an converges, then

a

q

n=1

a

1 + sin (an)

2 b

n

converges.

- Find a value for the constant b that will make the radius of convergence

of the power series

a

q

n=2

bnxn

ln n

equal to 5.

- How do you know that the functions sin x, ln x, and ex are not

polynomials? Give reasons for your answer.

- Find the value of a for which the limit

lim

xS0

sin (ax) – sin x – x

x3

is finite and evaluate the limit.

- Find values of a and b for which

lim

xS0

cos (ax) – b

2×2 = -1.

- Raabe’s (or Gauss’s) Test The following test, which we state

without proof, is an extension of the Ratio Test.

Raabe’s Test: If gqn

=1 un is a series of positive constants and

there exist constants C, K, and N such that

un

un+1 = 1 + Cn

+

ƒ(n)

n2 ,

where _ ƒ(n) _ 6 K for n Ú N, then gqn

=1 un converges if C 7 1

and diverges if C … 1.

Show that the results of Raabe’s Test agree with what you

know about the series gqn

=1 (1>n2) and gqn

=1 (1>n).

- (Continuation of Exercise 25.) Suppose that the terms of gqn

=1 un

are defined recursively by the formulas

u1 = 1, un+1 =

(2n – 1)2

(2n)(2n + 1)

un.

Apply Raabe’s Test to determine whether the series converges.

- If gqn

=1 an converges, and if an _ 1 and an 7 0 for all n,

- Show that gqn

=1 an

2 converges.

- Does gqn

=1 an>(1 – an) converge? Explain.

- (Continuation of Exercise 27.) If gqn

=1 an converges, and if

1 7 an 7 0 for all n, show that gqn

=1 ln (1 – an) converges.

(Hint: First show that _ ln (1 – an) _ … an>(1 – an).)

- Nicole Oresme’s Theorem Prove Nicole Oresme’s Theorem that

1 + 1

2

# 2 + 1

4

# 3 + g + n

2n-1 + g = 4.

(Hint: Differentiate both sides of the equation 1>(1 – x) =

1 + gqn

=1 xn.)

- a. Show that

a

q

n=1

n(n + 1)

xn = 2×2

(x – 1)3

for _ x _ 7 1 by differentiating the identity

a

q

n=1

xn+1 = x2

1 – x

twice, multiplying the result by x, and then replacing x by 1 > x.

- Use part (a) to find the real solution greater than 1 of the

equation

x = a

q

n=1

n(n + 1)

xn .

- Quality control
- Differentiate the series

1

1 – x = 1 + x + x2 + g+ x

n + g

to obtain a series for 1>(1 – x)2.

- In one throw of two dice, the probability of getting a roll of

7 is p = 1>6. If you throw the dice repeatedly, the probability

that a 7 will appear for the first time at the nth throw is

q

n-1p, where q = 1 – p = 5>6. The expected number of

throws until a 7 first appears is gqn

=1nq

n-1p. Find the sum of

this series.

- As an engineer applying statistical control to an industrial operation,

you inspect items taken at random from the assembly

line. You classify each sampled item as either “good” or “bad.”

If the probability of an item’s being good is p and of an item’s

being bad is q = 1 – p, the probability that the first bad item

found is the nth one inspected is p

n-1q. The average number

inspected up to and including the first bad item found is

gqn

=1np

n-1q. Evaluate this sum, assuming 0 6 p 6 1.

- Expected value Suppose that a random variable X may assume

the values 1, 2, 3, . . . , with probabilities p1, p2, p3, . . . , where pk

is the probability that X equals k (k = 1, 2, 3, c). Suppose also

that pk Ú 0 and that gqk=1 pk = 1. The expected value of X,

denoted by E(X), is the number gqk

=1 k pk, provided the series

converges. In each of the following cases, show that gqk

=1 pk = 1

and find E(X) if it exists. (Hint: See Exercise 31.)

- pk = 2-k b. pk = 5k-1

6k

- pk = 1

k(k + 1) = 1

k – 1

k + 1

- Safe and effective dosage The concentration in the blood

resulting from a single dose of a drug normally decreases with

time as the drug is eliminated from the body. Doses may therefore

need to be repeated periodically to keep the concentration

from dropping below some particular level. One model for the

effect of repeated doses gives the residual concentration just

before the (n + 1)st dose as

Rn = C0e-k t0 + C0e-2k t0 + g + C0e-nk t0 ,

where Co = the change in concentration achievable by a single

dose (mg>mL), k = the elimination constant (h–1), and t0 = time

between doses (h). See the accompanying figure.

- Write Rn in closed from as a single fraction, and find

R = limnSq Rn.

- Calculate R1 and R10 for C0 = 1 mg>mL, k = 0.1 h-1, and

t0 = 10 h. How good an estimate of R is R10?

- If k = 0.01 h-1 and t0 = 10 h, find the smallest n such that

Rn 7 (1>2)R. Use C0 = 1 mg>mL.

(Source: Prescribing Safe and Effective Dosage, B. Horelick and

- Koont, COMAP, Inc., Lexington, MA.)
- Time between drug doses (Continuation of Exercise 33.) If a

drug is known to be ineffective below a concentration CL and

harmful above some higher concentration CH, one need to find values

of C0 and t0 that will produce a concentration that is safe (not

above CH ) but effective (not below CL ). See the accompanying

figure. We therefore want to find values for C0 and t0 for which

R = CL and C0 + R = CH.

t0

CL

0 Time

Concentration in blood

C0

Highest safe level

CH

Lowest effective level

t

C

Thus C0 = CH – CL . When these values are substituted in the equation

for R obtained in part (a) of Exercise 33, the resulting equation

simplifies to

t0 = 1

k

ln

CH

CL

.

To reach an effective level rapidly, one might administer a “loading”

dose that would produce a concentration of CH mg>mL. This could be

followed every t0 hours by a dose that raises the concentration by

C0 = CH – CL mg>mL.

- Verify the preceding equation for t0.
- If k = 0.05 h-1 and the highest safe concentration is e times the

lowest effective concentration, find the length of time between

doses that will ensure safe and effective concentrations.

- Given CH = 2 mg>mL, CL = 0.5 mg>mL, and k = 0.02 h-1,

determine a scheme for administering the drug.

- Suppose that k = 0.2 h-1 and that the smallest effective concentration

is 0.03 mg>mL. A single dose that produces a concentration

of 0.1 mg>mL is administered. About how long will the drug

remain effective?

Chapter 10 Parametric Equations and Polar Coordinates

Chapter 10 Questions to Guide Your Review

- What is a parametrization of a curve in the xy-plane? Does a function

y = ƒ(x) always have a parametrization? Are parametrizations

of a curve unique? Give examples.

- Give some typical parametrizations for lines, circles, parabolas,

ellipses, and hyperbolas. How might the parametrized curve differ

from the graph of its Cartesian equation?

- What is a cycloid? What are typical parametric equations for

cycloids? What physical properties account for the importance of

cycloids?

- What is the formula for the slope dy>dx of a parametrized curve

x = ƒ(t), y = g(t)? When does the formula apply? When can you

expect to be able to find d2y>dx2 as well? Give examples.

- How can you sometimes find the area bounded by a parametrized

curve and one of the coordinate axes?

- How do you find the length of a smooth parametrized curve

x = ƒ(t), y = g(t), a … t … b? What does smoothness have to

do with length? What else do you need to know about the parametrization

in order to find the curve’s length? Give examples.

- What is the arc length function for a smooth parametrized curve?

What is its arc length differential?

- Under what conditions can you find the area of the surface generated

by revolving a curve x = ƒ(t), y = g(t), a … t … b, about

the x-axis? the y-axis? Give examples.

- What are polar coordinates? What equations relate polar coordinates

to Cartesian coordinates? Why might you want to change

from one coordinate system to the other?

- What consequence does the lack of uniqueness of polar coordinates

have for graphing? Give an example.

- How do you graph equations in polar coordinates? Include in

your discussion symmetry, slope, behavior at the origin, and the

use of Cartesian graphs. Give examples.

- How do you find the area of a region 0 … r1(u) … r … r2(u),

a … u … b, in the polar coordinate plane? Give examples.

- Under what conditions can you find the length of a curve

r = ƒ(u), a … u … b, in the polar coordinate plane? Give an

example of a typical calculation.

- What is the eccentricity of a conic section? How can you classify

conic sections by eccentricity? How does eccentricity change the

shape of ellipses and hyperbolas?

Chapter 10 Practice Exercises

Identifying Parametric Equations in the Plane

Exercises 1–6 give parametric equations and parameter intervals for

the motion of a particle in the xy-plane. Identify the particle’s path by

finding a Cartesian equation for it. Graph the Cartesian equation and

indicate the direction of motion and the portion traced by the particle.

- x = t>2, y = t + 1; -q 6 t 6 q
- x = 2t, y = 1 – 2t; t Ú 0
- x = (1>2) tan t, y = (1>2) sec t; -p>2 6 t 6 p>2
- x = -2 cos t, y = 2 sin t; 0 … t … p
- x = -cos t, y = cos2 t; 0 … t … p
- x = 4 cos t, y = 9 sin t; 0 … t … 2p

Finding Parametric Equations and Tangent Lines

- Find parametric equations and a parameter interval for the

motion of a particle in the xy-plane that traces the ellipse

16×2 + 9y2 = 144 once counterclockwise. (There are many ways

to do this.)

- Find parametric equations and a parameter interval for the motion

of a particle that starts at the point (-2, 0) in the xy-plane and traces

the circle x2 + y2 = 4 three times clockwise. (There are many

ways to do this.)

In Exercises 9 and 10, find an equation for the line in the xy-plane that is

tangent to the curve at the point corresponding to the given value of t.

Also, find the value of d2y>dx2 at this point.

- x = (1>2) tan t, y = (1>2) sec t; t = p>3
- x = 1 + 1>t2, y = 1 – 3>t; t = 2
- Eliminate the parameter to express the curve in the form y = ƒ(x) .
- x = 4t2, y = t3 – 1
- x = cos t, y = tan t
- Find parametric equations for the given curve.
- Line through (1, -2) with slope 3
- (x – 1)2 + ( y + 2)2 = 9
- y = 4×2 – x
- 9×2 + 4y2 = 36

Lengths of Curves

Find the lengths of the curves in Exercises 13–19.

- y = x1>2 – (1>3)x3>2, 1 … x … 4
- x = y2>3, 1 … y … 8
- y = (5>12)x6>5 – (5>8)x4>5, 1 … x … 32
- x = (y3>12) + (1>y), 1 … y … 2
- x = 5 cos t – cos 5t, y = 5 sin t – sin 5t, 0 … t … p>2
- x = t3 – 6t2, y = t3 + 6t2, 0 … t … 1
- x = 3 cos u, y = 3 sin u, 0 … u … 3p

2

- Find the length of the enclosed loop x = t2, y = (t3>3) – t

shown here. The loop starts at t = -23 and ends at t = 23.

y

0

1

1

_1

2 4

x

t _ 0 t _ _”3

t > 0

t < 0

Surface Areas

Find the areas of the surfaces generated by revolving the curves in

Exercises 21 and 22 about the indicated axes.

- x = t2>2, y = 2t, 0 … t … 25; x-axis
- x = t2 + 1>(2t), y = 42t, 1>22 … t … 1; y-axis

Polar to Cartesian Equations

Sketch the lines in Exercises 23–28. Also, find a Cartesian equation

for each line.

- r cos au + p

3 b = 223 24. r cos au – 3p

4 b =

22

2

- r = 2 sec u 26. r = -22 sec u
- r = -(3>2) csc u 28. r = 13232 csc u

Find Cartesian equations for the circles in Exercises 29–32. Sketch

each circle in the coordinate plane and label it with both its Cartesian

and polar equations.

- r = -4 sin u 30. r = 323 sin u
- r = 222 cos u 32. r = -6 cos u

Cartesian to Polar Equations

Find polar equations for the circles in Exercises 33–36. Sketch each

circle in the coordinate plane and label it with both its Cartesian and

polar equations.

- x2 + y2 + 5y = 0 34. x2 + y2 – 2y = 0
- x2 + y2 – 3x = 0 36. x2 + y2 + 4x = 0

Graphs in Polar Coordinates

Sketch the regions defined by the polar coordinate inequalities in

Exercises 37 and 38.

- 0 … r … 6 cos u 38. -4 sin u … r … 0

Match each graph in Exercises 39–46 with the appropriate equation

(a)–(l). There are more equations than graphs, so some equations will

not be matched.

- r = cos 2u b. r cos u = 1 c. r = 6

1 – 2 cos u

- r = sin 2u e. r = u f. r2 = cos 2u
- r = 1 + cos u h. r = 1 – sin u i. r = 2

1 – cos u

- r2 = sin 2u k. r = -sin u l. r = 2 cos u + 1
- Four-leaved rose 40. Spiral

x

y

x

y

- Limaçon 42. Lemniscate

x

y

x

y

- Circle 44. Cardioid

x

y

x

y

- Parabola 46. Lemniscate

x

y

x

y

Area in Polar Coordinates

Find the areas of the regions in the polar coordinate plane described in

Exercises 47–50.

- Enclosed by the limaçon r = 2 – cos u
- Enclosed by one leaf of the three-leaved rose r = sin 3u
- Inside the “figure eight” r = 1 + cos 2u and outside the circle

r = 1

- Inside the cardioid r = 2(1 + sin u) and outside the circle

r = 2 sin u

Length in Polar Coordinates

Find the lengths of the curves given by the polar coordinate equations

in Exercises 51–54.

- r = -1 + cos u
- r = 2 sin u + 2 cos u, 0 … u … p>2
- r = 8 sin3 (u>3), 0 … u … p>4
- r = 21 + cos 2u, -p>2 … u … p>2

Conics in Polar Coordinates

Sketch the conic sections whose polar coordinate equations are given

in Exercises 55–58. Give polar coordinates for the vertices and, in the

case of ellipses, for the centers as well.

- r = 2

1 + cos u

- r = 8

2 + cos u

- r = 6

1 – 2 cos u

- r = 12

3 + sin u

Exercises 59–62 give the eccentricities of conic sections with one

focus at the origin of the polar coordinate plane, along with the directrix

for that focus. Find a polar equation for each conic section.

- e = 2, r cos u = 2
- e = 1, r cos u = -4
- e = 1>2, r sin u = 2
- e = 1>3, r sin u = -6

Chapter 10 A dditional and Advanced Exercises

Polar Coordinates

- a. Find an equation in polar coordinates for the curve

x = e2t cos t, y = e2t sin t; -q 6 t 6 q.

- Find the length of the curve from t = 0 to t = 2p.
- Find the length of the curve r = 2 sin3 (u>3), 0 … u … 3p, in the

polar coordinate plane.

Exercises 3–6 give the eccentricities of conic sections with one focus

at the origin of the polar coordinate plane, along with the directrix for

that focus. Find a polar equation for each conic section.

- e = 2, r cos u = 2 4. e = 1, r cos u = -4
- e = 1>2, r sin u = 2 6. e = 1>3, r sin u = -6

Theory and Examples

- Epicycloids When a circle rolls externally along the circumference

of a second, fixed circle, any point P on the circumference

of the rolling circle describes an epicycloid, as shown here. Let

the fixed circle have its center at the origin O and have radius a.

x

y

O

u

b

C

P

A(a, 0)

Let the radius of the rolling circle be b and let the initial position

of the tracing point P be A(a, 0). Find parametric equations for

the epicycloid, using as the parameter the angle u from the positive

x-axis to the line through the circles’ centers.

- Find the centroid of the region enclosed by the x-axis and the

cycloid arch

x = a(t – sin t), y = a(1 – cos t); 0 … t … 2p.

The Angle Between the Radius Vector and the Tangent Line to a

Polar Coordinate Curve In Cartesian coordinates, when we want

to discuss the direction of a curve at a point, we use the angle f measured

counterclockwise from the positive x-axis to the tangent line. In

polar coordinates, it is more convenient to calculate the angle c from

the radius vector to the tangent line (see the accompanying figure).

The angle f can then be calculated from the relation

f = u + c, (1)

which comes from applying the Exterior Angle Theorem to the triangle

in the accompanying figure.

x

y

0

u f

c

r

r _ f (u)

P(r, u)

Suppose the equation of the curve is given in the form r = ƒ(u),

where ƒ(u) is a differentiable function of u. Then

x = r cos u and y = r sin u (2)

are differentiable functions of u with

dx

du = -r sin u + cos u

dr

du

,

dy

du = r cos u + sin u

dr

du

. (3)

Since c = f – u from (1),

tan c = tan (f – u) =

tan f – tan u

1 + tan f tan u

.

Furthermore,

tan f =

dy

dx =

dy>du

dx>du

because tan f is the slope of the curve at P. Also,

tan u =

y

x.

Hence

tan c =

dy>du

dx>du

–

y

x

1 +

y

x

dy>du

dx>du

=

x

dy

du – y

dx

du

x

dx

du + y

dy

du

. (4)

- From Equations (2), (3), and (4), show that

tan c = r

dr>du

. (5)

This is the equation we use for finding c as a function of u.

- Find the value of tan c for the curve r = sin4 (u>4).
- Find the angle between the radius vector to the curve r =

2a sin 3u and its tangent when u = p>6.

- a. Graph the hyperbolic spiral ru = 1. What appears to happen

to c as the spiral winds in around the origin?

- Confirm your finding in part (a) analytically.

Chapter 11 Vectors and the Geometry of Space

Chapter 11 Questions to Guide Your Review

- When do directed line segments in the plane represent the same

vector?

- How are vectors added and subtracted geometrically? Algebraically?
- How do you find a vector’s magnitude and direction?
- If a vector is multiplied by a positive scalar, how is the result

related to the original vector? What if the scalar is zero? Negative?

- Define the dot product (scalar product) of two vectors. Which

algebraic laws are satisfied by dot products? Give examples.

When is the dot product of two vectors equal to zero?

- What geometric interpretation does the dot product have? Give

examples.

- What is the vector projection of a vector u onto a vector v? Give

an example of a useful application of a vector projection.

- Define the cross product (vector product) of two vectors. Which

algebraic laws are satisfied by cross products, and which are not?

Give examples. When is the cross product of two vectors equal to

zero?

- What geometric or physical interpretations do cross products

have? Give examples.

- What is the determinant formula for calculating the cross product

of two vectors relative to the Cartesian i, j, k-coordinate system?

Use it in an example.

- How do you find equations for lines, line segments, and planes in

space? Give examples. Can you express a line in space by a single

equation? A plane?

- How do you find the distance from a point to a line in space?

From a point to a plane? Give examples.

- What are box products? What significance do they have? How

are they evaluated? Give an example.

- How do you find equations for spheres in space? Give examples.
- How do you find the intersection of two lines in space? A line and

a plane? Two planes? Give examples.

- What is a cylinder? Give examples of equations that define cylinders

in Cartesian coordinates.

- What are quadric surfaces? Give examples of different kinds of

ellipsoids, paraboloids, cones, and hyperboloids (equations and

sketches).

Chapter 11 Practice Exercises

Vector Calculations in Two Dimensions

In Exercises 1–4, let u = 8-3, 49 and v = 82, -59. Find (a) the

component form of the vector and (b) its magnitude.

- 5u + 9v 2. u – v
- -7u 4. 9v

In Exercises 5–8, find the component form of the vector.

- The vector obtained by rotating 80, 19 through an angle of 3p>4

radians

- The unit vector that makes an angle of p>3 radian with the positive

x-axis

- The vector 2 units long in the direction 4i – j
- The vector 5 units long in the direction opposite to the direction

of (3>5)i + (4>5)j

Express the vectors in Exercises 9–12 in terms of their lengths and

directions.

- 23i + 23j 10. -3i + 2j
- Velocity vector v = (-5 sin t)i + (5 cos t)j when t = p>4.
- Velocity vector v = (et sin t + et cos t)i + (et cos t – et sin t)j

when t = ln 3.

Vector Calculations in Three Dimensions

Express the vectors in Exercises 13 and 14 in terms of their lengths

and directions.

- 4i + 5j – 3k 14. i – 5j + k
- Find a vector 8 units long in the direction of v = 3i + 2j – k.
- Find a vector 6 units long in the direction along the direction of

v = (2>5)i + (3>5)j + (4>5)k.

In Exercises 17 and 18, find 0 v 0 , 0 u 0 , v # u, u # v, v * u, u * v,

0 v * u 0 , the angle between v and u, the scalar component of u in the

direction of v, and the vector projection of u onto v.

- v = i – j

u = 3i – j + 5k

In Exercises 25 and 26, find (a) the area of the parallelogram determined

by vectors u and v and (b) the volume of the parallelepiped

determined by the vectors u, v, and w.

- u = i – j – k, v = 3i + 4j – 5k, w = 2i + 2j – k
- u = j – k, v = k, w = i – j – k

Lines, Planes, and Distances

- Suppose that n is normal to a plane and that v is parallel to the

plane. Describe how you would find a vector n that is both perpendicular

to v and parallel to the plane.

- Find a vector in the plane parallel to the line ax + by = c.

In Exercises 29 and 30, find the distance from the point to the line.

- (2, 2, 0); x = -t, y = t, z = -1 + t
- (0, 4, 1); x = 2 + t, y = 2 + t, z = t
- Parametrize the line that passes through the point (1, 2, 3) parallel

to the vector v = -3i + 7k.

- Parametrize the line segment joining the points P(1, 2, 0) and

Q(1, 3, -1) .

In Exercises 33 and 34, find the distance from the point to the plane.

- (6, 0, -6), x – y = 4
- (3, 0, 10), 2x + 3y + z = 2
- Find an equation for the plane that passes through the point

(3, -2, 1) normal to the vector n = 2i + j + k.

- Find an equation for the plane that passes through the point

(-1, 6, 0) perpendicular to the line x = -1 + t, y = 6 – 2t,

z = 3t .

In Exercises 37 and 38, find an equation for the plane through points

P, Q, and R.

- P(1, -1, 2), Q(2, 1, 3), R(-1, 2, -1)
- P(1, 0, 0), Q(0, 1, 0), R(0, 0, 1)
- Find the points in which the line x = 1 + 2t, y = -1 – t,

z = 3t meets the three coordinate planes.

- Find the point in which the line through the origin perpendicular

to the plane 2x – y – z = 4 meets the plane 3x – 5y +

2z = 6.

- Find the acute angle between the planes x = 7 and x + y +

22z = -3.

- Find the acute angle between the planes x + y = 1 and y + z = 1.
- Find parametric equations for the line in which the planes

x + 2y + z = 1 and x – y + 2z = -8 intersect.

- Show that the line in which the planes

x + 2y – 2z = 5 and 5x – 2y – z = 0

intersect is parallel to the line

x = -3 + 2t, y = 3t, z = 1 + 4t .

- The planes 3x + 6z = 1 and 2x + 2y – z = 3 intersect in a line.
- Show that the planes are orthogonal.
- Find equations for the line of intersection.
- Find an equation for the plane that passes through the point

(1, 2, 3) parallel to u = 2i + 3j + k and v = i – j + 2k.

- Is v = 2i – 4j + k related in any special way to the plane

2x + y = 5? Give reasons for your answer.

- The equation n # r P0 P = 0 represents the plane through P0 normal

to n. What set does the inequality n # r P0 P 7 0 represent?

- Find the distance from the point P(1, 4, 0) to the plane through

A(0, 0, 0), B(2, 0, -1), and C(2, -1, 0) .

- Find the distance from the point (2, 2, 3) to the plane

2x + 3y + 5z = 0.

- Find a vector parallel to the plane 2x – y – z = 4 and orthogonal

to i + j + k.

- Find a unit vector orthogonal to A in the plane of B and C if

A = 2i – j + k, B = i + 2j + k, and C = i + j – 2k.

- Find a vector of magnitude 2 parallel to the line of intersection of

the planes x + 2y + z – 1 = 0 and x – y + 2z + 7 = 0.

- Find the point in which the line through the origin perpendicular

to the plane 2x – y – z = 4 meets the plane 3x – 5y +

2z = 6.

- Find the point in which the line through P(3, 2, 1) normal to the

plane 2x – y + 2z = -2 meets the plane.

- What angle does the line of intersection of the planes

2x + y – z = 0 and x + y + 2z = 0 make with the positive

x-axis?

- The line

L: x = 3 + 2t, y = 2t, z = t

intersects the plane x + 3y – z = -4 in a point P. Find the

coordinates of P and find equations for the line in the plane

through P perpendicular to L.

- Show that for every real number k the plane

x – 2y + z + 3 + k(2x – y – z + 1) = 0

contains the line of intersection of the planes

x – 2y + z + 3 = 0 and 2x – y – z + 1 = 0.

- Find an equation for the plane through A(-2, 0, -3) and

B(1, -2, 1) that lies parallel to the line through

C(-2, -13>5, 26>5) and D(16>5, -13>5, 0) .

- Is the line x = 1 + 2t, y = -2 + 3t, z = -5t related in any way

to the plane -4x – 6y + 10z = 9? Give reasons for your answer.

- Which of the following are equations for the plane through the

points P(1, 1, -1), Q(3, 0, 2), and R(-2, 1, 0)?

- (2i – 3j + 3k) # ((x + 2)i + (y – 1)j + zk) = 0
- x = 3 – t, y = -11t, z = 2 – 3t
- (x + 2) + 11(y – 1) = 3z
- (2i – 3j + 3k) * ((x + 2)i + ( y – 1)j + zk) = 0
- (2i – j + 3k) * (-3i + k) # ((x + 2)i + ( y – 1)j + zk)

= 0

- The parallelogram shown here has vertices at A(2, -1, 4),

B(1, 0, -1), C(1, 2, 3), and D. Find

z

y

x

D

C(1, 2, 3)

A(2, _1, 4)

B(1, 0, _1)

- the coordinates of D,
- the cosine of the interior angle at B,
- the vector projection of rBA onto rBC,
- the area of the parallelogram,
- an equation for the plane of the parallelogram,
- the areas of the orthogonal projections of the parallelogram

on the three coordinate planes.

- Distance between skew lines Find the distance between the

line L1 through the points A(1, 0, -1) and B(-1, 1, 0) and the

line L2 through the points C(3, 1, -1) and D(4, 5, -2) . The distance

is to be measured along the line perpendicular to the two

lines. First find a vector n perpendicular to both lines. Then project

rAC onto n.

- (Continuation of Exercise 63.) Find the distance between the line

through A(4, 0, 2) and B(2, 4, 1) and the line through C(1, 3, 2)

and D(2, 2, 4).

Quadric Surfaces

Identify and sketch the surfaces in Exercises 65–76.

- x2 + y2 + z2 = 4 66. x2 + (y – 1)2 + z2 = 1
- 4×2 + 4y2 + z2 = 4 68. 36×2 + 9y2 + 4z2 = 36
- z = – (x2 + y2) 70. y = – (x2 + z2)
- x2 + y2 = z2 72. x2 + z2 = y2
- x2 + y2 – z2 = 4 74. 4y2 + z2 – 4×2 = 4
- y2 – x2 – z2 = 1 76. z2 – x2 – y2 = 1

Chapter 11 Additional and Advanced Exercises

- Submarine hunting Two surface ships on maneuvers are trying

to determine a submarine’s course and speed to prepare for an aircraft

intercept. As shown here, ship A is located at (4, 0, 0), whereas

ship B is located at (0, 5, 0). All coordinates are given in thousands

of feet. Ship A locates the submarine in the direction of the vector

2i + 3j – (1>3)k, and ship B locates it in the direction of the vector

18i – 6j – k. Four minutes ago, the submarine was located at

(2, -1, -1>3) . The aircraft is due in 20 min. Assuming that the

submarine moves in a straight line at a constant speed, to what

position should the surface ships direct the aircraft?

z

y

x

(4, 0, 0)

Submarine

(0, 5, 0)

Ship A

Ship B

NOT TO SCALE

- A helicopter rescue Two helicopters, H1 and H2, are traveling

together. At time t = 0, they separate and follow different

straight-line paths given by

H1: x = 6 + 40t, y = -3 + 10t, z = -3 + 2t

H2: x = 6 + 110t, y = -3 + 4t, z = -3 + t .

Time t is measured in hours, and all coordinates are measured in

miles. Due to system malfunctions, H2 stops its flight at (446,

13, 1) and, in a negligible amount of time, lands at (446, 13, 0).

Two hours later, H1 is advised of this fact and heads toward H2 at

150 mph. How long will it take H1 to reach H2?

- Torque The operator’s manual for the Toro® 21-in. lawnmower

says “tighten the spark plug to 15 ft@lb (20.4 N # m) .” If you are

installing the plug with a 10.5-in. socket wrench that places the

center of your hand 9 in. from the axis of the spark plug, about

how hard should you pull? Answer in pounds.

9 in.

- Rotating body The line through the origin and the point A(1, 1, 1)

is the axis of rotation of a rigid body rotating with a constant

angular speed of 3 >2 rad > sec. The rotation appears to be

clockwise when we look toward the origin from A. Find the

velocity v of the point of the body that is at the position B(1, 3, 2).

y

z

O

x

1

1

3

v

A(1, 1, 1) B(1, 3, 2)

- Consider the weight suspended by two wires in each diagram.

Find the magnitudes and components of vectors F1 and F2, and

angles a and b.

a.

F1 F2

100 lbs

5 ft

a b

3 ft 4 ft

b.

F1 F2

200 lbs

13 ft

a b

12 ft

5 ft

(Hint: This triangle is a right triangle.)

- Consider a weight of w N suspended by two wires in the diagram,

where T1 and T2 are force vectors directed along the wires.

T1 T2

a b

w

a b

- Find the vectors T1 and T2 and show that their magnitudes are

0 T1 0 =

w cos b

sin (a + b)

and

0 T2 0 = w cos a

sin (a + b)

.

- For a fixed b determine the value of a which minimizes the

magnitude 0 T1 0 .

- For a fixed a determine the value of b which minimizes the

magnitude 0 T2 0 .

- Determinants and planes
- Show that

3

x1 – x y1 – y z1 – z

x2 – x y2 – y z2 – z

x3 – x y3 – y z3 – z

3 = 0

is an equation for the plane through the three noncollinear

points P1(x1, y1, z1), P2(x2, y2, z2), and P3(x3, y3, z3) .

- What set of points in space is described by the equation

4

x y z 1

x1 y1 z1 1

x2 y2 z2 1

x3 y3 z3 1

4 = 0 ?

- Determinants and lines Show that the lines

x = a1 s + b1, y = a2 s + b2, z = a3 s + b3, -q 6 s 6 q

and

x = c1 t + d1, y = c2 t + d2, z = c3 t + d3, -q 6 t 6 q,

intersect or are parallel if and only if

3

a1 c1 b1 – d1

a2 c2 b2 – d2

a3 c3 b3 – d3

3 = 0.

- Use vectors to show that the distance from P1(x1, y1) to the line

ax + by = c is

d =

0 ax1 + by1 – c 0

2a2 + b2

.

- a. Use vectors to show that the distance from P1(x1, y1, z1) to the

plane Ax + By + Cz = D is

d =

0 Ax1 + By1 + Cz1 – D0

2A2 + B2 + C2

.

- Find an equation for the sphere that is tangent to the planes

x + y + z = 3 and x + y + z = 9 if the planes 2x – y = 0

and 3x – z = 0 pass through the center of the sphere.

- a. Distance between parallel planes Show that the distance

between the parallel planes Ax + By + Cz = D1 and

Ax + By + Cz = D2 is

d =

0D1 – D2 0

0 Ai + Bj + C k 0 .

- Find the distance between the planes 2x + 3y – z = 6 and

2x + 3y – z = 12.

- Find an equation for the plane parallel to the plane

2x – y + 2z = -4 if the point (3, 2, -1) is equidistant from

the two planes.

- Write equations for the planes that lie parallel to and 5 units

away from the plane x – 2y + z = 3.

- Prove that four points A, B, C, and D are coplanar (lie in a common

plane) if and only if rAD # (rAB * rBC) = 0.

- Triple vector products The triple vector products

(u * v) * w and u * (v * w) are usually not equal, although

the formulas for evaluating them from components are similar:

(u * v) * w = (u # w)v – (v # w)u.

u * (v * w) = (u # w)v – (u # v)w.

Verify each formula for the following vectors by evaluating its

two sides and comparing the results.

u v w

- 2i 2j 2k
- i – j + k 2i + j – 2k -i + 2j – k
- 2i + j 2i – j + k i + 2k
- i + j – 2k -i – k 2i + 4j – 2k
- Cross and dot products Show that if u, v, w, and r are any

vectors, then

- u * (v * w) + v * (w * u) + w * (u * v) = 0
- u * v = (u # v * i)i + (u # v * j)j + (u # v * k)k
- (u * v) # (w * r) = `

u # w v # w

u # r v # r

` .

- Cross and dot products Prove or disprove the formula

u * (u * (u * v)) # w = – 0 u 0 2 u # v * w.

- By forming the cross product of two appropriate vectors, derive

the trigonometric identity

sin (A – B) = sin A cos B – cos A sin B.

- Use vectors to prove that

(a2 + b2)(c2 + d2) Ú (ac + bd )2

for any four numbers a, b, c, and d. (Hint: Let u = ai + bj and

v = ci + dj .)

- Dot multiplication is positive definite Show that dot multiplication

of vectors is positive definite; that is, show u ~ u Ú 0 for

every vector u and that u # u = 0 if and only if u _ 0.

Chapter 12 Vector-Valued Functions and Motion in Space

Chapter 12 Questions to Guide Your Review

- State the rules for differentiating and integrating vector functions.

Give examples.

- How do you define and calculate the velocity, speed, direction of

motion, and acceleration of a body moving along a sufficiently

differentiable space curve? Give an example.

- What is special about the derivatives of vector functions of constant

length? Give an example.

- What are the vector and parametric equations for ideal projectile

motion? How do you find a projectile’s maximum height, flight

time, and range? Give examples.

- How do you define and calculate the length of a segment of a

smooth space curve? Give an example. What mathematical assumptions

are involved in the definition?

- How do you measure distance along a smooth curve in space

from a preselected base point? Give an example.

- What is a differentiable curve’s unit tangent vector? Give an

example.

- Define curvature, circle of curvature (osculating circle), center of

curvature, and radius of curvature for twice-differentiable curves

in the plane. Give examples. What curves have zero curvature?

Constant curvature?

- What is a plane curve’s principal normal vector? When is it

defined? Which way does it point? Give an example.

- How do you define N and k for curves in space?
- What is a curve’s binormal vector? Give an example.
- What formulas are available for writing a moving object’s acceleration

as a sum of its tangential and normal components? Give

an example. Why might one want to write the acceleration this

way? What if the object moves at a constant speed? At a constant

speed around a circle?

- State Kepler’s laws.

Chapter 12 Practice Exercises

Motion in the Plane

In Exercises 1 and 2, graph the curves and sketch their velocity and

acceleration vectors at the given values of t. Then write a in the form

a = aTT + aNN without finding T and N, and find the value of k at

the given values of t.

- Find the point on the curve y = ex where the curvature is greatest.
- A particle moves around the unit circle in the xy-plane. Its position

at time t is r = xi + yj, where x and y are differentiable

functions of t. Find dy > dt if v # i = y. Is the motion clockwise or

counterclockwise?

- You send a message through a pneumatic tube that follows the

curve 9y = x3 (distance in meters). At the point (3, 3), v # i = 4

and a # i = -2. Find the values of v # j and a # j at (3, 3).

- Characterizing circular motion A particle moves in the plane

so that its velocity and position vectors are always orthogonal.

Show that the particle moves in a circle centered at the origin.

- Speed along a cycloid A circular wheel with radius 1 ft and

center C rolls to the right along the x-axis at a half-turn per second.

(See the accompanying figure.) At time t seconds, the position

vector of the point P on the wheel’s circumference is

r = (pt – sin pt)i + (1 – cos pt)j.

- r(t) = (4 cos t)i + 122 sin t2j, t = 0 and p>4
- r(t) = 123 sec t2i + 123 tan t2j, t = 0
- The position of a particle in the plane at time t is

r = 1

21 + t2

i + t

21 + t2

j.

Find the particle’s highest speed.

- Suppose r(t) = (et cos t)i + (et sin t)j. Show that the angle

between r and a never changes. What is the angle?

- Finding curvature At point P, the velocity and acceleration of

a particle moving in the plane are v = 3i + 4j and a = 5i + 15j.

Find the curvature of the particle’s path at P.

- Sketch the curve traced by P during the interval 0 … t … 3.
- Find v and a at t = 0, 1, 2, and 3 and add these vectors to

your sketch.

- At any given time, what is the forward speed of the topmost

point of the wheel? Of C?

x

y

1

C

P

pt

r

0

Projectile Motion

- Shot put A shot leaves the thrower’s hand 6.5 ft above the

ground at a 45° angle at 44 ft > sec. Where is it 3 sec later?

- Javelin A javelin leaves the thrower’s hand 7 ft above the

ground at a 45° angle at 80 ft > sec. How high does it go?

- A golf ball is hit with an initial speed y0 at an angle a to the horizontal

from a point that lies at the foot of a straight-sided hill that

is inclined at an angle f to the horizontal, where

0 6 f 6 a 6 p

2

.

Show that the ball lands at a distance

2y0 2 cos a

g cos2 f

sin (a – f),

measured up the face of the hill. Hence, show that the great-est

range that can be achieved for a given y0 occurs when

a = (f>2) + (p>4), i.e., when the initial velocity vector bisects

the angle between the vertical and the hill.

- Javelin In Potsdam in 1988, Petra Felke of (then) East Germany

set a women’s world record by throwing a javelin 262 ft 5 in.

- Assuming that Felke launched the javelin at a 40° angle to the

horizontal 6.5 ft above the ground, what was the javelin’s initial

speed?

- How high did the javelin go?

Motion in Space

Find the lengths of the curves in Exercises 15 and 16.

- r(t) = (2 cos t)i + (2 sin t)j + t2k, 0 … t … p>4
- r(t) = (3 cos t)i + (3 sin t)j + 2t3>2k, 0 … t … 3

T

In Exercises 17–20, find T, N, B, and k, at the given value of t.

- r(t) = 4

9

(1 + t)3>2 i + 4

9

(1 – t)3>2 j + 1

3

t k, t = 0

- r(t) = (et sin 2t)i + (et cos 2t)j + 2et

k, t = 0

- r(t) = t i + 1

2

e2t

j, t = ln 2

- r(t) = (3 cosh 2t)i + (3 sinh 2t)j + 6t k, t = ln 2

In Exercises 21 and 22, write a in the form a = aTT + aNN at t = 0

without finding T and N.

- r(t) = (2 + 3t + 3t2)i + (4t + 4t2)j – (6 cos t)k
- r(t) = (2 + t)i + (t + 2t2)j + (1 + t2)k
- Find T, N, B, and k, as functions of t if

r(t) = (sin t)i + 122 cos t2j + (sin t)k.

- At what times in the interval 0 … t … p are the velocity and

acceleration vectors of the motion r(t) = i + (5 cos t)j +

(3 sin t)k orthogonal?

- The position of a particle moving in space at time t Ú 0 is

r(t) = 2i + a4 sin

t

2bj + a3 – t

pbk.

Find the first time r is orthogonal to the vector i – j.

- Find equations for the osculating, normal, and rectifying planes

of the curve r(t) = t i + t2

j + t3 k at the point (1, 1, 1).

- Find parametric equations for the line that is tangent to the curve

r(t) = et i + (sin t)j + ln (1 – t)k at t = 0.

- Find parametric equations for the line tangent to the helix r(t) =

122 cos t2i + 122 sin t2j + t k at the point where t = p>4.

Theory and Examples

- Synchronous curves By eliminating a from the ideal projectile

equations

x = (y0 cos a)t, y = (y0 sin a)t – 1

2

gt2,

show that x2 + (y + gt2>2)2 = y0 2 t2. This shows that projectiles

launched simultaneously from the origin at the same initial

speed will, at any given instant, all lie on the circle of radius y0 t

centered at (0, -gt2>2), regardless of their launch angle. These

circles are the synchronous curves of the launching.

- Radius of curvature Show that the radius of curvature of a

twice-differentiable plane curve r(t) = ƒ(t)i + g(t)j is given by

the formula

r =

x #

2 + y # 2

2x $ 2 + y $ 2 – s $ 2

, where s $ = d

dt

2x # 2 + y # 2.

Chapter 12 A dditional and Advanced Exercises

Applications

- A frictionless particle P, starting from rest at time t = 0 at the

point (a, 0, 0), slides down the helix

r(u) = (a cos u)i + (a sin u)j + buk (a, b 7 0)

under the influence of gravity, as in the accompanying figure. The u

in this equation is the cylindrical coordinate u and the helix is the

curve r = a, z = bu, u Ú 0, in cylindrical coordinates. We assume

u to be a differentiable function of t for the motion. The law of

conservation of energy tells us that the particle’s speed after it has

fallen straight down a distance z is 22gz, where g is the constant

acceleration of gravity.

- Find the angular velocity du>dt when u = 2p.
- Express the particle’s u@ and z-coordinates as functions of t.
- Express the tangential and normal components of the velocity

dr > dt and acceleration d2r>dt2 as functions of t. Does the

acceleration have any nonzero component in the direction of

the binormal vector B?

x

The helix

r _ a, z _ bu

z

Positive z-axis

points down.

a

P

r

y

- Suppose the curve in Exercise 1 is replaced by the conical helix

r = au, z = bu shown in the accompanying figure.

- Express the angular velocity du>dt as a function of u.
- Express the distance the particle travels along the helix as a

function of u.

Conical helix

r _ au, z _ bu

Positive z-axis points down.

Cone z _ r ba

z

x

y

P

Motion in Polar and Cylindrical Coordinates

- Deduce from the orbit equation

r =

(1 + e)r0

1 + e cos u

that a planet is closest to its sun when u = 0 and show that

r = r0 at that time.

- A Kepler equation The problem of locating a planet in its orbit

at a given time and date eventually leads to solving “Kepler”

equations of the form

ƒ(x) = x – 1 – 1

2

sin x = 0.

- Show that this particular equation has a solution between

x = 0 and x = 2.

- With your computer or calculator in radian mode, use Newton’s

method to find the solution to as many places as you can.

- In Section 12.6, we found the velocity of a particle moving in the

plane to be

v = x # i + y # j = r # ur + ru #

uu .

- Express x # and y # in terms of r # and ru #

by evaluating the dot

products v # i and v # j.

- Express r # and ru #

in terms of x # and y # by evaluating the dot

products v # ur and v # uu .

- Express the curvature of a twice-differentiable curve r = ƒ(u) in

the polar coordinate plane in terms of ƒ and its derivatives.

- A slender rod through the origin of the polar coordinate plane

rotates (in the plane) about the origin at the rate of 3 rad > min. A

beetle starting from the point (2, 0) crawls along the rod toward

the origin at the rate of 1 in. > min.

- Find the beetle’s acceleration and velocity in polar form when

it is halfway to (1 in. from) the origin.

- To the nearest tenth of an inch, what will be the length of

the path the beetle has traveled by the time it reaches the

origin?

- Conservation of angular momentum Let r(t) denote the position

in space of a moving object at time t. Suppose the force acting

on the object at time t is

F(t) = –

c

0 r(t) 0 3 r(t),

where c is a constant. In physics the angular momentum of an

object at time t is defined to be L(t) = r(t) * mv(t), where m is

the mass of the object and v(t) is the velocity. Prove that angular

momentum is a conserved quantity; i.e., prove that L(t) is a constant

vector, independent of time. Remember Newton’s law

F = ma. (This is a calculus problem, not a physics problem.)

Chapter 13 Partial Derivatives

Chapter 13 Questions to Guide Your Review

- What is a real-valued function of two independent variables?

Three independent variables? Give examples.

- What does it mean for sets in the plane or in space to be open?

Closed? Give examples. Give examples of sets that are neither

open nor closed.

- How can you display the values of a function ƒ(x, y) of two independent

variables graphically? How do you do the same for a

function ƒ(x, y, z) of three independent variables?

- What does it mean for a function ƒ(x, y) to have limit L as

(x, y) S (x0 , y0)? What are the basic properties of limits of functions

of two independent variables?

- When is a function of two (three) independent variables continuous

at a point in its domain? Give examples of functions that are

continuous at some points but not others.

- What can be said about algebraic combinations and composites of

continuous functions?

- Explain the two-path test for nonexistence of limits.
- How are the partial derivatives 0ƒ>0x and 0ƒ>0y of a function

ƒ(x, y) defined? How are they interpreted and calculated?

- How does the relation between first partial derivatives and continuity

of functions of two independent variables differ from the

relation between first derivatives and continuity for real-valued

functions of a single independent variable? Give an example.

- What is the Mixed Derivative Theorem for mixed second-order

partial derivatives? How can it help in calculating partial derivatives

of second and higher orders? Give examples.

- What does it mean for a function ƒ(x, y) to be differentiable?

What does the Increment Theorem say about differentiability?

- How can you sometimes decide from examining ƒx and ƒy that a

function ƒ(x, y) is differentiable? What is the relation between the

differentiability of ƒ and the continuity of ƒ at a point?

- What is the general Chain Rule? What form does it take for functions

of two independent variables? Three independent variables?

Functions defined on surfaces? How do you diagram these different

forms? Give examples. What pattern enables one to remember

all the different forms?

- What is the derivative of a function ƒ(x, y) at a point P0 in the

direction of a unit vector u? What rate does it describe? What

geometric interpretation does it have? Give examples.

- What is the gradient vector of a differentiable function ƒ(x, y)?

How is it related to the function’s directional derivatives?

State the analogous results for functions of three independent

variables.

- How do you find the tangent line at a point on a level curve of a

differentiable function ƒ(x, y)? How do you find the tangent plane

and normal line at a point on a level surface of a differentiable

function ƒ(x, y, z)? Give examples.

- How can you use directional derivatives to estimate change?
- How do you linearize a function ƒ(x, y) of two independent variables

at a point (x0, y0)? Why might you want to do this? How do

you linearize a function of three independent variables?

- What can you say about the accuracy of linear approximations of

functions of two (three) independent variables?

- If (x, y) moves from (x0, y0) to a point (x0 + dx, y0 + dy) nearby,

how can you estimate the resulting change in the value of a differentiable

function ƒ(x, y)? Give an example.

- How do you define local maxima, local minima, and saddle points

for a differentiable function ƒ(x, y)? Give examples.

- What derivative tests are available for determining the local

extreme values of a function ƒ(x, y)? How do they enable you to

narrow your search for these values? Give examples.

- How do you find the extrema of a continuous function ƒ(x, y) on a

closed bounded region of the xy-plane? Give an example.

- Describe the method of Lagrange multipliers and give examples.

Chapter 13 P ractice Exercises

Domain, Range, and Level Curves

In Exercises 1–4, find the domain and range of the given function and

identify its level curves. Sketch a typical level curve.

- ƒ(x, y) = 9×2 + y2 2. ƒ(x, y) = ex+y
- g(x, y) = 1>xy 4. g(x, y) = 2×2 – y

In Exercises 5–8, find the domain and range of the given function and

identify its level surfaces. Sketch a typical level surface.

- ƒ(x, y, z) = x2 + y2 – z 6. g(x, y, z) = x2 + 4y2 + 9z2
- h(x, y, z) = 1

x2 + y2 + z2 8. k(x, y, z) = 1

x2 + y2 + z2 + 1

Evaluating Limits

Find the limits in Exercises 9–14.

- lim

(x,y)S(p, ln 2)

ey cos x 10. lim

(x,y)S(0,0)

2 + y

x + cos y

- lim

(x,y)S(1,1)

x – y

x2 – y2 12. lim

(x,y)S(1,1)

x3y3 – 1

xy – 1

- lim

PS(1, -1, e)

ln 0 x + y + z 0 14. lim

PS(1,-1,-1)

tan-1 (x + y + z)

By considering different paths of approach, show that the limits in

Exercises 15 and 16 do not exist.

- lim

(x,y)S(0,0)

y

x2 – y

- lim

(x,y)S(0,0)

x2 + y2

xy

y_x2 xy_0

- Continuous extension Let ƒ(x, y) = (x2 – y2) > (x2 + y2) for

(x, y) _ (0, 0). Is it possible to define ƒ(0, 0) in a way that makes

ƒ continuous at the origin? Why?

- Continuous extension Let

ƒ(x, y) = •

sin (x – y)

0 x 0 + 0 y 0 , 0 x 0 + 0 y 0 _ 0

0, (x, y) = (0, 0).

Is ƒ continuous at the origin? Why?

Partial Derivatives

In Exercises 19–24, find the partial derivative of the function with

respect to each variable.

- ƒ(r, u) = r sin u – r cos u
- ƒ(x, y) = 1

2

ln (x2 – y2) + sin-1

y

x

- ƒ(R1, R2, R3) = 1

R1

+ 1

R2

+ 1

R3

- ƒ(x, y, z) = cos (4px – y + 5z)
- P(n, R, T, V ) = nRT

V

(the ideal gas law)

- ƒ(r, l, T, w) = 1

2rl

A

T

Pw

Second-Order Partials

Find the second-order partial derivatives of the functions in Exercises

25–28.

- g(x, y) = y + xy
- ƒ(x, y) = ey – x cos y
- g(x, y) = y – xy – 8y3 + ln (y2 – 1)
- g(x, y) = x2 – 9xy – sin y + 5ex

Chain Rule Calculations

- Find dw> dt at t = 0 if w = sin (xy + p), x = et, and y =

ln (t + 1).

- Find dw> dt at t = 1 if w = xey + y sin z – cos z, x = 22t,

y = t – 1 + ln t, and z = pt.

- Find 0w>0r and 0w>0s when r = p and s = 0 if w =

sin (2x – y), x = r + sin s, y = rs.

- Find 0w>0u and 0w>0y when u = y = 0 if w =

ln21 + x2 – tan-1 x and x = 2eu cos y.

- Find the value of the derivative of ƒ(x, y, z) = xy + yz + xz

with respect to t on the curve x = cos t, y = sin t, z = cos 2t at

t = 1.

- Show that if w = ƒ(s) is any differentiable function of s and if

s = y + 5x, then

0w

0x – 5 0w

0y = 0.

Implicit Differentiation

Assuming that the equations in Exercises 35 and 36 define y as a differentiable

function of x, find the value of dy > dx at point P.

- 1 – x – y2 – sin xy = 0, P(0, 1)
- 2xy + ex+y – 2 = 0, P(0, ln 2)

Directional Derivatives

In Exercises 37–40, find the directions in which ƒ increases and

decreases most rapidly at P0 and find the derivative of ƒ in each direction.

Also, find the derivative of ƒ at P0 in the direction of the vector v.

- ƒ(x, y) = cos x cos y, P0(p>4, p>4), v = 3i + 4j
- ƒ(x, y) = x2e-2y, P0(1, 0), v = i + j
- ƒ(x, y, z) = ln (2x + 3y + 6z), P0(-1, -1, 1),

v = 2i + 3j + 6k

- ƒ(x, y, z) = x2 + 3xy – z2 + 2y + z + 4, P0(0, 0, 0),

v = i + j + k

- Derivative in velocity direction Find the derivative of

ƒ(x, y, z) = xyz in the direction of the velocity vector of the helix

r(t) = (cos 3t)i + (sin 3t)j + 3t k

at t = p>3.

- Maximum directional derivative What is the largest value

that the directional derivative of ƒ(x, y, z) = xyz can have at the

point (1, 1, 1)?

- Directional derivatives with given values At the point (1, 2),

the function ƒ(x, y) has a derivative of 2 in the direction toward

(2, 2) and a derivative of -2 in the direction toward (1, 1).

- Find ƒx(1, 2) and ƒy(1, 2).
- Find the derivative of ƒ at (1, 2) in the direction toward the

point (4, 6).

- Which of the following statements are true if ƒ(x, y) is differentiable

at (x0 , y0)? Give reasons for your answers.

- If u is a unit vector, the derivative of ƒ at (x0 , y0) in the direction

of u is (ƒx(x0 , y0)i + ƒy(x0 , y0)j) # u.

- The derivative of ƒ at (x0 , y0) in the direction of u is a vector.
- The directional derivative of ƒ at (x0 , y0) has its greatest value

in the direction of _ƒ.

- At (x0 , y0), vector _ƒ is normal to the curve ƒ(x, y) = ƒ(x0 , y0).

Gradients, Tangent Planes, and Normal Lines

In Exercises 45 and 46, sketch the surface ƒ(x, y, z) = c together with

_ƒ at the given points.

- x2 + y + z2 = 0; (0, -1, {1), (0, 0, 0)
- y2 + z2 = 4; (2, {2, 0), (2, 0, {2)

In Exercises 47 and 48, find an equation for the plane tangent to the

level surface ƒ(x, y, z) = c at the point P0 . Also, find parametric

equations for the line that is normal to the surface at P0 .

- x2 – y – 5z = 0, P0(2, -1, 1)
- x2 + y2 + z = 4, P0(1, 1, 2)

In Exercises 49 and 50, find an equation for the plane tangent to the

surface z = ƒ(x, y) at the given point.

- z = ln (x2 + y2), (0, 1, 0)
- z = 1> (x2 + y2), (1, 1, 1>2)

In Exercises 51 and 52, find equations for the lines that are tangent

and normal to the level curve ƒ(x, y) = c at the point P0 . Then sketch

the lines and level curve together with _ƒ at P0 .

- y – sin x = 1, P0(p, 1) 52.

y2

2 – x2

2 = 3

2

, P0(1, 2)

Tangent Lines to Curves

In Exercises 53 and 54, find parametric equations for the line that is

tangent to the curve of intersection of the surfaces at the given point.

- Surfaces: x2 + 2y + 2z = 4, y = 1

Point: (1, 1, 1 > 2)

- Surfaces: x + y2 + z = 2, y = 1

Point: (1 > 2, 1, 1 > 2)

Linearizations

In Exercises 55 and 56, find the linearization L(x, y) of the function ƒ(x, y)

at the point P0 . Then find an upper bound for the magnitude of the error

E in the approximation ƒ(x, y) _ L(x, y) over the rectangle R.

- ƒ(x, y) = sin x cos y, P0(p>4, p>4)

R: ` x – p

4 ` … 0.1, ` y – p

4 ` … 0.1

- ƒ(x, y) = xy – 3y2 + 2, P0(1, 1)

R: 0 x – 1 0 … 0.1, 0 y – 1 0 … 0.2

Find the linearizations of the functions in Exercises 57 and 58 at the

given points.

- ƒ(x, y, z) = xy + 2yz – 3xz at (1, 0, 0) and (1, 1, 0)
- ƒ(x, y, z) = 22 cos x sin ( y + z) at (0, 0, p>4) and (p>4,

p>4, 0)

Estimates and Sensitivity to Change

- Measuring the volume of a pipeline You plan to calculate the

volume inside a stretch of pipeline that is about 36 in. in diameter

and 1 mile long. With which measurement should you be more

careful, the length or the diameter? Why?

- Sensitivity to change Is ƒ(x, y) = x2 – xy + y2 – 3 more

sensitive to changes in x or to changes in y when it is near the

point (1, 2)? How do you know?

- Change in an electrical circuit Suppose that the current I

(amperes) in an electrical circuit is related to the voltage V (volts)

and the resistance R (ohms) by the equation I = V>R. If the voltage

drops from 24 to 23 volts and the resistance drops from 100 to

80 ohms, will I increase or decrease? By about how much? Is the

change in I more sensitive to change in the voltage or to change in

the resistance? How do you know?

- Maximum error in estimating the area of an ellipse If

a = 10 cm and b = 16 cm to the nearest millimeter, what should

you expect the maximum percentage error to be in the calculated

area A = pab of the ellipse x2>a2 + y2>b2 = 1?

- Error in estimating a product Let y = uy and z = u + y,

where u and y are positive independent variables.

- If u is measured with an error of 2% and y with an error of 3%,

about what is the percentage error in the calculated value of y?

- Show that the percentage error in the calculated value of z is

less than the percentage error in the value of y.

- Cardiac index To make different people comparable in studies

of cardiac output, researchers divide the measured cardiac output

by the body surface area to find the cardiac index C:

C =

cardiac output

body surface area

.

The body surface area B of a person with weight w and height h is

approximated by the formula

B = 71.84w0.425h0.725,

which gives B in square centimeters when w is measured in kilograms

and h in centimeters. You are about to calculate the cardiac

index of a person 180 cm tall, weighing 70 kg, with cardiac output

of 7 L>min. Which will have a greater effect on the calculation,

a 1-kg error in measuring the weight or a 1-cm error in measuring

the height?

Local Extrema

Test the functions in Exercises 65–70 for local maxima and minima

and saddle points. Find each function’s value at these points.

- ƒ(x, y) = x2 – xy + y2 + 2x + 2y – 4
- ƒ(x, y) = 5×2 + 4xy – 2y2 + 4x – 4y
- ƒ(x, y) = 2×3 + 3xy + 2y3
- ƒ(x, y) = x3 + y3 – 3xy + 15
- ƒ(x, y) = x3 + y3 + 3×2 – 3y2
- ƒ(x, y) = x4 – 8×2 + 3y2 – 6y

Absolute Extrema

In Exercises 71–78, find the absolute maximum and minimum values

of ƒ on the region R.

- ƒ(x, y) = x2 + xy + y2 – 3x + 3y

R: The triangular region cut from the first quadrant by the line

x + y = 4

- ƒ(x, y) = x2 – y2 – 2x + 4y + 1

R: The rectangular region in the first quadrant bounded by the

coordinate axes and the lines x = 4 and y = 2

- ƒ(x, y) = y2 – xy – 3y + 2x

R: The square region enclosed by the lines x = {2 and y = {2

- ƒ(x, y) = 2x + 2y – x2 – y2

R: The square region bounded by the coordinate axes and the

lines x = 2, y = 2 in the first quadrant

- ƒ(x, y) = x2 – y2 – 2x + 4y

R: The triangular region bounded below by the x-axis, above by

the line y = x + 2, and on the right by the line x = 2

- ƒ(x, y) = 4xy – x4 – y4 + 16

R: The triangular region bounded below by the line y = -2,

above by the line y = x, and on the right by the line x = 2

- ƒ(x, y) = x3 + y3 + 3×2 – 3y2

R: The square region enclosed by the lines x = {1 and y = {1

- ƒ(x, y) = x3 + 3xy + y3 + 1

R: The square region enclosed by the lines x = {1 and y = {1

Lagrange Multipliers

- Extrema on a circle Find the extreme values of ƒ(x, y) =

x3 + y2 on the circle x2 + y2 = 1.

- Extrema on a circle Find the extreme values of ƒ(x, y) = xy

on the circle x2 + y2 = 1.

- Extrema in a disk Find the extreme values of ƒ(x, y) =

x2 + 3y2 + 2y on the unit disk x2 + y2 … 1.

- Extrema in a disk Find the extreme values of ƒ(x, y) =

x2 + y2 – 3x – xy on the disk x2 + y2 … 9.

- Extrema on a sphere Find the extreme values of ƒ(x, y, z) =

x – y + z on the unit sphere x2 + y2 + z2 = 1.

- Minimum distance to origin Find the points on the surface

x2 – zy = 4 closest to the origin.

- Minimizing cost of a box A closed rectangular box is to have

volume V cm3. The cost of the material used in the box is

a cents>cm2 for top and bottom, b cents>cm2 for front and back,

and c cents>cm2 for the remaining sides. What dimensions minimize

the total cost of materials?

- Least volume Find the plane x>a + y>b + z>c = 1 that passes

through the point (2, 1, 2) and cuts off the least volume from the

first octant.

- Extrema on curve of intersecting surfaces Find the extreme

values of ƒ(x, y, z) = x( y + z) on the curve of intersection of the

right circular cylinder x2 + y2 = 1 and the hyperbolic cylinder

xz = 1.

- Minimum distance to origin on curve of intersecting plane and

cone Find the point closest to the origin on the curve of intersection

of the plane x + y + z = 1 and the cone z2 = 2×2 + 2y2.

Theory and Examples

- Let w = ƒ(r, u), r = 2×2 + y2, and u = tan-1 ( y>x). Find

0w>0x and 0w>0y and express your answers in terms of r and u.

- Let z = ƒ(u, y), u = ax + by, and y = ax – by. Express zx and

zy in terms of fu , fy , and the constants a and b.

- If a and b are constants, w = u3 + tanh u + cos u, and u =

ax + by, show that

a 0w

0y = b 0w

0x .

- Using the Chain Rule If w = ln (x2 + y2 + 2z), x = r + s,

y = r – s, and z = 2rs, find wr and ws by the Chain Rule. Then

check your answer another way.

- Angle between vectors The equations eu cos y – x = 0 and

eu sin y – y = 0 define u and y as differentiable functions of x

and y. Show that the angle between the vectors

0u

0x i + 0u

0y j and 0y

0x i + 0y

0y j

is constant.

- Polar coordinates and second derivatives Introducing polar

coordinates x = r cos u and y = r sin u changes ƒ(x, y) to

g(r, u). Find the value of 02g>0u2 at the point (r, u) = (2, p>2),

given that

0ƒ

0x =

0ƒ

0y =

02ƒ

0x2 =

02ƒ

0y2 = 1

at that point.

- Normal line parallel to a plane Find the points on the surface

(y + z)2 + (z – x)2 = 16

where the normal line is parallel to the yz-plane.

- Tangent plane parallel to xy-plane Find the points on the

surface

xy + yz + zx – x – z2 = 0

where the tangent plane is parallel to the xy-plane.

- When gradient is parallel to position vector Suppose that

_ƒ(x, y, z) is always parallel to the position vector x i + y j + z k.

Show that ƒ(0, 0, a) = ƒ(0, 0, -a) for any a.

- One-sided directional derivative in all directions, but no gradient

The one-sided directional derivative of ƒ at P(x0 , y0 , z0)

in the direction u = u1i + u2 j + u3 k is the number

lim

sS0 +

f (x0 + su1, y0 + su2 , z0 + su3) – f (x0 , y0 , z0)

s .

Show that the one-sided directional derivative of

ƒ(x, y, z) = 2×2 + y2 + z2

at the origin equals 1 in any direction but that ƒ has no gradient

vector at the origin.

- Normal line through origin Show that the line normal to the

surface xy + z = 2 at the point (1, 1, 1) passes through the origin.

- Tangent plane and normal line
- Sketch the surface x2 – y2 + z2 = 4.
- Find a vector normal to the surface at (2, -3, 3). Add the

vector to your sketch.

- Find equations for the tangent plane and normal line at

(2, -3, 3).

Chapter 13 A dditional and Advanced Exercises

Partial Derivatives

- Function with saddle at the origin If you did Exercise 60 in

Section 13.2, you know that the function

ƒ(x, y) = • xy

x2 – y2

x2 + y2 , (x, y) _ (0, 0)

0, (x, y) = (0, 0)

(see the accompanying figure) is continuous at (0, 0). Find

ƒxy(0, 0) and ƒyx(0, 0).

z

y

x

- Finding a function from second partials Find a function

w = ƒ(x, y) whose first partial derivatives are 0w>0x = 1 +

ex cos y and 0w>0y = 2y – ex sin y and whose value at the point

(ln 2, 0) is ln 2.

- A proof of Leibniz’s Rule Leibniz’s Rule says that if ƒ is continuous

on 3a, b4 and if u(x) and y(x) are differentiable functions

of x whose values lie in 3a, b4 , then

d

dxL

y(x)

u(x)

ƒ(t) dt = ƒ(y(x))

dy

dx – ƒ(u(x))

du

dx

.

Prove the rule by setting

g(u, y) =

L

y

u

ƒ(t) dt, u = u(x), y = y(x)

and calculating dg > dx with the Chain Rule.

- Finding a function with constrained second partials Suppose

that ƒ is a twice-differentiable function of r, that r =

2×2 + y2 + z2, and that

ƒxx + ƒyy + ƒzz = 0.

Show that for some constants a and b,

ƒ(r) = ar

+ b.

- Homogeneous functions A function ƒ(x, y) is homogeneous of

degree n (n a nonnegative integer) if ƒ(tx, ty) = tnƒ(x, y) for all t,

x, and y. For such a function (sufficiently differentiable), prove

that

- x

0ƒ

0x + y

0ƒ

0y = nƒ(x, y)

- x2 a02ƒ

0x2b + 2xya 02ƒ

0x0yb + y2 a02ƒ

0y2b = n(n – 1)ƒ.

- Surface in polar coordinates Let

ƒ(r, u) = •

sin 6r

6r

, r _ 0

1, r = 0,

where r and u are polar coordinates. Find

- lim

rS0

ƒ(r, u) b. ƒr(0, 0) c. ƒu(r, u), r _ 0.

z _ f (r, u)

Gradients and Tangents

- Properties of position vectors Let r = xi + yj + zk and let

r = 0 r 0 .

- Show that _r = r>r.
- Show that _(rn) = nrn-2r.
- Find a function whose gradient equals r.
- Show that r # dr = r dr.
- Show that _(A # r) = A for any constant vector A.
- Gradient orthogonal to tangent Suppose that a differentiable

function ƒ(x, y) has the constant value c along the differentiable

curve x = g(t), y = h(t); that is,

ƒ(g(t), h(t)) = c

for all values of t. Differentiate both sides of this equation with

respect to t to show that _ƒ is orthogonal to the curve’s tangent

vector at every point on the curve.

- Curve tangent to a surface Show that the curve

r(t) = (ln t)i + (t ln t)j + t k

is tangent to the surface

xz2 – yz + cos xy = 1

at (0, 0, 1).

- Curve tangent to a surface Show that the curve

r(t) = at3

4 – 2bi + a4t

– 3bj + cos (t – 2)k

is tangent to the surface

x3 + y3 + z3 – xyz = 0

at (0, -1, 1).

Extreme Values

- Extrema on a surface Show that the only possible maxima and

minima of z on the surface z = x3 + y3 – 9xy + 27 occur at

(0, 0) and (3, 3). Show that neither a maximum nor a minimum

occurs at (0, 0). Determine whether z has a maximum or a minimum

at (3, 3).

- Maximum in closed first quadrant Find the maximum value

of ƒ(x, y) = 6xye-(2x+3y) in the closed first quadrant (includes the

nonnegative axes).

- Minimum volume cut from first octant Find the minimum

volume for a region bounded by the planes x = 0, y = 0, z = 0

and a plane tangent to the ellipsoid

x2

a2 +

y2

b2 +

z2

c2 = 1

at a point in the first octant.

- Minimum distance from a line to a parabola in xy-plane By

minimizing the function ƒ(x, y, u, y) = (x – u)2 + (y – y)2

subject to the constraints y = x + 1 and u = y2, find the minimum

distance in the xy-plane from the line y = x + 1 to the

parabola y2 = x.

Theory and Examples

- Boundedness of first partials implies continuity Prove the

following theorem: If ƒ(x, y) is defined in an open region R of the

xy-plane and if ƒx and ƒy are bounded on R, then ƒ(x, y) is continuous

on R. (The assumption of boundedness is essential.)

- Suppose that r(t) = g(t)i + h(t)j + k(t)k is a smooth curve in

the domain of a differentiable function ƒ(x, y, z). Describe the

relation between dƒ > dt, _ƒ, and v = dr>dt. What can be said

about _ƒ and v at interior points of the curve where ƒ has extreme

values relative to its other values on the curve? Give reasons for

your answer.

- Finding functions from partial derivatives Suppose that ƒ

and g are functions of x and y such that

0ƒ

0y =

0g

0x and

0ƒ

0x =

0g

0y ,

and suppose that

0ƒ

0x = 0, ƒ(1, 2) = g(1, 2) = 5, and ƒ(0, 0) = 4.

Find ƒ(x, y) and g(x, y).

- Rate of change of the rate of change We know that if ƒ(x, y) is a

function of two variables and if u = ai + bj is a unit vector, then

Du ƒ(x, y) = ƒx(x, y)a + ƒy(x, y)b is the rate of change of ƒ(x, y) at

(x, y) in the direction of u. Give a similar formula for the rate of

change of the rate of change of ƒ(x, y) at (x, y) in the direction u.

- Path of a heat-seeking particle A heat-seeking particle has the

property that at any point (x, y) in the plane it moves in the direction

of maximum temperature increase. If the temperature at (x, y)

is T(x, y) = -e-2y cos x, find an equation y = ƒ(x) for the path of

a heat-seeking particle at the point (p>4, 0).

- Velocity after a ricochet A particle traveling in a straight line

with constant velocity i + j – 5k passes through the point (0, 0,

30) and hits the surface z = 2×2 + 3y2. The particle ricochets off

the surface, the angle of reflection being equal to the angle of

incidence. Assuming no loss of speed, what is the velocity of the

particle after the ricochet? Simplify your answer.

- Directional derivatives tangent to a surface Let S be the surface

that is the graph of ƒ(x, y) = 10 – x2 – y2. Suppose that the

temperature in space at each point (x, y, z) is T(x, y, z) = x2y +

y2z + 4x + 14y + z.

- Among all the possible directions tangential to the surface S at

the point (0, 0, 10), which direction will make the rate of

change of temperature at (0, 0, 10) a maximum?

- Which direction tangential to S at the point (1, 1, 8) will make

the rate of change of temperature a maximum?

- Drilling another borehole On a flat surface of land, geologists

drilled a borehole straight down and hit a mineral deposit at

1000 ft. They drilled a second borehole 100 ft to the north of the

first and hit the mineral deposit at 950 ft. A third borehole 100 ft

east of the first borehole struck the mineral deposit at 1025 ft.

The geologists have reasons to believe that the mineral deposit is

in the shape of a dome, and for the sake of economy, they would

like to find where the deposit is closest to the surface. Assuming

the surface to be the xy-plane, in what direction from the first

borehole would you suggest the geologists drill their fourth

borehole?

The one-dimensional heat equation If w(x, t) represents the temperature

at position x at time t in a uniform wire with perfectly insulated

sides, then the partial derivatives wxx and wt satisfy a differential

equation of the form

wxx = 1

c2 wt .

This equation is called the one-dimensional heat equation. The value

of the positive constant c2 is determined by the material from which

the wire is made.

- Find all solutions of the one-dimensional heat equation of the

form w = ert sin px, where r is a constant.

- Find all solutions of the one-dimensional heat equation that have

the form w = ert sin kx and satisfy the conditions that w(0, t) = 0

and w(L, t) = 0. What happens to these solutions as tS q?

Chapter 14 Multiple Integrals

Chapter 14 Questions to Guide Your Review

- Define the double integral of a function of two variables over a

bounded region in the coordinate plane.

- How are double integrals evaluated as iterated integrals? Does the

order of integration matter? How are the limits of integration

determined? Give examples.

- How are double integrals used to calculate areas and average values.

Give examples.

- How can you change a double integral in rectangular coordinates

into a double integral in polar coordinates? Why might it be

worthwhile to do so? Give an example.

- Define the triple integral of a function ƒ(x, y, z) over a bounded

region in space.

- How are triple integrals in rectangular coordinates evaluated?

How are the limits of integration determined? Give an example.

- How are double and triple integrals in rectangular coordinates

used to calculate volumes, average values, masses, moments, and

centers of mass? Give examples.

- How are triple integrals defined in cylindrical and spherical coordinates?

Why might one prefer working in one of these coordinate

systems to working in rectangular coordinates?

- How are triple integrals in cylindrical and spherical coordinates

evaluated? How are the limits of integration found? Give

examples.

- How are substitutions in double integrals pictured as transformations

of two-dimensional regions? Give a sample calculation.

- How are substitutions in triple integrals pictured as transformations

of three-dimensional regions? Give a sample calculation.

Chapter 14 Practice Exercises

Evaluating Double Iterated Integrals

In Exercises 1–4, sketch the region of integration and evaluate the

double integral.

1.

L

10

1 L

1>y

0

yexy dx dy 2.

L

1

0 L

x3

0

ey>x dy dx

3.

L

3>2

0 L

29-4t2

-29-4t2

t ds dt 4.

L

1

0 L

2-2y

2y

xy dx dy

In Exercises 5–8, sketch the region of integration and write an equivalent

integral with the order of integration reversed. Then evaluate both

integrals.

5.

L

4

0 L

(y-4)>2

-24-y

dx dy 6.

L

1

0 L

x

x2

2x dy dx

7.

L

3>2

0 L

29-4y2

-29-4y2

y dx dy 8.

L

2

0 L

4-x2

0

2x dy dx

Evaluate the integrals in Exercises 9–12.

9.

L

1

0 L

2

2y

4 cos (x2) dx dy 10.

L

2

0 L

1

y>2

ex2 dx dy

11.

L

8

0 L

2

2 3

x

dy dx

y4 + 1

12.

L

1

0 L

1

2 3

y

2p sin px2

x2 dx dy

Areas and Volumes Using Double Integrals

- Area between line and parabola Find the area of the region

enclosed by the line y = 2x + 4 and the parabola y = 4 – x2 in

the xy-plane.

- Area bounded by lines and parabola Find the area of the “triangular”

region in the xy-plane that is bounded on the right by the

parabola y = x2, on the left by the line x + y = 2, and above by

the line y = 4.

- Volume of the region under a paraboloid Find the volume

under the paraboloid z = x2 + y2 above the triangle enclosed by

the lines y = x, x = 0, and x + y = 2 in the xy-plane.

- Volume of the region under a parabolic cylinder Find the volume

under the parabolic cylinder z = x2 above the region enclosed

by the parabola y = 6 – x2 and the line y = x in the xy-plane.

Average Values

Find the average value of ƒ(x, y) = xy over the regions in Exercises

17 and 18.

- The square bounded by the lines x = 1, y = 1 in the first quadrant
- The quarter circle x2 + y2 … 1 in the first quadrant

Polar Coordinates

Evaluate the integrals in Exercises 19 and 20 by changing to polar

coordinates.

19.

L

1

-1

L

21-x2

-21-x2

2 dy dx

(1 + x2 + y2)2

20.

L

1

-1

L

21-y2

-21-y2

ln (x2 + y2 + 1) dx dy

- Integrating over a lemniscate Integrate the function ƒ(x, y) =

1> (1 + x2 + y2)2 over the region enclosed by one loop of the

lemniscate (x2 + y2)2 – (x2 – y2) = 0.

- Integrate ƒ(x, y) = 1> (1 + x2 + y2)2 over
- Triangular region The triangle with vertices (0, 0), (1, 0),

and 11, 232.

- First quadrant The first quadrant of the xy-plane.

Evaluating Triple Iterated Integrals

Evaluate the integrals in Exercises 23–26.

23.

L

p

0 L

p

0 L

p

0

cos (x + y + z) dx dy dz

24.

L

ln 7

ln 6 L

ln 2

0 L

ln 5

ln 4

e(x+y+z) dz dy dx

25.

L

1

0 L

x2

0 L

x+y

0

(2x – y – z) dz dy dx

26.

L

e

1 L

x

1 L

z

0

2y

z3 dy dz dx

Volumes and Average Values Using Triple Integrals

- Volume Find the volume of the wedge-shaped region enclosed

on the side by the cylinder x = -cos y, -p>2 … y … p>2, on

the top by the plane z = -2x, and below by the xy-plane.

z

y

x

p2

_

2

x _ _cos y

z _ _2x

p

- Volume Find the volume of the solid that is bounded above by

the cylinder z = 4 – x2, on the sides by the cylinder x2 +

y2 = 4, and below by the xy-plane.

x

x2 + y2 _ 4

y

z

z _ 4 _ x2

- Average value Find the average value of ƒ(x, y, z) =

30xz 2×2 + y over the rectangular solid in the first octant bounded

by the coordinate planes and the planes x = 1, y = 3, z = 1.

- Average value Find the average value of r over the solid

sphere r … a (spherical coordinates).

Cylindrical and Spherical Coordinates

- Cylindrical to rectangular coordinates Convert

L

2p

0 L

22

0 L

24-r2

r

3 dz r dr du, r Ú 0

to (a) rectangular coordinates with the order of integration

dz dx dy and (b) spherical coordinates. Then (c) evaluate one

of the integrals.

- Rectangular to cylindrical coordinates (a) Convert to cylindrical

coordinates. Then (b) evaluate the new integral.

L

1

0 L

21-x2

-21-x2L

(x2+y2)

-(x2+y2)

21xy2 dz dy dx

- Rectangular to spherical coordinates (a) Convert to spherical

coordinates. Then (b) evaluate the new integral.

L

1

-1L

21-x2

-21-x2L

1

2×2+y2

dz dy dx

- Rectangular, cylindrical, and spherical coordinates Write an

iterated triple integral for the integral of ƒ(x, y, z) = 6 + 4y over

the region in the first octant bounded by the cone z = 2×2 + y2,

the cylinder x2 + y2 = 1, and the coordinate planes in (a) rectangular

coordinates, (b) cylindrical coordinates, and (c) spherical

coordinates. Then (d) find the integral of ƒ by evaluating one of

the triple integrals.

- Cylindrical to rectangular coordinates Set up an integral in

rectangular coordinates equivalent to the integral

L

p>2

0 L

23

1 L

24-r2

1

r3(sin u cos u)z2 dz dr du.

Arrange the order of integration to be z first, then y, then x.

- Rectangular to cylindrical coordinates The volume of a solid is

L

2

0 L

22x-x2

0 L

24-x2-y2

-24-x2-y2

dz dy dx.

- Describe the solid by giving equations for the surfaces that

form its boundary.

- Convert the integral to cylindrical coordinates but do not

evaluate the integral.

- Spherical versus cylindrical coordinates Triple integrals

involving spherical shapes do not always require spherical coordinates

for convenient evaluation. Some calculations may be

accomplished more easily with cylindrical coordinates. As a case

in point, find the volume of the region bounded above by the

sphere x2 + y2 + z2 = 8 and below by the plane z = 2 by using

(a) cylindrical coordinates and (b) spherical coordinates.

Masses and Moments

- Finding Iz in spherical coordinates Find the moment of inertia

about the z-axis of a solid of constant density d = 1 that is

bounded above by the sphere r = 2 and below by the cone

f = p>3 (spherical coordinates).

- Moment of inertia of a “thick” sphere Find the moment of

inertia of a solid of constant density d bounded by two concentric

spheres of radii a and b (a 6 b) about a diameter.

- Moment of inertia of an apple Find the moment of inertia

about the z-axis of a solid of density d = 1 enclosed by the

spherical coordinate surface r = 1 – cos f. The solid is the red

curve rotated about the z-axis in the accompanying figure.

z

y

x

r _ 1 _ cos f

- Centroid Find the centroid of the “triangular” region bounded

by the lines x = 2, y = 2 and the hyperbola xy = 2 in the

xy-plane.

- Centroid Find the centroid of the region between the parabola

x + y2 – 2y = 0 and the line x + 2y = 0 in the xy-plane.

- Polar moment Find the polar moment of inertia about the origin

of a thin triangular plate of constant density d = 3 bounded

by the y-axis and the lines y = 2x and y = 4 in the xy-plane.

- Polar moment Find the polar moment of inertia about the center

of a thin rectangular sheet of constant density d = 1 bounded

by the lines

- x = {2, y = {1 in the xy-plane
- x = {a, y = {b in the xy-plane.

(Hint: Find Ix . Then use the formula for Ix to find Iy , and add the

two to find I0 .)

- Inertial moment Find the moment of inertia about the x-axis of

a thin plate of constant density d covering the triangle with vertices

(0, 0), (3, 0), and (3, 2) in the xy-plane.

- Plate with variable density Find the center of mass and the

moments of inertia about the coordinate axes of a thin plate

bounded by the line y = x and the parabola y = x2 in the xyplane

if the density is d(x, y) = x + 1.

- Plate with variable density Find the mass and first moments

about the coordinate axes of a thin square plate bounded by the

lines x = {1, y = {1 in the xy-plane if the density is d(x, y) =

x2 + y2 + 1>3.

- Triangles with same inertial moment Find the moment of

inertia about the x-axis of a thin triangular plate of constant density

d whose base lies along the interval 30, b4 on the x-axis and

whose vertex lies on the line y = h above the x-axis. As you will

see, it does not matter where on the line this vertex lies. All such

triangles have the same moment of inertia about the x-axis.

- Centroid Find the centroid of the region in the polar coordinate

plane defined by the inequalities 0 … r … 3, -p>3 … u … p>3.

- Centroid Find the centroid of the region in the first quadrant

bounded by the rays u = 0 and u = p>2 and the circles r = 1

and r = 3.

- a. Centroid Find the centroid of the region in the polar coordinate

plane that lies inside the cardioid r = 1 + cos u and

outside the circle r = 1.

- Sketch the region and show the centroid in your sketch.
- a. Centroid Find the centroid of the plane region defined by

the polar coordinate inequalities 0 … r … a, -a … u … a

(0 6 a … p). How does the centroid move as a S p-?

- Sketch the region for a = 5p>6 and show the centroid in

your sketch.

Substitutions

- Show that if u = x – y and y = y, then for any continuous ƒ

L

q

0 L

x

0

e-sx ƒ(x – y, y) dy dx =

L

q

0 L

q

0

e-s(u+y) ƒ(u, y) du dy.

- What relationship must hold between the constants a, b, and c to

make

L

q

-q

L

q

-q

e-(ax2+2bxy+cy2) dx dy = 1?

(Hint: Let s = ax + by and t = gx + dy, where (ad – bg)2 =

ac – b2. Then ax2 + 2bxy + cy2 = s2 + t2.)

Chapter 14 Additional and Advanced Exercises

Volumes

- Sand pile: double and triple integrals The base of a sand pile

covers the region in the xy-plane that is bounded by the parabola

x2 + y = 6 and the line y = x. The height of the sand above the

point (x, y) is x2. Express the volume of sand as (a) a double integral,

(b) a triple integral. Then (c) find the volume.

- Water in a hemispherical bowl A hemispherical bowl of

radius 5 cm is filled with water to within 3 cm of the top. Find the

volume of water in the bowl.

- Solid cylindrical region between two planes Find the volume

of the portion of the solid cylinder x2 + y2 … 1 that lies between

the planes z = 0 and x + y + z = 2.

- Sphere and paraboloid Find the volume of the region bounded

above by the sphere x2 + y2 + z2 = 2 and below by the paraboloid

z = x2 + y2.

- Two paraboloids Find the volume of the region bounded above

by the paraboloid z = 3 – x2 – y2 and below by the paraboloid

z = 2×2 + 2y2.

- Spherical coordinates Find the volume of the region enclosed

by the spherical coordinate surface r = 2 sin f (see accompanying

figure).

z

x

y

r _ 2 sin f

- Hole in sphere A circular cylindrical hole is bored through a

solid sphere, the axis of the hole being a diameter of the sphere.

The volume of the remaining solid is

V = 2

L

2p

0 L

23

0 L

24-z2

1

r dr dz du.

- Find the radius of the hole and the radius of the sphere.
- Evaluate the integral.
- Sphere and cylinder Find the volume of material cut from the

solid sphere r2 + z2 … 9 by the cylinder r = 3 sin u.

- Two paraboloids Find the volume of the region enclosed by

the surfaces z = x2 + y2 and z = (x2 + y2 + 1)>2.

- Cylinder and surface z _ xy Find the volume of the region in

the first octant that lies between the cylinders r = 1 and r = 2

and that is bounded below by the xy-plane and above by the surface

z = xy.

Changing the Order of Integration

- Evaluate the integral

L

q

0

e-ax – e-bx

x dx.

(Hint: Use the relation

e-ax – e-bx

x =

L

b

a

e-xy dy

to form a double integral and evaluate the integral by changing

the order of integration.)

- a. Polar coordinates Show, by changing to polar coordinates,

that

L

a sin b

0

L

2a2-y2

y cot b

ln (x2 + y2) dx dy = a2b aln a – 1

2b,

where a 7 0 and 0 6 b 6 p>2.

- Rewrite the Cartesian integral with the order of integration

reversed.

- Reducing a double to a single integral By changing the order

of integration, show that the following double integral can be

reduced to a single integral:

L

x

0

L

u

0

em(x-t) ƒ(t) dt du =

L

x

0

(x – t)em(x-t) ƒ(t) dt.

Similarly, it can be shown that

L

x

0

L

y

0

L

u

0

em(x-t) ƒ(t) dt du dy =

L

x

0

(x – t)2

2

em(x-t) ƒ(t) dt.

- Transforming a double integral to obtain constant limits

Sometimes a multiple integral with variable limits can be changed

into one with constant limits. By changing the order of integration,

show that

L

1

0

ƒ(x)a

L

x

0

g(x – y)ƒ( y) dyb dx

=

L

1

0

ƒ( y)a

L

1

y

g(x – y)ƒ(x) dxb dy

= 1

2

L

1

0 L

1

0

g( 0 x – y 0 )ƒ(x)ƒ(y) dx dy.

Masses and Moments

- Minimizing polar inertia A thin plate of constant density is to

occupy the triangular region in the first quadrant of the xy-plane

having vertices (0, 0), (a, 0), and (a, 1 > a). What value of a will

minimize the plate’s polar moment of inertia about the origin?

- Polar inertia of triangular plate Find the polar moment of

inertia about the origin of a thin triangular plate of constant

density d = 3 bounded by the y-axis and the lines y = 2x and

y = 4 in the xy-plane.

- Mass and polar inertia of a counterweight The counterweight

of a flywheel of constant density 1 has the form of the smaller

segment cut from a circle of radius a by a chord at a distance b

from the center (b 6 a). Find the mass of the counterweight and

its polar moment of inertia about the center of the wheel.

- Centroid of a boomerang Find the centroid of the boomerangshaped

region between the parabolas y2 = -4(x – 1) and

y2 = -2(x – 2) in the xy-plane.

Theory and Examples

- Evaluate

L

a

0 L

b

0

emax (b2x2, a2y2) dy dx,

where a and b are positive numbers and

max (b2x2, a2y2) = e

b2x2 if b2x2 Ú a2y2

a2y2 if b2x2 6 a2y2.

- Show that

O

02F(x, y)

0x 0y dx dy

over the rectangle x0 … x … x1, y0 … y … y1, is

F(x1 , y1) – F(x0 , y1) – F(x1 , y0) + F(x0 , y0).

- Suppose that ƒ(x, y) can be written as a product ƒ(x, y) = F(x)G(y)

of a function of x and a function of y. Then the integral of ƒ over

the rectangle R: a … x … b, c … y … d can be evaluated as a

product as well, by the formula

OR

ƒ(x, y) dA = a

L

b

a

F(x) dxb a

L

d

c

G(y) dyb. (1)

The argument is that

OR

ƒ(x, y) dA =

L

d

c

a L

b

a

F(x)G( y) dxb dy (i)

=

L

d

c

aG( y)

L

b

a

F(x) dxb dy (ii)

=

L

d

c

a

L

b

a

F(x) dxbG( y) dy (iii)

= a

L

b

a

F(x) dxb

L

d

c

G( y) dy. (iv)

- Give reasons for steps (i) through (iv).

When it applies, Equation (1) can be a time-saver. Use it to evaluate

the following integrals.

b.

L

ln 2

0 L

p>2

0

ex cos y dy dx c.

L

2

1 L

1

-1

x

y2 dx dy

- Let Du ƒ denote the derivative of ƒ(x, y) = (x2 + y2) >2 in the

direction of the unit vector u = u1 i + u2 j.

- Finding average value Find the average value of Du ƒ

over the triangular region cut from the first quadrant by the

line x + y = 1.

- Average value and centroid Show in general that the

average value of Du ƒ over a region in the xy-plane is the

value of Du ƒ at the centroid of the region.

- The value of _(1,2) The gamma function,

_(x) =

L

q

0

tx-1 e-t dt,

extends the factorial function from the nonnegative integers to

other real values. Of particular interest in the theory of differential

equations is the number

_a1

2b =

L

q

0

t(1>2)-1 e-t dt =

L

q

0

e-t

2t

- (2)
- If you have not yet done Exercise 41 in Section 14.4, do it

now to show that

I =

L

q

0

e-y2 dy =

2p

2

.

- Substitute y = 2t in Equation (2) to show that

_(1>2) = 2I = 2p.

- Total electrical charge over circular plate The electrical

charge distribution on a circular plate of radius R meters is

s(r, u) = kr(1 – sin u) coulomb>m2 (k a constant). Integrate s

over the plate to find the total charge Q.

- A parabolic rain gauge A bowl is in the shape of the graph of

z = x2 + y2 from z = 0 to z = 10 in. You plan to calibrate the

bowl to make it into a rain gauge. What height in the bowl would

correspond to 1 in. of rain? 3 in. of rain?

- Water in a satellite dish A parabolic satellite dish is 2 m wide

and 1 > 2 m deep. Its axis of symmetry is tilted 30 degrees from the

vertical.

- Set up, but do not evaluate, a triple integral in rectangular coordinates

that gives the amount of water the satellite dish will

hold. (Hint: Put your coordinate system so that the satellite dish

is in “standard position” and the plane of the water level is

slanted.) (Caution: The limits of integration are not “nice.”)

- What would be the smallest tilt of the satellite dish so that it

holds no water?

- An infinite half-cylinder Let D be the interior of the infinite

right circular half-cylinder of radius 1 with its single-end face

suspended 1 unit above the origin and its axis the ray from (0, 0,

1) to q. Use cylindrical coordinates to evaluate

lD

z(r2 + z2)-5>2 dV.

- Hypervolume We have learned that 1

b

a 1 dx is the length of the

interval 3a, b4 on the number line (one-dimensional space),

4R 1 dA is the area of region R in the xy-plane (two-dimensional

space), and 7D 1 dV is the volume of the region D in threedimensional

space (xyz-space). We could continue: If Q is a

region in 4-space (xyzw-space), then |Q 1 dV is the “hypervolume”

of Q. Use your generalizing abilities and a Cartesian

coordinate system of 4-space to find the hypervolume inside the

unit 4-dimensional sphere x2 + y2 + z2 + w2 = 1.

Chapter 15 Integrals and Vector Fields

Chapter 15 Questions to Guide Your Review

- What are line integrals of scalar functions? How are they evaluated?

Give examples.

- How can you use line integrals to find the centers of mass of

springs or wires? Explain.

- What is a vector field? What is the line integral of a vector field?

What is a gradient field? Give examples.

- What is the flow of a vector field along a curve? What is the work

done by vector field moving an object along a curve? How do you

calculate the work done? Give examples.

- What is the Fundamental Theorem of line integrals? Explain how

it relates to the Fundamental Theorem of Calculus.

- Specify three properties that are special about conservative fields.

How can you tell when a field is conservative?

- What is special about path independent fields?
- What is a potential function? Show by example how to find a

potential function for a conservative field.

- What is a differential form? What does it mean for such a form to

be exact? How do you test for exactness? Give examples.

- What is Green’s Theorem? Discuss how the two forms of Green’s

Theorem extend the Net Change Theorem in Chapter 5.

- How do you calculate the area of a parametrized surface in space?

Of an implicitly defined surface F(x, y, z) = 0? Of the surface

which is the graph of z = ƒ(x, y)? Give examples.

- How do you integrate a scalar function over a parametrized surface?

Of surfaces that are defined implicitly or in explicit form?

Give examples.

- What is an oriented surface? What is the surface integral of a

vector field in three-dimensional space over an oriented surface?

How is it related to the net outward flux of the field? Give

examples.

- What is the curl of a vector field? How can you interpret it?
- What is Stokes’ Theorem? Explain how it generalizes Green’s

Theorem to three dimensions.

- What is the divergence of a vector field? How can you interpret it?
- What is the Divergence Theorem? Explain how it generalizes

Green’s Theorem to three dimensions.

- How do Green’s Theorem, Stokes’ Theorem, and the Divergence

Theorem relate to the Fundamental Theorem of Calculus for ordinary

single integrals?

Chapter 15 Practice Exercises

Evaluating Line Integrals

- The accompanying figure shows two polygonal paths in space

joining the origin to the point (1, 1, 1). Integrate ƒ(x, y, z) =

2x – 3y2 – 2z + 3 over each path.

z

y

x

(0, 0, 0) (1, 1, 1)

(1, 1, 0)

Path 1

z

y

x

(0, 0, 0) (1, 1, 1)

(1, 1, 0)

Path 2

- The accompanying figure shows three polygonal paths joining

the origin to the point (1, 1, 1). Integrate ƒ(x, y, z) = x2 + y – z

over each path.

- Integrate ƒ(x, y, z) = 2×2 + z2 over the circle

r(t) = (a cos t)j + (a sin t)k, 0 … t … 2p.

- Integrate ƒ(x, y, z) = 2×2 + y2 over the involute curve

r(t) = (cos t + t sin t)i + (sin t – t cos t)j, 0 … t … 23.

Evaluate the integrals in Exercises 5 and 6.

5.

L

(4,-3,0)

(-1,1,1)

dx + dy + dz

2x + y + z

6.

L

(10,3,3)

(1,1,1)

dx – A

zy

dy – A

y

z dz

- Integrate F = -(y sin z)i + (x sin z)j + (xy cos z)k around the

circle cut from the sphere x2 + y2 + z2 = 5 by the plane

z = -1, clockwise as viewed from above.

- Integrate F = 3x2yi + (x3 + 1)j + 9z2k around the circle cut

from the sphere x2 + y2 + z2 = 9 by the plane x = 2.

Evaluate the integrals in Exercises 9 and 10.

9.

LC

8x sin y dx – 8y cos x dy

C is the square cut from the first quadrant by the lines x = p>2

and y = p>2.

10.

LC

y2 dx + x2 dy

C is the circle x2 + y2 = 4.

Finding and Evaluating Surface Integrals

- Area of an elliptical region Find the area of the elliptical

region cut from the plane x + y + z = 1 by the cylinder

x2 + y2 = 1.

- Area of a parabolic cap Find the area of the cap cut from the

paraboloid y2 + z2 = 3x by the plane x = 1.

- Area of a spherical cap Find the area of the cap cut from the

top of the sphere x2 + y2 + z2 = 1 by the plane z = 22>2.

- a. Hemisphere cut by cylinder Find the area of the surface cut

from the hemisphere x2 + y2 + z2 = 4, z Ú 0, by the cylinder

x2 + y2 = 2x.

- Find the area of the portion of the cylinder that lies inside the

hemisphere. (Hint: Project onto the xz-plane. Or evaluate the

integral 1h ds, where h is the altitude of the cylinder and ds

is the element of arc length on the circle x2 + y2 = 2x in the

xy-plane.)

z

x

Cylinder r _ 2 cos u y

Hemisphere

z _ “4 _ r2

- Area of a triangle Find the area of the triangle in which the

plane (x>a) + ( y>b) + (z>c) = 1 (a, b, c 7 0) intersects the first

octant. Check your answer with an appropriate vector calculation.

- Parabolic cylinder cut by planes Integrate
- g(x, y, z) =

yz

24y2 + 1

- g(x, y, z) =

z

24y2 + 1

over the surface cut from the parabolic cylinder y2 – z = 1 by

the planes x = 0, x = 3, and z = 0.

- Circular cylinder cut by planes Integrate g(x, y, z) =

x4y(y2 + z2) over the portion of the cylinder y2 + z2 = 25 that

lies in the first octant between the planes x = 0 and x = 1 and

above the plane z = 3.

- Area of Wyoming The state of Wyoming is bounded by the

meridians 111_3_ and 104_3_ west longitude and by the circles

41° and 45° north latitude. Assuming that Earth is a sphere of

radius R = 3959 mi, find the area of Wyoming.

Parametrized Surfaces

Find parametrizations for the surfaces in Exercises 19–24. (There are

many ways to do these, so your answers may not be the same as those

in the back of the book.)

- Spherical band The portion of the sphere x2 + y2 + z2 = 36

between the planes z = -3 and z = 323

- Parabolic cap The portion of the paraboloid z = – (x2 + y2) >2

above the plane z = -2

- Cone The cone z = 1 + 2×2 + y2, z … 3
- Plane above square The portion of the plane 4x + 2y + 4z =

12 that lies above the square 0 … x … 2, 0 … y … 2 in the first

quadrant

- Portion of paraboloid The portion of the paraboloid y =

2(x2 + z2), y … 2, that lies above the xy-plane

- Portion of hemisphere The portion of the hemisphere

x2 + y2 + z2 = 10, y Ú 0, in the first octant

- Surface area Find the area of the surface

r(u, y) = (u + y)i + (u – y)j + yk,

0 … u … 1, 0 … y … 1.

- Surface integral Integrate ƒ(x, y, z) = xy – z2 over the surface

in Exercise 25.

- Area of a helicoid Find the surface area of the helicoid

r(r, u) = (r cos u)i + (r sin u)j + uk, 0 … u … 2p, 0 … r … 1,

in the accompanying figure.

y

z

x

(1, 0, 0)

(1, 0, 2p)

2p

- Surface integral Evaluate the integral 4S 2×2 + y2 + 1 ds,

where S is the helicoid in Exercise 27.

Conservative Fields

Which of the fields in Exercises 29–32 are conservative, and which

are not?

- F = x i + y j + z k
- F = (x i + y j + zk )>(x2 + y2 + z2)3>2
- F = xeyi + yezj + zexk
- F = (i + zj + yk)>(x + yz)

Find potential functions for the fields in Exercises 33 and 34.

- F = 2i + (2y + z)j + (y + 1)k
- F = (z cos xz)i + eyj + (x cos xz)k

Work and Circulation

In Exercises 35 and 36, find the work done by each field along the

paths from (0, 0, 0) to (1, 1, 1) in Exercise 1.

- F = 2xy i + j + x2 k 36. F = 2xy i + x2 j + k
- Finding work in two ways Find the work done by

F =

xi + yj

(x2 + y2)3>2

over the plane curve r(t) = (et cos t)i + (et sin t)j from the point

(1, 0) to the point (e2p, 0) in two ways:

- By using the parametrization of the curve to evaluate the work

integral.

- By evaluating a potential function for F.
- Flow along different paths Find the flow of the field F =

_(x2zey)

- once around the ellipse C in which the plane x + y + z = 1

intersects the cylinder x2 + z2 = 25, clockwise as viewed

from the positive y-axis.

- along the curved boundary of the helicoid in Exercise 27 from

(1, 0, 0) to (1, 0, 2p).

In Exercises 39 and 40, use the curl integral in Stokes’ Theorem to find the

circulation of the field F around the curve C in the indicated direction.

- Circulation around an ellipse F = y2 i – y j + 3z2 k

C: The ellipse in which the plane 2x + 6y – 3z = 6 meets the

cylinder x2 + y2 = 1, counterclockwise as viewed from above

- Circulation around a circle F = (x2 + y)i + (x + y)j +

(4y2 – z)k

C: The circle in which the plane z = -y meets the sphere

x2 + y2 + z2 = 4, counterclockwise as viewed from above

Masses and Moments

- Wire with different densities Find the mass of a thin wire lying

along the curve r(t) = 22t i + 22t j + (4 – t2)k, 0 … t … 1,

if the density at t is (a) d = 3t and (b) d = 1.

- Wire with variable density Find the center of mass of a thin

wire lying along the curve r(t) = t i + 2t j + (2>3)t3>2 k,

0 … t … 2, if the density at t is d = 325 + t.

- Wire with variable density Find the center of mass and the

moments of inertia about the coordinate axes of a thin wire lying

along the curve

r(t) = t i +

222

3

t3>2 j + t2

2

k, 0 … t … 2,

if the density at t is d = 1>(t + 1).

- Center of mass of an arch A slender metal arch lies along the

semicircle y = 2a2 – x2 in the xy-plane. The density at the point

(x, y) on the arch is d(x, y) = 2a – y. Find the center of mass.

- Wire with constant density A wire of constant density d = 1

lies along the curve r(t) = (et cos t)i + (et sin t)j + et k, 0 …

t … ln 2. Find z and Iz .

- Helical wire with constant density Find the mass and center

of mass of a wire of constant density d that lies along the helix

r(t) = (2 sin t)i + (2 cos t)j + 3t k, 0 … t … 2p.

- Inertia and center of mass of a shell Find Iz and the center of

mass of a thin shell of density d(x, y, z) = z cut from the upper

portion of the sphere x2 + y2 + z2 = 25 by the plane z = 3.

- Moment of inertia of a cube Find the moment of inertia about

the z-axis of the surface of the cube cut from the first octant by

the planes x = 1, y = 1, and z = 1 if the density is d = 1.

Flux Across a Plane Curve or Surface

Use Green’s Theorem to find the counterclockwise circulation and

outward flux for the fields and curves in Exercises 49 and 50.

- Square F = (2xy + x)i + (xy – y)j

C: The square bounded by x = 0, x = 1, y = 0, y = 1

- Triangle F = (y – 6×2)i + (x + y2)j

C: The triangle made by the lines y = 0, y = x, and x = 1

- Zero line integral Show that

F

C

ln x sin y dy –

cos y

x dx = 0

for any closed curve C to which Green’s Theorem applies.

- a. Outward flux and area Show that the outward flux of the

position vector field F = xi + yj across any closed curve to

which Green’s Theorem applies is twice the area of the region

enclosed by the curve.

- Let n be the outward unit normal vector to a closed curve to

which Green’s Theorem applies. Show that it is not possible

for F = x i + y j to be orthogonal to n at every point of C.

In Exercises 53–56, find the outward flux of F across the boundary

of D.

- Cube F = 2xyi + 2yzj + 2xzk

D: The cube cut from the first octant by the planes x = 1, y = 1,

z = 1

- Spherical cap F = xz i + yz j + k

D: The entire surface of the upper cap cut from the solid sphere

x2 + y2 + z2 … 25 by the plane z = 3

- Spherical cap F = -2x i – 3y j + z k

D: The upper region cut from the solid sphere x2 + y2 + z2 … 2

by the paraboloid z = x2 + y2

- Cone and cylinder F = (6x + y)i – (x + z)j + 4yz k

D: The region in the first octant bounded by the cone z = 2×2 + y2,

the cylinder x2 + y2 = 1, and the coordinate planes

- Hemisphere, cylinder, and plane Let S be the surface that is

bounded on the left by the hemisphere x2 + y2 + z2 = a2, y … 0,

in the middle by the cylinder x2 + z2 = a2, 0 … y … a, and on

the right by the plane y = a. Find the flux of F = y i + z j + x k

outward across S.

- Cylinder and planes Find the outward flux of the field

F = 3xz2

i + y j – z3

k across the surface of the solid in the first

octant that is bounded by the cylinder x2 + 4y2 = 16 and the

planes y = 2z, x = 0, and z = 0.

- Cylindrical can Use the Divergence Theorem to find the flux of

F = xy2i + x2yj + yk outward through the surface of the region

Chapter 15 Additional and Advanced Exercises

Finding Areas with Green’s Theorem

Use the Green’s Theorem area formula in Exercises 15.4 to find the

areas of the regions enclosed by the curves in Exercises 1– 4.

- The limaçon x = 2 cos t – cos 2t, y = 2 sin t – sin 2t,

0 … t … 2p

y

x

0 1

- The deltoid x = 2 cos t + cos 2t, y = 2 sin t – sin 2t,

0 … t … 2p

y

x

0 3

- The eight curve x = (1>2) sin 2t, y = sin t, 0 … t … p (one loop)

y

x

1

_1

- The teardrop x = 2a cos t – a sin 2t, y = b sin t, 0 … t … 2p

y

x

0

b

2a

Theory and Applications

- a. Give an example of a vector field F (x, y, z) that has value 0 at

only one point and such that curl F is nonzero everywhere. Be

sure to identify the point and compute the curl.

- Give an example of a vector field F (x, y, z) that has value 0 on

precisely one line and such that curl F is nonzero everywhere.

Be sure to identify the line and compute the curl.

- Give an example of a vector field F (x, y, z) that has value 0 on

a surface and such that curl F is nonzero everywhere. Be sure

to identify the surface and compute the curl.

- Find all points (a, b, c) on the sphere x2 + y2 + z2 = R2 where

the vector field F = yz2i + xz2j + 2xyzk is normal to the surface

and F(a, b, c) _ 0.

- Find the mass of a spherical shell of radius R such that at each

point (x, y, z) on the surface the mass density d(x, y, z) is its distance

to some fixed point (a, b, c) of the surface.

- Find the mass of a helicoid

r(r, u) = (r cos u)i + (r sin u)j + u k,

0 … r … 1, 0 … u … 2p, if the density function is d(x, y, z) =

22×2 + y2. See Practice Exercise 27 for a figure.

- Among all rectangular regions 0 … x … a, 0 … y … b, find the

one for which the total outward flux of F = (x2 + 4xy)i – 6yj

across the four sides is least. What is the least flux?

- Find an equation for the plane through the origin such that the

circulation of the flow field F = z i + x j + y k around the circle

of intersection of the plane with the sphere x2 + y2 + z2 = 4 is a

maximum.

- A string lies along the circle x2 + y2 = 4 from (2, 0) to (0, 2) in

the first quadrant. The density of the string is r (x, y) = xy.

- Partition the string into a finite number of subarcs to show that

the work done by gravity to move the string straight down to

the x-axis is given by

Work = lim

nSq

a

n

k=1

g xk yk 2_sk =

LC

g xy2 ds,

where g is the gravitational constant.

- Find the total work done by evaluating the line integral in part (a).
- Show that the total work done equals the work required to move

the string’s center of mass (x, y) straight down to the x-axis.

- A thin sheet lies along the portion of the plane x + y + z = 1 in

the first octant. The density of the sheet is d (x, y, z) = xy.

- Partition the sheet into a finite number of subpieces to show

that the work done by gravity to move the sheet straight down

to the xy-plane is given by

Work = lim

nSq

a

n

k=1

g xk yk zk _sk =

OS

g xyz ds,

where g is the gravitational constant.

- Find the total work done by evaluating the surface integral in

part (a).

- Show that the total work done equals the work required to

move the sheet’s center of mass (x, y, z) straight down to the

xy-plane.

- Archimedes’ principle If an object such as a ball is placed in a

liquid, it will either sink to the bottom, float, or sink a certain distance

and remain suspended in the liquid. Suppose a fluid has

constant weight density w and that the fluid’s surface coincides

with the plane z = 4. A spherical ball remains suspended in the

fluid and occupies the region x2 + y2 + (z – 2)2 … 1.

- Show that the surface integral giving the magnitude of the

total force on the ball due to the fluid’s pressure is

Force = lim

nSq a

n

k=1

w(4 – zk) _sk =

OS

w(4 – z) ds.

- Since the ball is not moving, it is being held up by the buoyant

force of the liquid. Show that the magnitude of the buoyant

force on the sphere is

Buoyant force =

OS

w(z – 4)k # n ds,

where n is the outer unit normal at (x, y, z). This illustrates

Archimedes’ principle that the magnitude of the buoyant force

on a submerged solid equals the weight of the displaced fluid.

- Use the Divergence Theorem to find the magnitude of the

buoyant force in part (b).

- Let

F = –

GmM

0 r 0 3 r

be the gravitational force field defined for r _ 0. Use Gauss’s

law in Section 15.8 to show that there is no continuously differentiable

vector field H satisfying F = _ * H.

- If ƒ(x, y, z) and g(x, y, z) are continuously differentiable scalar

functions defined over the oriented surface S with boundary curve

C, prove that

OS

(_ƒ * _g) # n ds =

FC

ƒ _g # dr.

- Suppose that _ # F1 = _ # F2 and _ * F1 = _ * F2 over a

region D enclosed by the oriented surface S with outward unit

normal n and that F1 # n = F2 # n on S. Prove that F1 = F2

throughout D.

- Prove or disprove that if _ # F = 0 and _ * F = 0, then F = 0.
- Let S be an oriented surface parametrized by r(u, y). Define the

notation dS = ru du * ry dy so that dS is a vector normal to the

surface. Also, the magnitude ds = 0 dS0 is the element of surface

area (by Equation 5 in Section 15.5). Derive the identity

ds = (EG – F2)1>2 du dy

where

E = 0 ru 0 2, F = ru # ry , and G = 0 ry 0 2.

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