# Solution Manual For University Calculus Early Transcendentals 3rd Edition, Global Edition By Joel R. Hass

$25.00 Category: ## Description COMPLETE TEXT BOOK SOLUTION WITH ANSWERS ORDER WILL BE DELIVER WITHIN FEW HOURS SAMPLE QUESTIONS Chapter 1 Functions Functions In Exercises 1–6, find the domain and range of each function. 1. ƒ(x) = 18 + x2 2. ƒ(x) = 1 – 2x 2. F(x) = 25x + 10 4. g(x) = 2×2 – 3x 3. ƒ(t) = 4 3 – t 1. 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. 1. a. x y 0 b. x y 0 1. a. x y 0 b. x y 0 Finding Formulas for Functions 1. Express the area and perimeter of an equilateral triangle as a function of the triangle’s side length x. 1. 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. 1. 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. 1. 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. 1. 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. 1. 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. 1. ƒ(x) = 5 – 2x 16. ƒ(x) = 1 – 2x – x2 2. g(x) = 20 x 0 18. g(x) = 2-x 3. F(t) = t> 0 t 0 20. G(t) = 1> 0 t 0 4. Find the domain of y = x + 7 12 – 2×2 – 25 . 1. Find the range of y = 2 + x2 x2 + 4 . 1. Graph the following equations and explain why they are not graphs of functions of x. 1. 0 y 0 = x b. y2 = x2 2. Graph the following equations and explain why they are not graphs of functions of x. 1. 0 x 0 + 0 y 0 = 1 b. 0 x + y 0 = 1 Piecewise-Defined Functions Graph the functions in Exercises 25–28. 1. ƒ(x) = e x, 0 … x … 1 2 – x, 1 6 x … 2 1. g(x) = e 1 – x, 0 … x … 1 2 – x, 1 6 x … 2 1. F(x) = e 4 – x2, x … 1 x2 + 2x, x 7 1 1. G(x) = e 1>x, x 6 0 x, 0 … x Find a formula for each function graphed in Exercises 29–32. 1. a. x y 0 1 2 (1, 1) b. t y 0 2 1 2 3 4 1. a. x y 2 5 2 (2, 1) b. _1 x y 3 1 2 2 1 _2 _3 _1 (2, _1) 1. a. x y 3 1 (_1, 1) (1, 1) b. x y 1 2 (_2, _1) (1, _1) (3, _1) 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 1. For what values of x is 2. :x; = -1? b. <x= = 0? 3. What real numbers x satisfy the equation :x; = <x=? 4. Does <-x= = -:x; for all real x? Give reasons for your answer. 5. 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. 1. y = -x3 38. y = – 1 x2 1. y = – 1x 1. y = 1 0 x 0 1. y = 20 x 0 42. y = 2-x 2. y = x3>8 44. y = -42x 3. 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. 1. ƒ(x) = -8 48. ƒ(x) = x-5 2. g(x) = 9×3 – 3 50. ƒ(x) = x2 + x 3. g(x) = x3 + x 52. g(x) = x4 + 3×2 – 1 4. g(x) = 1 x2 – 4 1. g(x) = x x2 – 1 1. h(t) = 1 t3 + 1 1. h(t) = _ t3 _ 2. h(t) = 5t – 1 58. h(t) = 2 _ t _ + 1 Theory and Examples 1. The variable s is proportional to t, and s = 15 when t = 105. Determine t when s = 40. 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? 1. The variables r and s are inversely proportional, and r = 7 when s = 4. Determine s when r = 10. 1. 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? 1. 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 1. The accompanying figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long. 1. Express the y-coordinate of P in terms of x. (You might start by writing an equation for the line AB.) 1. 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. 1. a. y = x4 b. y = x7 c. y = x10 x y f g h 0 1. a. y = 5x b. y = 5x c. y = x5 x y f h g 0 1. 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 . 1. Confirm your findings in part (a) algebraically. 2. 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 . 1. Confirm your findings in part (a) algebraically. 2. 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. 1. 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.

1. 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.

1. 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

1. 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.

1. 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.

1. a.

x

y

3

1

(_1, 1) (1, 1)

b.

x

y

1

2

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

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

1. For what values of x is
2. :x; = -1? b. <x= = 0?
3. What real numbers x satisfy the equation :x; = <x=?
4. Does <-x= = -:x; for all real x? Give reasons for your answer.
5. 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.

1. y = -x3 38. y = –

1

x2

1. y = –

1x

1. y = 1

0 x 0

1. y = 20 x 0 42. y = 2-x
2. y = x3>8 44. y = -42x
3. 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.

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

x2 – 4

1. g(x) = x

x2 – 1

1. h(t) = 1

t3 + 1

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

Theory and Examples

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

Determine t when s = 40.

1. 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.

1. 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.

1. 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.

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

12 – 2×2 – 25

.

1. Find the range of y = 2 + x2

x2 + 4 .

1. Graph the following equations and explain why they are not

graphs of functions of x.

1. 0 y 0 = x b. y2 = x2
2. Graph the following equations and explain why they are not

graphs of functions of x.

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

Piecewise-Defined Functions

Graph the functions in Exercises 25–28.

1. ƒ(x) = e

x, 0 … x … 1

2 – x, 1 6 x … 2

1. g(x) = e

1 – x, 0 … x … 1

2 – x, 1 6 x … 2

1. F(x) = e

4 – x2, x … 1

x2 + 2x, x 7 1

1. G(x) = e

1>x, x 6 0

x, 0 … x

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

1. a.

x

y

0

1

2

(1, 1)

b.

t

y

0

2

1 2 3 4

1. a.

x

y

2 5

2

(2, 1)

b.

_1

x

y

3

1 2

2

1

_2

_3

_1

(2, _1)

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?

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

s = 4. Determine s when r = 10.

1. 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?

1. 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

1. The accompanying figure shows a rectangle inscribed in an isosceles

right triangle whose hypotenuse is 2 units long.

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

by writing an equation for the line AB.)

1. 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

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

x

y

f

g

h

0

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

x

y

f

h

g

0

1. 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

.

1. Confirm your findings in part (a) algebraically.
2. 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

.

1. Confirm your findings in part (a) algebraically.
2. 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.

1. 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. 1. 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. 1. 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 1. 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. 1. 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. 1. ƒ(x) = x4 – 7×2 + 6x 2. 3-1, 14 by 3-1, 14 b. 3-2, 24 by 3-5, 54 3. 3-10, 104 by 3-10, 104 d. 3-5, 54 by 3-25, 154 4. ƒ(x) = x3 – 4×2 – 4x + 16 5. 3-1, 14 by 3-5, 54 b. 3-3, 34 by 3-10, 104 6. 3-5, 54 by 3-10, 204 d. 3-20, 204 by 3-100, 1004 7. ƒ(x) = 5 + 12x – x3 8. 3-1, 14 by 3-1, 14 b. 3-5, 54 by 3-10, 104 9. 3-4, 44 by 3-20, 204 d. 3-4, 54 by 3-15, 254 10. ƒ(x) = 25 + 4x – x2 11. 3-2, 24 by 3-2, 24 b. 3-2, 64 by 3-1, 44 12. 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. 1. ƒ(x) = x4 – 4×3 + 15 6. ƒ(x) = x3 3 – x2 2 – 2x + 1 1. ƒ(x) = x5 – 5×4 + 10 8. ƒ(x) = 4×3 – x4 2. ƒ(x) = x29 – x2 10. ƒ(x) = x2(6 – x3) T T 1. y = 2x – 3×2>3 12. y = x1>3(x2 – 8) 2. y = 5×2>5 – 2x 14. y = x2>3(5 – x) 3. y = 0 x2 – 1 0 16. y = 0 x2 – x 0 4. y = x + 3 x + 2 1. y = 1 – 1 x + 3 1. ƒ(x) = x2 + 2 x2 + 1 1. ƒ(x) = x2 – 1 x2 + 1 1. ƒ(x) = x – 1 x2 – x – 6 1. ƒ(x) = 8 x2 – 9 1. ƒ(x) = 6×2 – 15x + 6 4×2 – 10x 1. ƒ(x) = x2 – 3 x – 2 1. y = sin 250x 26. y = 3 cos 60x 2. y = cos a x 50 b 28. y = 1 10 sin a x 10b 1. 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. 1. Graph the lower half of the circle defined by the equation x2 + 2x = 4 + 4y – y2. 1. Graph the upper branch of the hyperbola y2 – 16×2 = 1. 2. Graph four periods of the function ƒ(x) = – tan 2x. 3. Graph two periods of the function ƒ(x) = 3 cot x 2 + 1. 1. Graph the function ƒ(x) = sin 2x + cos 3x. 2. 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. 1. y = 2x, y = 4x, y = 3-x, y = (1>5)x 2. y = 3x, y = 8x, y = 2-x, y = (1>4)x 3. y = 2-t and y = -2t 4. y = 3-t and y = -3t 4. y = ex and y = 1>ex 6. y = -ex and y = -e-x In each of Exercises 7–10, sketch the shifted exponential curves. 1. y = 2x – 1 and y = 2-x – 1 2. y = 3x + 2 and y = 3-x + 2 3. y = 1 – ex and y = 1 – e-x 4. y = -1 – ex and y = -1 – Applying the Laws of Exponents Use the laws of exponents to simplify the expressions in Exercises 11–20. 1. 497 # 49-6.5 12. 91>3 # 91>6 2. 44.2 43.7 14. 35>3 32>3 1. 1641>1222 16. 11322222>2 2. 223 # 723 18. 12321>2 # 121221>2 3. a 2 22 b 4 1. a26 3 b 2 Composites Involving Exponential Functions Find the domain and range for each of the functions in Exercises 21–24. 1. ƒ(x) = 1 2 + ex 22. g(t) = cos (e-t) 1. g(t) = 21 + 3-t 24. ƒ(x) = 3 1 – e2x Applications In Exercises 25–28, use graphs to find approximate solutions. 1. 2x = 5 26. ex = 4 2. 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. 1. 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? 1. 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. 1. Estimate the population in 1915 and 1940. 2. Approximately when did the population reach 50,000? 3. Radioactive decay The half-life of phosphorus-32 is about 14 days. There are 6.6 grams present initially. 1. Express the amount of phosphorus-32 remaining as a function of time t. 1. When will there be 1 gram remaining? 2. 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? 1. 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. 1. 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. 1. 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? 1. 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 1. to reduce the number of cases to 1000? 2. 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 1. 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. 1. ƒ(x) = e 3 – x, x 6 0 3, x Ú 0 1. ƒ(x) = e 2x + 6, x … -3 x + 4, x 7 -3 1. ƒ(x) = d 1 – x 2 , x … 0 x x + 2 , x 7 0 1. 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. 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 1. 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 _ 1. 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 1. a. Graph the function ƒ(x) = 21 – x2, 0 … x … 1. What symmetry does the graph have? 1. Show that ƒ is its own inverse. (Remember that 2×2 = x if x Ú 0.) 1. a. Graph the function ƒ(x) = 1>x. What symmetry does the graph have? 1. 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. 1. 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) 1. 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) 1. 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. 1. ƒ(x) = x5 26. ƒ(x) = x4, x Ú 0 2. ƒ(x) = x3 + 1 28. ƒ(x) = (1>2)x – 7>2 3. ƒ(x) = 1>x2, x 7 0 30. ƒ(x) = 1>x3, x _ 0 4. ƒ(x) = x + 3 x – 2 1. ƒ(x) = 2x 2x – 3 1. ƒ(x) = x2 – 2x, x … 1 34. ƒ(x) = (2×3 + 1)1>5 (Hint: Complete the square.) 1. ƒ(x) = x + b x – 2 , b 7 -2 and constant 1. ƒ(x) = x2 – 2bx, b 7 0 and constant, x … b Inverses of Lines 1. a. Find the inverse of the function ƒ(x) = mx, where m is a constant different from zero. 1. 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? 1. 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. 1. 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. 1. Find the inverse of ƒ(x) = x + b (b constant). How is the graph of ƒ -1 related to the graph of ƒ? 1. What can you conclude about the inverses of functions whose graphs are lines parallel to the line y = x? 1. 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? 1. Find the inverse of ƒ(x) = -x + b (b constant). What angle does the line y = -x + b make with the line y = x? 1. What can you conclude about the inverses of functions whose graphs are lines perpendicular to the line y = x? Logarithms and Exponentials 1. Express the following logarithms in terms of ln 2 and ln 3. 2. ln 0.75 b. ln (4>9) 3. ln (1>2) d. ln23 9 1. ln 322 f. ln 213.5 2. Express the following logarithms in terms of ln 5 and ln 7. 3. ln (1>125) b. ln 9.8 4. ln 727 d. ln 1225 5. 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. 1. a. ln sin u – ln asin u 2 b b. ln (3×2 – 9x) + ln a 1 3xb 1. 1 2 ln (4t4) – ln b 1. a. ln sec u + ln cos u b. ln (8x + 4) – 2 ln c 2. 3 ln23 t2 – 1 – ln (t + 1) Find simpler expressions for the quantities in Exercises 45–48. 1. a. eln 8.3 b. e-ln 6×6 c. eln 3x-ln 5y 2. a. eln (x2+y2) b. e-ln 0.3 c. eln px-ln 2 3. a. 2 ln 2e b. ln (ln ee) c. ln (e-x2-y2) 4. 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. 1. ln y = 4t + 5 50. ln y = -t + 5 2. ln (y – 30) = 5t 52. ln (c – 2y) = t 3. ln (y – 4) – ln 5 = x + ln x 4. ln (y2 – 1) – ln (y + 1) = ln (sin x) In Exercises 55 and 56, solve for k. 1. a. e3k = 27 b. 35e8k = 175 c. ek>8 = a 2. a. e5k = 1 4 1. 80ek = 1 c. e(ln 0.8)k = 0.8 In Exercises 57–60, solve for t. 1. a. e-0.3t = 64 b. ekt = 1 7 1. e(ln 0.7)t = 0.8 2. a. e-0.01t = 1000 b. ekt = 1 10 1. e(ln 2)t = 1 2 1. e2t = x6 60. e(x2)e(2x+1) = et Simplify the expressions in Exercises 61–64. 1. a. 5log5 7 b. 8log822 c. 1.3log1.3 75 2. log4 16 e. log323 f. log4 a1 4b 1. a. 2log2 3 b. 10log10 (1>2) c. plogp 7 2. log11 121 e. log121 11 f. log3 a1 9b 1. a. 2log4 x b. 9log3 x c. log2 (e(ln 2)(sin x)) 2. 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. 1. a. log11 x log12 x b. log5 x log125 x c. logx a logx5 a 1. 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. 1. a. sin-1 a1 2b b. sin-1 a 1 22 b c. sin-1 a-23 2 b 1. a. cos-1 a1 2b b. cos-1 a -1 22 b c. cos-1 a23 2 b 1. a. arccos (-1) b. arccos (0) 2. a. arcsin (-1) b. arcsin a- 1 22 b Theory and Examples 1. 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. 1. 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. 1. 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. 1. If a composite ƒ _ g is one-to-one, must g be one-to-one? Give reasons for your answer. 1. Find a formula for the inverse function ƒ -1 and verify that (ƒ _ ƒ -1)(x) = (ƒ -1 _ ƒ)(x) = x. 1. ƒ(x) = 100 1 + 2-x b. ƒ(x) = 50 1 + 1.1-x 1. 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). 1. Start with the graph of y = ln x. Find an equation of the graph that results from 1. shifting down 3 units. 2. shifting right 1 unit. 3. shifting left 1, up 3 units. 4. shifting down 4, right 2 units. 5. reflecting about the y-axis. 6. reflecting about the line y = x. 7. Start with the graph of y = ln x. Find an equation of the graph that results from 1. vertical stretching by a factor of 2. 2. horizontal stretching by a factor of 3. 3. vertical compression by a factor of 4. 4. horizontal compression by a factor of 2. 5. 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. 1. Could xln 2 possibly be the same as 2ln x for x 7 0? Graph the two functions and explain what you see. 1. Radioactive decay The half-life of a certain radioactive substance is 36 hours. There are 12 grams present initially. 1. Express the amount of substance remaining as a function of time t. 1. When will there be 1 gram remaining? 2. 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.

1. 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.

1. 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.

1. ƒ(x) = 8×3 + 8
2. 35, 74 b. 3-5, 54
3. g(x) = x2 – 2x
4. 31, 34 b. 3-2, 44
5. h(t) = cot t
6. 33p>4, 5p>44 b. 3p>3, 3p>24
7. g(t) = 2 + cos t
8. 30, p4 b. 3-p, p4
9. R(u) = 23u + 1; 30, 54
10. 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.

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

Instantaneous Rates of Change

1. 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

1. 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?

1. Then estimate the car’s speed at time t = 20 sec.
2. 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.

1. Estimate the slopes of the secants PQ1, PQ2, PQ3, and PQ4,

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

1. 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

1. 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 1. Plot points representing the profit as a function of year, and join them by as smooth a curve as you can 1. What is the average rate of increase of the profits between 2012 and 2014? 1. Use your graph to estimate the rate at which the profits were changing in 2012. 1. 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. 1. Find the average rate of change of F(x) over the intervals 31, x4 for each x _ 1 in your table. 1. Extending the table if necessary, try to determine the rate of change of F(x) at x = 1. 1. Let g(x) = 2x for x Ú 0. 2. 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. 1. 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. 1. What does your table indicate is the rate of change of g(x) with respect to x at x = 1? 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. 1. Let ƒ(t) = 1>t for t _ 0. 2. 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. 1. 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. 1. What does your table indicate is the rate of change of ƒ with respect to t at t = 2? 1. 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. 1. 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 1. Estimate the bicyclist’s average speed over the time intervals 30, 14, 31, 2.54 , and 32.5, 3.54 . 1. Estimate the bicyclist’s instantaneous speed at the times t = 12 , t = 2, and t = 3. 1. Estimate the bicyclist’s maximum speed and the specific time at which it occurs. 1. The accompanying graph shows the total amount of gasoline A in the gas tank of an automobile after being driven for t days. 1. Estimate the average rate of gasoline consumption over the time intervals 30, 34, 30, 54, and 37, 104 . 1. Estimate the instantaneous rate of gasoline consumption at the times t = 1, t = 4, and t = 8. 1. Estimate the maximum rate of gasoline consumption and the specific time at which it occurs Exercises 2.2 Limits from Graphs 1. For the function g(x) graphed here, find the following limits or explain why they do not exist. 1. 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) 1. For the function ƒ(t) graphed here, find the following limits or explain why they do not exist. 1. 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 1. Which of the following statements about the function y = ƒ(x) graphed here are true, and which are false? 1. lim xS0 ƒ(x) exists. 1. lim xS0 ƒ(x) = 0 1. lim xS0 ƒ(x) = 1 1. lim xS1 ƒ(x) = 1 1. lim xS1 ƒ(x) = 0 1. lim xSc ƒ(x) exists at every point c in (-1, 1). 1. lim xS1 ƒ(x) does not exist. x y _1 1 2 1 _1 y _ f (x) 1. Which of the following statements about the function y = ƒ(x) graphed here are true, and which are false? 1. lim xS2 ƒ(x) does not exist. 1. lim xS2 ƒ(x) = 2 1. lim xS1 ƒ(x) does not exist. 1. lim xSc ƒ(x) exists at every point c in (-1, 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. 1. lim xS0 x 0 x 0 6. lim xS1 1 x – 1 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. 1. 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. 1. 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. 1. 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. 1. lim xS -3 (x2 – 13) 12. lim xS3 (-x2 + 8x – 7) 1. lim tS6 8(t – 5)(t – 7) 14. lim xS -1 (2×3 – 5×2 + 3x + 5) 1. lim xS2 x + 2 x + 5 1. lim sS2>3 (8 – 3s)(2s – 1) 1. lim xS-1>4 16x(13x + 16)2 18. lim yS2 y + 2 y2 + 5y + 6 1. lim yS -9 (18 – y)7>3 20. lim zS4 2z2 – 10 1. lim hS0 5 25h + 4 + 4 1. lim hS0 24h + 1 – 1 h Limits of quotients Find the limits in Exercises 23–42. 1. lim xS9 x – 9 x2 – 81 1. lim xS -3 x + 3 x2 + 4x + 3 1. lim xS 8 x2 – 2x – 48 x – 8 1. lim xS6 x2 – 4x – 12 x – 6 1. lim tS1 t2 + t – 2 t2 – 1 1. lim tS -1 t2 + 3t + 2 t2 – t – 2 1. lim xS -2 -2x – 4 x3 + 2×2 30. lim yS0 5y3 + 8y2 3y4 – 16y2 1. lim xS1 x-1 – 1 x – 1 1. lim xS0 1 x – 1 + 1 x + 1 x 1. lim uS2 u4 – 16 u3 – 8 1. lim uS3 u4 – 81 u3 – 27 1. lim xS25 2x – 5 x – 25 1. lim xS4 4x – x2 2 – 2x 1. lim xS87 x – 87 2x + 13 – 10 1. lim xS -1 2×2 + 8 – 3 x + 1 1. lim xS2 2×2 + 12 – 4 x – 2 1. lim xS -2 x + 2 2×2 + 5 – 3 1. lim xS -24 23 – 2×2 – 47 x + 24 1. lim xS4 4 – x 5 – 2×2 + 9 Limits with trigonometric functions Find the limits in Exercises 43–50. 1. lim xS0 (2 sin x – 1) 44. lim xSp>4 sin2 x 1. lim xS0 sec x 46. lim xSp>3 tan x 1. lim xS0 1 + x + sin x 3 cos x 1. lim xS0 (x2 – 1)(2 – cos x) 1. lim xS -p 2x + 4 cos (x + p) 50. lim xS0 27 + sec2 x Using Limit Rules 1. 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 1. 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 1. Suppose limxS3 ƒ(x) = 3 and limxS3 g(x) = -8. Find 2. lim xS3 ƒ(x)g(x) b. lim xS3 3ƒ(x)g(x) 1. lim xS3 (ƒ(x) + 5g(x)) d. lim xS3 ƒ(x) ƒ(x) – g(x) 1. Suppose limxS4 ƒ(x) = 0 and limxS4 g(x) = -3. Find 2. lim xS4 (g(x) + 3) b. lim xS4 xƒ(x) 1. lim xS4 (g(x))2 d. lim xS4 g(x) ƒ(x) – 1 1. Suppose limxSb ƒ(x) = 10 and limxSb g(x) = -3. Find 2. lim xSb (ƒ(x) + g(x)) b. lim xSb 6(x) 1. lim xSb ƒ(x) # g(x) d. lim xSb ƒ(x)>g(x) 1. Suppose that limxS -2 p(x) = 4, limxS -2 r(x) = 0, and limxS -2 s(x) = -3. Find 1. lim xS -2 (p(x) + r(x) + s(x)) 1. lim xS -2 p(x) # r(x) # s(x) 1. 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. 1. ƒ(x) = x2, x = 1 2. ƒ(x) = x2, x = -2 3. ƒ(x) = 3x – 4, x = 2 4. ƒ(x) = 1>x, x = -2 5. ƒ(x) = 2x, x = 7 6. ƒ(x) = 23x + 1, x = 0 Using the Sandwich Theorem 1. If 25 – 2×2 … ƒ(x) … 25 – x2 for -1 … x … 1, find limxS0 ƒ(x). 1. If 2 – x2 … g(x) … 2 cos x for all x, find limxS0 g(x). 2. 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. 1. 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. 1. 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. 1. 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. 1. Let ƒ(x) = (x2 – 9)>(x + 3). 2. 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? 1. 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. 1. Find limxS -3 ƒ(x) algebraically, as in Example 7. 2. Let g(x) = (x2 – 2) >(x – 22). 3. 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). 1. 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. 1. Find limxS22 g(x) algebraically. 2. Let G(x) = (x + 6)> (x2 + 4x – 12). 3. 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? 1. Support your conclusions in part (a) by graphing G and using Zoom and Trace to estimate y-values on the graph as xS -6. 1. Find limxS -6 G(x) algebraically. 2. Let h(x) = (x2 – 2x – 3) > (x2 – 4x + 3). 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? 1. 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. 1. Find limxS3 h(x) algebraically. 2. Let ƒ(x) = (x2 – 1) > ( 0 x 0 – 1). 3. 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 1. 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. 1. Find limxS -1 ƒ(x) algebraically. 2. Let F(x) = (x2 + 3x + 2) > (2 – 0 x 0 ). 3. Make tables of values of F at values of x that approach c = -2 from above and below. Then estimate limxS -2 F(x). 1. 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. 1. Find limxS -2 F(x) algebraically. 2. Let g(u) = (sin u)>u. 3. 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). 1. Support your conclusion in part (a) by graphing g near u0 = 0. 1. Let G(t) = (1 – cos t)>t2. 2. Make tables of values of G at values of t that approach t0 = 0 from above and below. Then estimate limtS0 G(t). 1. Support your conclusion in part (a) by graphing G near t0 = 0. 1. Let ƒ(x) = x1>(1-x). 2. 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? 1. Support your conclusions in part (a) by graphing ƒ near c = 1. 2. Let ƒ(x) = (3x – 1)>x. 3. 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? 1. Support your conclusions in part (a) by graphing ƒ near c = 0. Theory and Examples 1. 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? 1. 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. 1. If lim xS4 ƒ(x) – 5 x – 2 = 1, find lim xS4 ƒ(x). 1. If lim xS -2 ƒ(x) x2 = 1, find 1. lim xS -2 ƒ(x) b. lim xS -2 ƒ(x) x 1. a. If lim xS2 ƒ(x) – 5 x – 2 = 3, find lim xS2 ƒ(x). 1. If lim xS2 ƒ(x) – 5 x – 2 = 4, find lim xS2 ƒ(x). 1. If lim xS0 ƒ(x) x2 = 1, find 1. lim xS0 ƒ(x) 1. lim xS0 ƒ(x) x 1. a. Graph g(x) = x sin (1>x) to estimate limxS0 g(x), zooming in on the origin as necessary. 1. Confirm your estimate in part (a) with a proof. 2. a. Graph h(x) = x2 cos (1>x3) to estimate limxS0 h(x), zooming in on the origin as necessary. 1. 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: 1. Plot the function near the point c being approached. 2. 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. 1. a = 1, b = 7, c = 5 2. a = 1, b = 7, c = 2 3. a = -7>2, b = -1>2, c = -3 4. a = -7>2, b = -1>2, c = -3>2 5. a = 4>9, b = 4>7, c = 1>2 6. 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. 1. 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. 1. ƒ(x) = x + 1, L = 5, c = 4, P = 0.01 2. ƒ(x) = 2x – 2, L = -6, c = -2, P = 0.02 3. ƒ(x) = 2x + 1, L = 1, c = 0, P = 0.1 4. ƒ(x) = 2x, L = 1>2, c = 1>4, P = 0.1 5. ƒ(x) = 219 – x, L = 3, c = 10, P = 1 6. ƒ(x) = 2x – 7, L = 4, c = 23, P = 1 7. ƒ(x) = 1>x, L = 1>4, c = 4, P = 0.05 8. ƒ(x) = x2, L = 3, c = 23, P = 0.1 9. ƒ(x) = x2, L = 4, c = -2, P = 0.5 10. ƒ(x) = 1>x, L = -1, c = -1, P = 0.1 11. ƒ(x) = x2 – 5, L = 11, c = 4, P = 1 12. ƒ(x) = 120>x, L = 5, c = 24, P = 1 13. ƒ(x) = mx, m 7 0, L = 2m, c = 2, P = 0.03 14. ƒ(x) = mx, m 7 0, L = 3m, c = 3, P = c 7 0 15. ƒ(x) = mx + b, m 7 0, L = (m>2) + b, c = 1>2, P = c 7 0 1. 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. 1. ƒ(x) = 2 – 4x, c = 4, P = 0.03 2. ƒ(x) = -3x – 2, c = -1, P = 0.03 3. ƒ(x) = x2 – 9 x – 3 , c = 3, P = 0.03 1. ƒ(x) = x2 + 6x + 5 x + 5 , c = -5, P = 0.05 1. ƒ(x) = 21 – x, c = -3, P = 0.5 2. ƒ(x) = 4>x, c = 2, P = 0.4 Prove the limit statements in Exercises 37–50. 1. lim xS4 (9 – x) = 5 38. lim xS3 (3x – 7) = 2 1. lim xS9 2x – 5 = 2 40. lim xS0 24 – x = 2 1. lim xS1 ƒ(x) = 1 if ƒ(x) = e x2, x _ 1 2, x = 1 1. lim xS -2 ƒ(x) = 4 if ƒ(x) = e x2, x _ -2 1, x = -2 1. lim xS1 1x = 1 44. lim xS23 1 x2 = 1 3 1. lim xS -3 x2 – 9 x + 3 = -6 46. lim xS1 x2 – 1 x – 1 = 2 1. lim xS1 ƒ(x) = 2 if ƒ(x) = e 4 – 2x, x 6 1 6x – 4, x Ú 1 1. lim xS0 ƒ(x) = 0 if ƒ(x) = e 2x, x 6 0 x>2, x Ú 0 1. lim xS0 x sin 1x = 0 1. 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 1. Define what it means to say that lim xS0 g(x) = k. 1. Prove that lim xSc ƒ(x) = L if and only if lim hS0 ƒ(h + c) = L. 1. 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. 1. 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. 1. 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? 1. 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) 1. 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 1. 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. 1. Show that limxS1 ƒ(x) _ 1. 2. 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. 1. ƒ(x) = x + 1, L = 5, c = 4, P = 0.01 2. ƒ(x) = 2x – 2, L = -6, c = -2, P = 0.02 3. ƒ(x) = 2x + 1, L = 1, c = 0, P = 0.1 4. ƒ(x) = 2x, L = 1>2, c = 1>4, P = 0.1 5. ƒ(x) = 219 – x, L = 3, c = 10, P = 1 6. ƒ(x) = 2x – 7, L = 4, c = 23, P = 1 7. ƒ(x) = 1>x, L = 1>4, c = 4, P = 0.05 8. ƒ(x) = x2, L = 3, c = 23, P = 0.1 9. ƒ(x) = x2, L = 4, c = -2, P = 0.5 10. ƒ(x) = 1>x, L = -1, c = -1, P = 0.1 11. ƒ(x) = x2 – 5, L = 11, c = 4, P = 1 12. ƒ(x) = 120>x, L = 5, c = 24, P = 1 13. ƒ(x) = mx, m 7 0, L = 2m, c = 2, P = 0.03 14. ƒ(x) = mx, m 7 0, L = 3m, c = 3, P = c 7 0 15. ƒ(x) = mx + b, m 7 0, L = (m>2) + b, c = 1>2, P = c 7 0 1. 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. 1. ƒ(x) = 2 – 4x, c = 4, P = 0.03 2. ƒ(x) = -3x – 2, c = -1, P = 0.03 3. ƒ(x) = x2 – 9 x – 3 , c = 3, P = 0.03 1. ƒ(x) = x2 + 6x + 5 x + 5 , c = -5, P = 0.05 1. ƒ(x) = 21 – x, c = -3, P = 0.5 2. ƒ(x) = 4>x, c = 2, P = 0.4 Prove the limit statements in Exercises 37–50. 1. lim xS4 (9 – x) = 5 38. lim xS3 (3x – 7) = 2 1. lim xS9 2x – 5 = 2 40. lim xS0 24 – x = 2 1. lim xS1 ƒ(x) = 1 if ƒ(x) = e x2, x _ 1 2, x = 1 1. lim xS -2 ƒ(x) = 4 if ƒ(x) = e x2, x _ -2 1, x = -2 1. lim xS1 1x = 1 44. lim xS23 1 x2 = 1 3 1. lim xS -3 x2 – 9 x + 3 = -6 46. lim xS1 x2 – 1 x – 1 = 2 1. lim xS1 ƒ(x) = 2 if ƒ(x) = e 4 – 2x, x 6 1 6x – 4, x Ú 1 1. lim xS0 ƒ(x) = 0 if ƒ(x) = e 2x, x 6 0 x>2, x Ú 0 1. lim xS0 x sin 1x = 0 x y y _ x sin1x 1p _ 1p 1 2p _ 1 2p 90 1. 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 1. Define what it means to say that lim xS0 g(x) = k. 1. Prove that lim xSc ƒ(x) = L if and only if lim hS0 ƒ(h + c) = L. 1. 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. 1. 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. 1. 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? 1. 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) 1. 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 1. 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. 1. Show that limxS1 ƒ(x) _ 1. 2. Show that limxS1 ƒ(x) _ 1.5. 3. 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 1. lim xS2 h(x) _ 4 1. lim xS2 h(x) _ 3 1. lim xS2 h(x) _ 2 1. For the function graphed here, explain why 2. lim xS3 ƒ(x) _ 4 1. lim xS3 ƒ(x) _ 4.8 1. lim xS3 ƒ(x) _ 3 x y 0 3 3 4 4.8 y _ f (x) 1. a. For the function graphed here, show that limxS -1 g(x) _ 2. 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: 1. Plot the function y = ƒ(x) near the point c being approached. 2. Guess the value of the limit L and then evaluate the limit symbolically to see if you guessed correctly. 1. Using the value P = 0.2, graph the banding lines y1 = L – P and y2 = L + P together with the function ƒ near c. 1. 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. 1. Repeat parts (c) and (d) successively for P = 0.1, 0.05, and 0.001. 2. ƒ(x) = x4 – 81 x – 3 , c = 3 62. ƒ(x) = 5×3 + 9×2 2×5 + 3×2 , c = 0 1. ƒ(x) = sin 2x 3x , c = 0 64. ƒ(x) = x(1 – cos x) x – sin x , c = 0 1. ƒ(x) = 23 x – 1 x – 1 , c = 1 1. ƒ(x) = 3×2 – (7x + 1)2x + 5 x – 1 , c = 1 Exercises 2.4 Finding Limits Graphically 1. 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) 1. lim xS -1+ ƒ(x) = 1 b. lim xS0- ƒ(x) = 0 1. lim xS0- ƒ(x) = 1 d. lim xS0- ƒ(x) = lim xS0+ ƒ(x) 1. lim xS0 ƒ(x) exists. f. lim xS0 ƒ(x) = 0 1. lim xS0 ƒ(x) = 1 h. lim xS1 ƒ(x) = 1 1. lim xS1 ƒ(x) = 0 j. lim xS2- ƒ(x) = 2 1. lim xS -1- ƒ(x) does not exist. l. lim xS2+ ƒ(x) = 0 1. 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) 1. lim xS -1+ ƒ(x) = 1 b. lim xS2 ƒ(x) does not exist. 1. lim xS2 ƒ(x) = 2 d. lim xS1- ƒ(x) = 2 1. lim xS1+ ƒ(x) = 1 f. lim xS1 ƒ(x) does not exist. 1. lim xS0+ ƒ(x) = lim xS0- ƒ(x) 1. lim xSc ƒ(x) exists at every c in the open interval (-1, 1). 1. lim xSc ƒ(x) exists at every c in the open interval (1, 3). 1. lim xS -1- ƒ(x) = 0 k. lim xS3+ ƒ(x) does not exist. 1. 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 1. Find limxS2+ ƒ(x) and limxS2- ƒ(x). 2. Does limxS2 ƒ(x) exist? If so, what is it? If not, why not? 3. Find limxS4- ƒ(x) and limxS4+ ƒ(x). 4. Does limxS4 ƒ(x) exist? If so, what is it? If not, why not? 5. 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 1. Find limxS2+ ƒ(x), limxS2- ƒ(x), and ƒ(2). 2. Does limxS2 ƒ(x) exist? If so, what is it? If not, why not? 3. Find limxS -1- ƒ(x) and limxS -1+ ƒ(x). 4. Does limxS -1 ƒ(x) exist? If so, what is it? If not, why not? 5. 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 1. Does limxS0+ ƒ(x) exist? If so, what is it? If not, why not? 2. Does limxS0- ƒ(x) exist? If so, what is it? If not, why not? 3. Does limxS0 ƒ(x) exist? If so, what is it? If not, why not? 4. 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 1. Does limxS0+ g(x) exist? If so, what is it? If not, why not? 2. Does limxS0- g(x) exist? If so, what is it? If not, why not? 3. Does limxS0 g(x) exist? If so, what is it? If not, why not? 4. a. Graph ƒ(x) = e x3, x _ 1 0, x = 1. 1. Find limxS1- ƒ(x) and limxS1+ ƒ(x). 2. Does limxS1 ƒ(x) exist? If so, what is it? If not, why not? 3. a. Graph ƒ(x) = e 1 – x2, x _ 1 2, x = 1. 1. Find limxS1+ ƒ(x) and limxS1- ƒ(x). 2. 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. 1. What are the domain and range of ƒ? 2. At what points c, if any, does limxSc ƒ(x) exist? 3. At what points does only the left-hand limit exist? 4. At what points does only the right-hand limit exist? 5. ƒ(x) = c 21 – x2, 0 … x 6 1 1, 1 … x 6 2 2, x = 2 1. 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. 1. lim xS -0.5-A x + 2 x + 1 1. lim xS1+ A x – 1 x + 2 1. lim xS -2+ a x x + 1b a2x + 5 x2 + x b 1. lim xS1- a 1 x + 1b ax + 6 x b a3 – x 7 b 1. lim hS0+ 2h2 + 6h + 13 – 213 1. lim hS0- 26 – 25h2 + 11h + 6 h 1. a. lim xS -2+ (x + 3) 0 x + 2 0 x + 2 1. lim xS -2- (x + 3) 0 x + 2 0 x + 2 1. a. lim xS1+ 22x (x – 1) 0 x – 1 0 1. 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. 1. a. lim uS3+ :u; u 1. lim uS3- :u; u 1. a. lim tS4+ (t – :t;) b. lim tS4- (t – :t;) Using lim Uu0 sin U U _ 1 Find the limits in Exercises 21–42. 1. lim uS0 2sin 29u 29u 1. lim tS0 sin kt t (k constant) 1. lim yS0 sin 19y 7y 1. lim hS0- h sin 3h 1. lim xS0 tan 3x x 26. lim tS0 2t tan t 1. lim xS0 x csc 5x cos 12x 1. lim xS0 6×2(cot x)(csc 2x) 1. lim xS0 11x + 11x cos 11x 8sin 11x cos 11x 1. lim xS0 x2 – x + sin x 2x 1. lim uS0 1 – cos u sin 2u 1. lim xS0 x – x cos x sin2 3x 1. lim tS0 sin (1 – cos t) 1 – cos t 1. lim hS0 sin (sin h) sin h 1. lim uS0 sin u sin 2u 1. lim xS0 sin 5x sin 4x 1. lim uS0 u cos u 38. lim uS0 sin u cot 2u 1. lim xS0 tan 3x sin 8x 1. lim yS0 sin 3y cot 5y y cot 4y 1. lim uS0 tan u u2 cot 3u 1. lim uS0 u cot 4u sin2 u cot2 2u Theory and Examples 1. 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. 1. If you know that limxSc ƒ(x) exists, can you find its value by calculating limxSc+ ƒ(x)? Give reasons for your answer. 1. 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. 1. 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 1. 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? 1. 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. 1. lim xS0- x 0 x 0 = -1 50. lim xS2+ x – 2 0 x – 2 0 = 1 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. 1. 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? 1. 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) 1. 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. 1. a. Does ƒ(-1) exist? 2. Does limxS -1+ ƒ(x) exist? 3. Does limxS -1+ ƒ(x) = ƒ(-1)? 4. Is ƒ continuous at x = -1? 5. a. Does ƒ(1) exist? 6. Does limxS1 ƒ(x) exist? 7. Does limxS1 ƒ(x) = ƒ(1)? 8. Is ƒ continuous at x = 1? 9. a. Is ƒ defined at x = 2? (Look at the definition of ƒ.) 10. Is ƒ continuous at x = 2? 11. At what values of x is ƒ continuous? 12. What value should be assigned to ƒ(2) to make the extended function continuous at x = 2? 1. 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. 1. Exercise 1, Section 2.4 12. Exercise 2, Section 2.4 At what points are the functions in Exercises 13–30 continuous? 1. y = 1 x – 2 – 3x 14. y = 1 (x + 2)2 + 4 1. y = x + 1 x2 – 4x + 3 1. y = x + 3 x2 – 3x – 10 1. y = 0 x – 1 0 + sin x 18. y = 1 0 x 0 + 1 – x2 2 1. y = cos x x 20. y = x + 2 cos x 1. y = csc 2x 22. y = tan px 2 1. y = x tan x x2 + 1 1. y = 2×4 + 1 1 + sin2 x 1. y = 22x + 3 26. y = 24 3x – 1 1. y = (2x – 1)1>3 28. y = (2 – x)1>5 2. g(x) = c x2 – x – 6 x – 3 , x _ 3 5, x = 3 1. 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? 1. lim xSp sin (x – sin x) 32. lim tS0 sinap 2 cos (tan t)b 1. lim yS1 sec (y sec2 y – tan2 y – 1) 1. lim xS0 tanap 4 cos (sin x1>3)b 1. lim tS0 cos a p 219 – 3 sec 2t b 36. lim xSp/6 2csc2 x + 513 tan x 1. lim xS0+ sin ap 2 e2xb 38. lim xS1 cos-1 1ln 2×2 Continuous Extensions 1. Define g(3) in a way that extends g(x) = (x2 – 9)>(x – 3) to be continuous at x = 3. 1. Define h(2) in a way that extends h(t) = (t2 + 3t – 10)>(t – 2) to be continuous at t = 2. 1. Define ƒ(1) in a way that extends ƒ(s) = (s3 – 1)>(s2 – 1) to be continuous at s = 1. 1. Define g(4) in a way that extends g(x) = (x2 – 16)> (x2 – 3x – 4) to be continuous at x = 4. 1. For what value of a is ƒ(x) = e x2 – 1, x 6 3 2ax, x Ú 3 continuous at every x? 1. For what value of b is g(x) = e x, x 6 -2 bx2, x Ú -2 continuous at every x? 1. For what values of a is ƒ(x) = b a2x – 2a, x Ú 2 12, x 6 2 continuous at every x? 1. 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? 1. 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? 1. 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? 1. ƒ(x) = 10 x – 1 x 50. ƒ(x) = 10 0 x 0 – 1 x 1. ƒ(x) = sin x 0 x 0 52. ƒ(x) = (1 + 2x)1>x Theory and Examples 1. 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. 1. Explain why the equation cos x = x has at least one solution. 2. Roots of a cubic Show that the equation x3 – 15x + 1 = 0 has three solutions in the interval 3-4, 4]. 1. 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. 1. 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. 1. Explain why the following five statements ask for the same information. 2. Find the roots of ƒ(x) = x3 – 3x – 1. 3. Find the x-coordinates of the points where the curve y = x3 crosses the line y = 3x + 1. 1. Find all the values of x for which x3 – 3x = 1. 2. Find the x-coordinates of the points where the cubic curve y = x3 – 3x crosses the line y = 1. 1. Solve the equation x3 – 3x – 1 = 0. 2. 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. 1. 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. 1. A function discontinuous at every point 2. 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. 1. Is ƒ right-continuous or left-continuous at any point? 2. 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. 1. 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. 1. 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. 1. 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. 1. 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. 1. 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). 1. 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). 1. Prove that ƒ is continuous at c if and only if lim hS0 ƒ(c + h) = ƒ(c). 1. 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. 1. x3 – 3x – 1 = 0 2. 2×3 – 2×2 – 2x + 1 = 0 3. x(x – 1)2 = 1 (one root) 4. xx = 2 5. 2x + 21 + x = 4 6. x3 – 15x + 1 = 0 (three roots) 7. cos x = x (one root). Make sure you are using radian mode. 8. 2 sin x = x (three roots). Make sure you are using radian mode. Exercises 2.6 Finding Limits 1. For the function ƒ whose graph is given, determine the following limits. 1. lim xS2 ƒ(x) b. lim xS -3 + ƒ(x) c. lim xS -3 – ƒ(x) 1. lim xS -3 ƒ(x) e. lim xS0 + ƒ(x) f. lim xS0 – ƒ(x) 1. 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 1. For the function ƒ whose graph is given, determine the following limits. 1. lim xS4 ƒ(x) b. lim xS2 + ƒ(x) c. lim xS2 – ƒ(x) 1. lim xS2 ƒ(x) e. lim xS -3 + ƒ(x) f. lim xS -3 – ƒ(x) 1. lim xS -3 ƒ(x) h. lim xS0 + ƒ(x) i. lim xS0 – ƒ(x) 1. 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.) 1. ƒ(x) = 2x – 3 4. ƒ(x) = p – 2 x2 1. g(x) = 1 5 + (1>x) 1. g(x) = 1 8 – (5>x2) 1. f(x) = -3 + (4>x) 4 – (3>x2) 1. h(x) = 3 – (2>x) 4 + (22>x2) Find the limits in Exercises 9–12. 1. lim xSq sin 2x x 10. lim uS -q cos u 3u 1. 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. 1. ƒ(x) = 2x + 3 5x + 7 1. ƒ(x) = 2×3 + 7 x3 – x2 + x + 7 1. ƒ(x) = x + 1 x2 + 3 1. ƒ(x) = 3x + 7 x2 – 2 1. h(x) = 7×3 x3 – 3×2 + 6x 1. h(x) = 9×4 + x 2×4 + 5×2 – x + 6 1. g(x) = 10×5 + x4 + 31 x6 20. g(x) = x3 + 7×2 – 2 x2 – x + 1 1. f(x) = 3×7 + 5×2 – 1 6×3 – 7x + 3 1. 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. 1. lim xSq A 8×2 – 3 2×2 + x 1. lim xS -q ¢x2 + x – 1 8×2 – 3 _ 1>3 1. lim xS – q ¢ 1 – x3 x2 + 7x _ 5 1. lim xS q A x2 – 5x x3 + x – 2 1. lim xS q 22x + x-1 3x – 7 1. lim xS q 2 + 2x 2 – 2x 1. lim xS – q 23 x – 25 x 23 x + 25 x 1. lim xS q x-1 + x-4 x-2 – x-3 1. lim xS q 2×5>3 – x1>3 + 7 x8>5 + 3x + 2x 1. lim xS – q 23 x – 5x + 3 2x + x2>3 – 4 1. lim xS q 2×2 + 1 x + 1 1. lim xS – q 2×2 + 1 x + 1 1. lim xS q x – 3 24×2 + 25 1. lim xS – q 4 – 3×3 2×6 + 9 Infinite Limits Find the limits in Exercises 37–48. 1. lim xS0+ 1 3x 1. lim xS0- 5 2x 1. lim xS2- 3 x – 2 1. lim xS3+ 1 x – 3 1. lim xS -8+ 2x x + 8 1. lim xS -5- 3x 2x + 10 1. lim xS7 4 (x – 7)2 44. lim xS0 -1 x2(x + 1) 1. a. lim xS0+ 2 3×1>3 b. lim xS0- 2 3×1>3 1. a. lim xS0+ 2 x1>5 b. lim xS0- 2 x1>5 1. lim xS0 4 x2>5 48. lim xS0 1 x2>3 Find the limits in Exercises 49–52. 1. lim xS(p>2)- tan x 50. lim xS(-p>2)+ sec x 1. lim uS0- (1 + csc u) 52. lim uS0 (2 – cot u) Find the limits in Exercises 53–58. 1. lim 1 x2 – 4 as 1. xS 2+ b. xS 2- 2. xS -2+ d. xS -2- 3. lim x x2 – 1 as 1. xS 1+ b. xS 1- 2. xS -1+ d. xS -1- 3. lim ax2 2 – 1x b as 1. xS 0+ b. xS 0- 2. xS 23 2 d. xS -1 1. lim x2 – 1 2x + 4 as 1. xS -2+ b. xS -2- 2. xS 1+ d. xS 0- 3. lim x2 – 3x + 2 x3 – 2×2 as 1. xS 0+ b. xS 2+ 2. xS 2- d. xS 2 3. What, if anything, can be said about the limit as xS 0? 4. lim x2 – 3x + 2 x3 – 4x as 1. xS 2+ b. xS -2+ 2. xS 0- d. xS 1+ 3. What, if anything, can be said about the limit as xS 0? Find the limits in Exercises 59–62. 1. lima2 – 3 t1>3b as 1. tS 0+ b. tS 0- 2. lima 1 t3>5 + 7b as 1. tS 0+ b. tS 0- 2. lima 1 x2>3 + 2 (x – 1)2>3b as 1. xS 0+ b. xS 0- 2. xS 1+ d. xS 1- 3. lima 1 x1>3 – 1 (x – 1)4>3b as 1. xS 0+ b. xS 0- 2. 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. 1. y = 1 x – 1 1. y = 1 x + 1 1. y = 1 2x + 4 1. y = -3 x – 3 1. y = x + 3 x + 2 1. 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.) 1. ƒ(0) = 0, ƒ(1) = 2, ƒ(-1) = -2, lim xS -q ƒ(x) = -1, and lim xSq ƒ(x) = 1 1. ƒ(0) = 0, lim xS{q ƒ(x) = 0, lim xS0+ ƒ(x) = 2, and lim xS0- ƒ(x) = -2 1. 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 1. 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.) 1. lim xS{q ƒ(x) = 0, lim xS2- ƒ(x) = q, and lim xS2+ ƒ(x) = q 1. lim xS{q g(x) = 0, lim xS3- g(x) = -q, and lim xS3+ g(x) = q 1. lim xS – q h(x) = -1, lim xS q h(x) = 1, lim xS0- h(x) = -1, and lim xS0+ h(x) = 1 1. lim xS{q k(x) = 1, lim xS1- k(x) = q, and lim xS1+ k(x) = -q 1. 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. 1. 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. 1. 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. 1. lim xS q (2x + 9 – 2x + 4 ) 1. lim xS q (2×2 + 25 – 2×2 – 1 ) 1. lim xS – q (2×2 + 3 + x ) 1. lim xS – q (2x + 24×2 + 3x – 2 ) 1. lim xS q (29×2 – x – 3x) 1. lim xS q (2×2 + 3x – 2×2 – 2x ) 1. 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. 1. If ƒ has the constant value ƒ(x) = k, then lim xSq ƒ(x) = k. 1. 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. 1. lim xS0 -1 x2 = -q 90. lim xS0 1 0 x 0 = q 1. lim xS3 -2 (x – 3)2 = -q 92. lim xS -5 1 (x + 5)2 = q 1. 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. 1. lim xScƒ( 1. x) = q 2. lim xSc+ ƒ(x) = -q 1. lim xScƒ( 1. x) = -q Use the formal definitions from Exercise 93 to prove the limit statements in Exercises 94–98. 1. lim xS0+ 1x = q 1. lim xS0- 1x = -q 1. lim xS2- 1 x – 2 = -q 1. lim xS2+ 1 x – 2 = q 1. lim xS1- 1 1 – x2 = q Oblique Asymptotes Graph the rational functions in Exercises 99–104. Include the graphs and equations of the asymptotes. 1. y = x2 x – 1 1. y = x2 + 1 x – 1 1. y = x2 – 4 x – 1 1. y = x2 – 1 2x + 4 1. y = x2 – 1 x 1. 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. 1. y = x 24 – x2 1. y = -1 24 – x2 1. y = x2>3 + 1 x1>3 1. y = sin a p x2 + 1 b Graph the functions in Exercises 109 and 110. Then answer the following questions. 1. How does the graph behave as xS 0+? 2. How does the graph behave as xS {q? 3. How does the graph behave near x = 1 and x = -1? Give reasons for your answers. 1. y = 3 2 ax – 1x b 2>3 1. y = 3 2 a x x – 1b 2>3 Chapter 2 Questions to Guide Your Review 1. 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? 1. What limit must be calculated to find the rate of change of a function g(t) at t = t0? 1. Give an informal or intuitive definition of the limit lim xSc ƒ(x) = L. Why is the definition “informal”? Give examples. 1. 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. 1. What function behaviors might occur for which the limit may fail to exist? Give examples. 1. What theorems are available for calculating limits? Give examples of how the theorems are used. 1. 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. 1. What is the value of limuS0 ((sin u)>u)? Does it matter whether u is measured in degrees or radians? Explain. 1. 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. 1. Give precise definitions of the following statements. 2. limxS2- ƒ(x) = 5 b. limxS2+ ƒ(x) = 5 3. limxS2 ƒ(x) = q d. limxS2 ƒ(x) = -q 4. What conditions must be satisfied by a function if it is to be continuous at an interior point of its domain? At an endpoint? 1. How can looking at the graph of a function help you tell where the function is continuous? 1. What does it mean for a function to be right-continuous at a point? Left-continuous? How are continuity and one-sided continuity related? 1. 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. 1. What are the basic types of discontinuity? Give an example of each. What is a removable discontinuity? Give an example. 1. 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? 1. Under what circumstances can you extend a function ƒ(x) to be continuous at a point x = c? Give an example. 1. What exactly do limxSq ƒ(x) = L and limxS -q ƒ(x) = L mean? Give examples. 1. What are limxS{q k (k a constant) and limxS{q (1>x)? How do you extend these results to other functions? Give examples. 1. How do you find the limit of a rational function as xS {q? Give examples. 1. What are horizontal and vertical asymptotes? Give examples. Chapter 2 Practice Exercises Limits and Continuity 1. 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. 1. 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. 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. 1. 5ƒ(x) b. (ƒ(x))3 2. ƒ(x) # g(x) d. ƒ(x) g(x) – 6 1. log g(x) f. 0 g(x) 0 2. ƒ(x) – g(x) h. 1>ƒ(x) 3. 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. 1. -g(x) b. g(x) # ƒ(x) 2. ƒ(x) + g(x) d. 1>ƒ(x) 3. 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. 1. lim xS0 a 4 – g(x) x b = 1 6. lim xS -4 ax lim xS0 g(x)b = 2 1. On what intervals are the following functions continuous? 2. ƒ(x) = x1>3 b. g(x) = x3>4 3. h(x) = x-2>3 d. k(x) = x-1>6 4. On what intervals are the following functions continuous? 5. ƒ(x) = tan x b. g(x) = csc x 6. 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. 1. lim x2 + 5x + 5 x3 + 3×2 – 11x 1. as xS 0 b. as xS 3 2. lim x3 + 1 x7 + 4×6 + x5 1. as xS 2 b. as xS -2 2. lim xS0 a21 + x – 1 x b 12. lim xS2 x5 – 25 x3 – 23 1. lim hS0 (x + h)3 – x3 h 1. lim xS0 (x + h)2 – x2 h 1. lim xS0 1 3 + x – 1 3 x 16. lim xS0 (3 + x)4 – 81 x 1. lim xS1 x2 – 2x 2x – 1 1. lim xS16 x2>3 – 162>3 2x – 4 1. lim xS0 sin mx tan nx 20. lim xSp sec x = -1 1. lim xSp cos ax 2 + sin xb 22. lim xSp sin2 (x – cot x) 1. lim xS0 7x 8 tan x – x 1. lim xS0 x2 – tan 2x tan x 1. lim xS0 (1 + x)1>x 26. lim tS4 t3 ln (3 – 2t ) 1. 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. 1. lim xS0+ [9g(x)]1>3 = 3 30. lim xS27 1 x + g(x) = 5 1. lim xS2 5×4 + 2 g(x) = q 32. lim xS -3 10 – x2 (g(x))1>3 = 0 Roots 1. Let ƒ(x) = x3 – x – 1. 2. Use the Intermediate Value Theorem to show that ƒ has a zero between -1 and 2. 1. Solve the equation ƒ(x) = 0 graphically with an error of magnitude at most 10-8. T 1. 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). 1. Let ƒ(u) = u3 – 2u + 2. 2. Use the Intermediate Value Theorem to show that ƒ has a zero between -2 and 0. 1. Solve the equation ƒ(u) = 0 graphically with an error of magnitude at most 10-4. 1. 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 1. 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.) 1. 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? 1. ƒ(x) = x – 1 x – 24 x , a = 1 1. g(u) = 5 cos u 4u – 2p , a = p>2 1. h(t) = (1 + 0 t 0 )1>t, a = 0 2. k(x) = x 1 – 20 x 0 , a = 0 Limits at Infinity Find the limits in Exercises 41–54. 1. lim xS q 5x + 8 7x + 4 1. lim xS q 3×4 + 8 5×4 + 9 1. lim xS – q x2 – 7x + 9 5×3 44. lim xS q 1 2×2 – 9x + 5 1. lim xS – q x2 + 11x x – 1 1. lim xS q x5 + x4 9×4 + 108 1. lim xS q sin x :x; (If you have a grapher, try graphing the function for -5 … x … 5.) 1. 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.) 1. lim xS q x + tan x + 32x x + tan x 50. lim xS q x5>7 + x-3 x5>7 + cos x 1. lim xS q e2>x sin 2x 1. lim tS q ln a1 – 2t b 1. lim xS – 1 sin-1 x 54. lim tS – q e3t sin-1 1t Horizontal and Vertical Asymptotes 1. Use limits to determine the equations for all vertical asymptotes. 2. y = x2 + 4 x – 3 1. ƒ(x) = x2 – x – 2 x2 – 2x + 1 1. y = x2 + x – 6 x2 + 2x – 8 1. Use limits to determine the equations for all horizontal asymptotes. 2. y = 1 – x2 x2 + 1 1. ƒ(x) = 2x + 4 2x + 4 1. g(x) = 2×2 + 4 x d. y = B x2 + 9 9×2 + 1 Chapter 2 Additional and Advanced Exercises 1. 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.) 1. 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? 1. 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. 1. 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. 1. Evaluate ƒ for x = 10, 100, 1000, and so on as far as your calculator can reasonably go. What pattern do you see? 1. 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? 1. 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 1. 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 1. within 0.2 ft3>min of the rate y0 = 1 ft3>min? 2. within 0.1 ft3>min of the rate y0 = 1 ft3>min? 3. 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? 1. 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. 1. ƒ(x) = x2 – 7, c = 1 8. g(x) = 1>(2x), c = 1>4 2. h(x) = 22x – 3, c = 2 10. F(x) = 29 – x, c = 5 3. 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. 1. Prove the limit Constant Multiple Rule: lim xSc kƒ(x) = k lim xSc ƒ(x) for any constant k. 1. One-sided limits If limxS0+ ƒ(x) = A and limxS0- ƒ(x) = B, find 1. limxS0+ ƒ(x3 – x) b. limxS0- ƒ(x3 – x) 2. limxS0+ ƒ(x2 – x4) d. limxS0- ƒ(x2 – x4) 3. 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). 1. If limxSc ƒ(x) exists but limxSc g(x) does not exist, then limxSc(ƒ(x) + g(x)) does not exist. 1. If neither limxSc ƒ(x) nor limxSc g(x) exists, then limxSc (ƒ(x) + g(x)) does not exist. 1. If ƒ is continuous at x, then so is 0 ƒ 0 . 2. 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. 1. ƒ(x) = x2 – 1 x + 1 , x = -1 16. g(x) = x2 – 2x – 3 2x – 6 , x = 3 1. A function continuous at only one point Let ƒ(x) = e x, if x is rational 0, if x is irrational. 1. Show that ƒ is continuous at x = 0. 2. 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. 1. 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 1. (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. 1. Show that ƒ is discontinuous at every rational number in 30, 14 . 2. 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.) 1. Sketch the graph of ƒ. Why do you think ƒ is called the “ruler function”? 1. 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. 1. If limxSc (ƒ(x) + g(x)) = 3 and limxSc (ƒ(x) – g(x)) = -1, find limxSc ƒ(x)g(x). 1. 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 , 1. What happens to r+(a) as aS 0? As aS -1+? 2. What happens to r-(a) as aS 0? As aS -1+? 3. Support your conclusions by graphing r+(a) and r-(a) as functions of a. Describe what you see. 1. For added support, graph ƒ(x) = ax2 + 2x – 1 simultaneously for a = 1, 0.5, 0.2, 0.1, and 0.05. 1. Root of an equation Show that the equation x + 2 cos x = 0 has at least one solution. 1. 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. 1. 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. 1. Suppose that ƒ is bounded from above by N. Show that if limxSc ƒ(x) = L, then L … N. 1. Suppose that ƒ is bounded from below by M. Show that if limxSc ƒ(x) = L, then L Ú M. 1. Max 5a, b6 and min 5a, b6 2. 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. 1. 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. 1. lim xS0 sin x2 x2 = 1 1. lim xS0 sin x2 x = lim xS0 sin x2 x2 lim xS0 x2 x = 1 # 0 = 0 1. 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 1. 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. 1. lim xS0 sin (1 – cos x) x 26. lim xS0+ sin x sin2x 1. lim xS0 sin (sin x) x 28. lim xS0 sin (x2 + x) x 1. lim xS2 sin (x2 – 4) x – 2 1. lim xS9 sin 12x – 32 x – 9 Oblique Asymptotes Find all possible oblique asymptotes in Exercises 31–34. 1. y = 2×3>2 + 2x – 3 2x + 1 1. y = x + x sin 1x 1. 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. 1. 2. x y 1 2 0 1 P1 P2 x y 0 1 2 2 1 _1 _2 P1 P2 _2 _1 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. 1. y = 4 – x2, (-1, 3) 6. y = (x – 1)2 + 1, (1, 1) 2. y = 22x, (1, 2) 8. y = 1 x2 , (-1, 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. 1. ƒ(x) = x2 + 1, (2, 5) 12. ƒ(x) = x – 2×2, (1, -1) 2. g(x) = x x – 2 , (3, 3) 14. g(x) = 8 x2 , (2, 2) 1. h(t) = t3, (2, 8) 16. h(t) = t3 + 3t, (1, 4) 2. 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. 1. y = 5x – 3×2, x = 1 20. y = x3 – 2x + 7, x = -2 2. y = 1 x – 1 , x = 3 22. y = x – 1 x + 1 , x = 0 Interpreting Derivative Values 1. 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 1. Explain what is meant by the derivative P_(5). What are its units? 1. Which is larger, P_(2) or P_(3)? Give a reason for your answer. 1. 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. 1. 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 1. At what times does the effectiveness appear to be increasing? What is true about the derivative at those times? 1. 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? 1. ƒ(x) = x2 + 4x – 1 26. g(x) = x3 – 3x 2. Find equations of all lines having slope -1 that are tangent to the curve y = 1>(x – 1). 1. Find an equation of the straight line having slope 1>4 that is tangent to the curve y = 2x. Rates of Change 1. 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? 1. 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? 1. 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? 1. 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? 1. Show that the line y = mx + b is its own tangent line at any point (x0, mx0 + b). 1. Find the slope of the tangent to the curve f(x) = 4>2x at the point where x = 1/4. Testing for Tangents 1. 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. 1. 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 1. 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. 1. 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. 1. Where do the graphs appear to have vertical tangents? T 1. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38. 1. y = x2>5 40. y = x4>5 2. y = x1>5 42. y = x3>5 3. y = 4×2>5 – 2x 44. y = x5>3 – 5×2>3 4. y = x2>3 – (x – 1)1>3 46. y = x1>3 + (x – 1)1>3 5. y = e-20 x 0 , x … 0 2x, x 7 0 1. y = 20 4 – x 0 Computer Explorations Use a CAS to perform the following steps for the functions in Exercises 49–52: 1. Plot y = ƒ(x) over the interval (x0 – 1>2) … x … (x0 + 3). 2. 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. 1. Find the limit of q as hS 0. 2. 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). 1. ƒ(x) = x3 + 2x, x0 = 0 50. ƒ(x) = x + 5x , x0 = 1 1. ƒ(x) = x + sin (2x), x0 = p>2 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. 1. ƒ(x) = 2 – x2; ƒ_(-7), ƒ_(0), ƒ_(5) 2. F(x) = (x – 1)2 + 1; F_(-1), F_(0), F_(2) 3. g(t) = 5 t4 ; g_(-4), g_(2), g_1262 1. k(z) = 1 – z 2z ; k_(-1), k_(1), k_1222 1. p(u) = 23u ; p_(1), p_(3), p_(2>3) 2. 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 1. ds dt if s = t 4t + 9 1. 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. 1. ƒ(x) = 3x + 2x , x = -1 14. k(x) = 1 2 + x , x = 2 1. 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. 1. y = ƒ(x) = 8 2x – 2 , (x, y) = (6, 4) 1. 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 1. dr du  u=0 if r = 2 24 – u 1. 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. 1. ƒ(x) = 1 x + 2 1. ƒ(x) = x2 – 3x + 4 2. g(x) = x x – 1 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) 1. 28. x y 0 y _ f1(x) x y 0 y _ f2(x) 1. 30. y 0 x y _ f3(x) y 0 x y _ f4(x) 1. 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) 1. Graph the derivative of ƒ. The graph should show a step function. 1. Recovering a function from its derivative 2. Use the following information to graph the function ƒ over the closed interval 3-2, 54. 1. i) The graph of ƒ is made of closed line segments joined end to end. 1. 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 1. Repeat part (a), assuming that the graph starts at (-2, 0) instead of (-2, 3). 1. 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% 1. 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. 1. 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 1. During what days does the population seem to be increasing fastest? Slowest? 1. 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 1. Estimate the rate of temperature change at the times 2. i) 7 a.m. ii) 9 a.m. iii) 2 p.m. iv) 4 p.m. 3. At what time does the temperature increase most rapidly? Decrease most rapidly? What is the rate for each of those times? 1. Use the graphical technique of Example 3 to graph the derivative of temperature T versus time t. 1. 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 1. Estimate Jared’s rate of weight loss when 2. i) t = 1 ii) t = 4 iii) t = 11 3. When does Jared lose weight most rapidly and what is this rate of weight loss? 1. 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. 1. 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) 1. 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. 1. ƒ(x) = e 2x – 1, x Ú 0 x2 + 2x + 7, x 6 0 1. 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 1. differentiable? 2. continuous but not differentiable? 3. 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 1. dr du  u=0 if r = 2 24 – u 1. 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. 1. ƒ(x) = 1 x + 2 1. ƒ(x) = x2 – 3x + 4 2. g(x) = x x – 1 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) 1. 28. x y 0 y _ f1(x) x y 0 y _ f2(x) 1. 30. y 0 x y _ f3(x) y 0 x y _ f4(x) 1. 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) 1. Graph the derivative of ƒ. The graph should show a step function. 1. Recovering a function from its derivative 2. Use the following information to graph the function ƒ over the closed interval 3-2, 54. 1. i) The graph of ƒ is made of closed line segments joined end to end. 1. 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 1. Repeat part (a), assuming that the graph starts at (-2, 0) instead of (-2, 3). 1. 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% 1. 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. 1. 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 1. During what days does the population seem to be increasing fastest? Slowest? 1. 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 1. Estimate the rate of temperature change at the times 2. i) 7 a.m. ii) 9 a.m. iii) 2 p.m. iv) 4 p.m. 3. At what time does the temperature increase most rapidly? Decrease most rapidly? What is the rate for each of those times? 1. Use the graphical technique of Example 3 to graph the derivative of temperature T versus time t. 1. 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 1. Estimate Jared’s rate of weight loss when 2. i) t = 1 ii) t = 4 iii) t = 11 3. When does Jared lose weight most rapidly and what is this rate of weight loss? 1. 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. 1. 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) 1. 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. 1. ƒ(x) = e 2x – 1, x Ú 0 x2 + 2x + 7, x 6 0 1. 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 1. differentiable? 2. continuous but not differentiable? 3. neither continuous nor differentiable? Give reasons for your answers. 1. 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 1. 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, 1. Find the derivative ƒ_(x) of the given function y = ƒ(x). 2. Graph y = ƒ(x) and y = ƒ_(x) side by side using separate sets of coordinate axes, and answer the following questions. 1. For what values of x, if any, is ƒ_ positive? Zero? Negative? 2. 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.) 1. y = -x2 50. y = -1>x 2. y = x3>3 52. y = x4>4 3. 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? 1. 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? 1. 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. 1. 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. 1. 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. 1. 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). 1. Show that ƒ(x) = c x2 sin 1x , x _ 0 0, x = 0 is differentiable at x = 0 and find ƒ_(0). 1. 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. 1. 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. 1. 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? 1. 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. 1. Plot y = ƒ(x) to see that function’s global behavior. 2. Define the difference quotient q at a general point x, with general step size h. 1. Take the limit as hS 0. What formula does this give? 2. Substitute the value x = x0 and plot the function y = ƒ(x) together with its tangent line at that point. 1. 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? 1. 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. 1. ƒ(x) = x3 + x2 – x, x0 = 1 2. ƒ(x) = x1>3 + x2>3, x0 = 1 3. ƒ(x) = 4x x2 + 1 , x0 = 2 1. ƒ(x) = x – 1 3×2 + 1 , x0 = -1 1. ƒ(x) = sin 2x, x0 = p>2 2. ƒ(x) = x2 cos x, x0 = p>4 Exercises 3.3 Derivative Calculations In Exercises 1–12, find the first and second derivatives. 1. y = -3×9 – 1 2. y = x2 + x + 8 2. s = 7t4 – 4t7 4. w = 3z7 – 7z3 + 21z2 3. y = 4×3 3 – x + 2ex 6. y = x3 3 + x2 2 + e-x 1. y = 8x-7 – 6x 2. s = -2t-1 + 4 t2 1. y = 5×2 – 15x – 4x-2 10. y = 4 – 2x – x-3 2. r = 1 3s2 – 5 2s 1. 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. 1. y = (3 – x2) (x3 – x + 1) 14. y = (2x + 3) (5×2 – 4x) 2. 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. 1. y = 3x – 1 7x + 2 1. z = 4 – 3x 3×2 + x 1. y = 9×2 + 2 x2 + 3 1. ƒ(t) = t2 – 1 t2 + t – 2 1. s = (5 – t) (1 + t2)-1 22. w = (2x – 7)-1(x + 5) 2. ƒ(s) = 1s – 3 1s + 3 1. u = 5x + 1 21x 1. y = 1 + x – 42x x 26. r = 2a 1 2u + 2ub 1. y = 1 (x2 – 1) (x2 + x + 1) 28. y = (x + 1) (x + 2) (x – 1) (x – 2) 1. y = 2e-x + e3x 30. y = x2 + 3ex 2ex – x 1. y = x3ex 32. w = re-r 2. y = x9>4 + e-2x 34. y = x-3>5 + p3>2 3. s = 5t8>5 + 5e4 36. w = 1 z1.4 + p 2z 1. y = 27 x2 – xe 38. y = 23 x9.6 + 2e1.3 1. r = es s 40. r = eua 1 u2 + u-p>2b Find the derivatives of all orders of the functions in Exercises 41–44. 1. y = x4 2 – 3 2 x2 – x 42. y = x5 120 1. 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. 1. y = x3 + 7 x 46. s = t2 + 5t – 1 t2 1. r = (u – 1)(u2 + u + 1) u3 48. u = (x2 + x)(x2 – x + 1) x4 1. w = a1 + 3z 3z b(3 – z) 50. p = q2 + 3 (q – 1)3 + (q + 1)3 1. w = 3z2e2z 52. w = ez(z – 1)(z2 + 1) 2. 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. 1. d dx (uy) b. d dx au yb c. d dx ayu b d. d dx (7y – 2u) 1. 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. 1. d dx (uy) b. d dx au yb c. d dx ayu b d. d dx (7y – 2u) Slopes and Tangents 1. 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). 1. Smallest slope What is the smallest slope on the curve? At what point on the curve does the curve have this slope? 1. Tangents having specified slope Find equations for the tangents to the curve at the points where the slope of the curve is 8. 1. 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. 1. 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. 1. 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 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 1. 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. 1. 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. 1. Find all points (x, y) on the graph of ƒ(x) = 3×2 – 4x with tangent lines parallel to the line y = 8x + 5. 1. 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. 1. Find all points (x, y) on the graph of y = x>(x – 2) with tangent lines perpendicular to the line y = 2x + 3. 1. 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) 1. a. Find an equation for the line that is tangent to the curve y = x3 – x at the point (-1, 0). 1. 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 1. Confirm your estimates of the coordinates of the second intersection point by solving the equations for the curve and tangent simultaneously (Solver key). 1. a. Find an equation for the line that is tangent to the curve y = x3 – 6×2 + 5x at the origin. 1. Graph the curve and tangent together. The tangent intersects the curve at another point. Use Zoom and Trace to estimate the point’s coordinates. 1. 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. 1. lim xS1 x50 – 1 x – 1 1. lim xS-1 x2>9 – 1 x + 1 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 1. 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 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). 1. 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. 1. 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? 1. The Reciprocal Rule 2. 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 1. Show that the Reciprocal Rule and the Derivative Product Rule together imply the Derivative Quotient Rule. 1. 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. 1. What is the analogous formula for the derivative of the product uyw of three differentiable functions of x? 1. What is the formula for the derivative of the product u1 u2 u3 u4 of four differentiable functions of x? 1. What is the formula for the derivative of a product u1 u2 u3gun of a finite number n of differentiable functions of x? 1. 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. 1. 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.) 1. 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. 1. Find the body’s displacement and average velocity for the given time interval. 1. Find the body’s speed and acceleration at the endpoints of the interval. 1. When, if ever, during the interval does the body change direction? 2. s = t2 – 3t + 2, 0 … t … 2 3. s = 6t – t2, 0 … t … 6 4. s = -t3 + 3t2 – 3t, 0 … t … 3 5. s = (t4>4) – t3 + t2, 0 … t … 3 6. s = 25 t2 – 5t , 1 … t … 5 1. s = 25 t + 5 , -4 … t … 0 1. Particle motion At time t, the position of a body moving along the s-axis is s = t3 – 6t2 + 9t m. 1. Find the body’s acceleration each time the velocity is zero. 2. Find the body’s speed each time the acceleration is zero. 3. Find the total distance traveled by the body from t = 0 to t = 2. 4. Particle motion At time t Ú 0, the velocity of a body moving along the horizontal s-axis is y = t2 – 4t + 3. 1. Find the body’s acceleration each time the velocity is zero. 2. When is the body moving forward? Backward? 3. When is the body’s velocity increasing? Decreasing? Free-Fall Applications 1. 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? 1. 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. 1. Find the rock’s velocity and acceleration at time t. (The acceleration in this case is the acceleration of gravity on the moon.) 1. How long does it take the rock to reach its highest point? 2. How high does the rock go? 3. How long does it take the rock to reach half its maximum height? 1. How long is the rock aloft? 2. 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? 1. 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? 1. 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. 1. What would have been the ball’s velocity, speed, and acceleration at time t? 1. About how long would it have taken the ball to hit the ground? 2. What would have been the ball’s velocity at the moment of impact? 1. 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 1. What is the equation for the ball’s velocity during free fall? 2. 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 1. 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) 1. When does the body reverse direction? 2. When (approximately) is the body moving at a constant speed? 3. Graph the body’s speed for 0 … t … 10. 4. Graph the acceleration, where defined. 5. 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) 1. When is P moving to the left? Moving to the right? Standing still? 1. Graph the particle’s velocity and speed (where defined). 2. 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. 1. How fast was the rocket climbing when the engine stopped? 2. 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) 1. When did the rocket reach its highest point? What was its velocity then? 1. When did the parachute pop out? How fast was the rocket falling then? 1. How long did the rocket fall before the parachute opened? 2. When was the rocket’s acceleration greatest? 3. When was the acceleration constant? What was its value then (to the nearest integer)? 1. 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) 1. When does the particle move forward? Move backward? Speed up? Slow down? 1. When is the particle’s acceleration positive? Negative? Zero? 2. When does the particle move at its greatest speed? 3. When does the particle stand still for more than an instant? 4. 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. 1. How long did it take the balls to fall the first 160 cm? What was their average velocity for the period? 1. How fast were the balls falling when they reached the 160-cm mark? What was their acceleration then? 1. About how fast was the light flashing (flashes per second)? 2. 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. 1. 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. 1. 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) 1. 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 1. 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 1. Marginal cost Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x – 0.1×2. 1. Find the average cost per machine of producing the first 100 washing machines. 1. Find the marginal cost when 100 washing machines are produced. 1. 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. 1. Marginal revenue Suppose that the revenue from selling x washing machines is r(x) = 20,000a1 – 1x b dollars. 1. Find the marginal revenue when 100 machines are produced. 2. 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. 1. Find the limit of r_(x) as xS q. How would you interpret this number? Additional Applications 1. 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 1. t = 0 hours. 2. t = 5 hours. 3. t = 10 hours. 4. 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. 1. 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. 1. Find the rate dy>dt (m > h) at which the tank is draining at time t. 1. When is the fluid level in the tank falling fastest? Slowest? What are the values of dy>dt at these times? 1. Graph y and dy>dt together and discuss the behavior of y in relation to the signs and values of dy>dt. 1. 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? 1. 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. 1. Inflating a balloon The volume V = (4>3)pr3 of a spherical balloon changes with the radius. 1. At what rate (ft3>ft) does the volume change with respect to the radius when r = 2 ft? 1. By approximately how much does the volume increase when the radius changes from 2 to 2.2 ft? 1. 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? 1. 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: 1. When is the object momentarily at rest? 2. When does it move to the left (down) or to the right (up)? 3. When does it change direction? 4. When does it speed up and slow down? 5. When is it moving fastest (highest speed)? Slowest? 6. When is it farthest from the axis origin? 7. s = 200t – 16t2, 0 … t … 12.5 (a heavy object fired straight up from Earth’s surface at 200 ft > sec) 1. s = t2 – 3t + 2, 0 … t … 5 2. s = t3 – 6t2 + 7t, 0 … t … 4 3. s = 4 – 7t + 6t2 – t3, 0 … t … 4 Exercises 3.5 Derivatives In Exercises 1–18, find dy>dx. 1. y = -16x + 9 cos x 2. y = 8x + 3 sin x 1. y = x5 cos x 4. y = 2x sec x + 3 2. y = csc x – 41x + 7 ex 6. y = x2 cot x – 1 x2 1. ƒ(x) = sin x tan x 8. g(x) = cos x sin2 x 1. y = xe-x sec x 10. y = (sin x + cos x) sec x 2. y = tan x 1 + tan x 1. y = 4 cos x 1 + sin x 1. y = 4 cos x + 1 tan x 14. y = cos x x + x cos x 1. y = 3(sec x + tan x) (sec x – tan x) 2. y = x2 cos x – 2x sin x – 2 cos x 3. ƒ(x) = x3 sin x cos x 18. g(x) = (2 – x) tan2 x In Exercises 19–22, find ds>dt. 1. s = tan t – e-t 20. s = t2 – sec t + 5et 2. s = 1 + csc t 1 – csc t 1. s = sin t 1 – cos t In Exercises 23–26, find dr>du. 1. r = 4 – u2 sin u 24. r = u sin u + cos u 2. r = sec u csc u 26. r = (1 + sec u) sin u In Exercises 27–32, find dp>dq. 1. p = 5 + 1 cot q 28. p = (1 + csc q) cos q 1. p = sin q + cos q cos q 30. p = tan q 1 + tan q 1. p = q sin q q2 – 1 1. p = 3q + tan q q sec q 1. Find y_ if 2. y = csc x. b. y = sec x. 3. Find y(4) = d4 y>dx4 if 4. 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. 1. y = sin x, -3p>2 … x … 2p x = -p, 0, 3p>2 1. y = tan x, -p>2 6 x 6 p>2 x = -p>3, 0, p>3 1. y = sec x, -p>2 6 x 6 p>2 x = -p>3, p>4 1. 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. 1. y = x + sin x 40. y = 2x + sin x 2. y = x – cot x 42. y = x + 2 cos x 3. 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. 1. 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. 1. lim xS2 sin a1x – 1 2b 48. lim xS -p>6 21 + cos (p csc x) 1. lim uSp>6 sin u – 12 u – p6 1. lim uSp>4 tan u – 1 u – p4 1. lim xS0 secc ex + p tan a p 4 sec xb – 1 d 1. lim xS0 sin a p + tan x tan x – 2 sec xb 1. 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. 1. s = 2 – 2 sin t 56. s = sin t + cos t 2. 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. 1. 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. 1. By computing the first few derivatives and looking for a pattern, find d 999>dx999 (cos x). 1. Derive the formula for the derivative with respect to x of 2. sec x. b. csc x. c. cot x. 3. 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 1. Find the spring’s displacement when t = 0, t = p>3, and t = 3p>4. 1. Find the spring’s velocity when t = 0, t = p>3, and t = 3p>4. 1. 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. 1. Find the particle’s position when t = 0, t = p>2, and t = p. 1. Find the particle’s velocity when t = 0, t = p>2, and t = p. 1. 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? 1. 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? 1. 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 . 1. 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. 1. 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. 1. 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. 1. 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. 1. 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). 1. y = 6u – 9, u = (1>2)x4 2. y = 2u3, u = 8x – 1 2. y = sin u, u = 3x + 1 4. y = cos u, u = e-x 3. y = 2u, u = sin x 6. y = sin u, u = x – cos x 4. 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. 1. y = (2x + 1)5 10. y = (4 – 3x)9 2. y = a1 – x 7b -7 1. y = a2x 2 – 1b -10 1. y = ax2 8 + x – 1x b 4 1. y = 23×2 – 4x + 6 2. y = sec (tan x) 16. y = cot ap – 1x b 1. y = tan3 x 18. y = 5 cos-4 x 2. y = e-5x 20. y = e2x>3 3. y = e5-7x 22. y = e142x+x22 Find the derivatives of the functions in Exercises 23–50. 1. p = 23 – t 24. q = 23 2r – r2 1. s = 4 3p sin 3t + 4 5p cos 5t 26. s = sin a3pt 2 b + cos a3pt 2 b 1. r = (csc u + cot u)-1 28. r = 6 (sec u – tan u)3>2 2. y = x2 sin4 x + x cos-2 x 30. y = 1x sin-5 x – x 3 cos3 x 1. y = 1 18 (3x – 2)6 + a4 – 1 2x2b -1 1. y = (5 – 2x)-3 + 1 8 a2x + 1b 4 1. y = (4x + 3)4(x + 1)-3 34. y = (2x – 5)-1(x2 – 5x)6 2. y = xe-x + ex3 36. y = (1 + 2x)e-2x 3. y = (x2 – 2x + 2)e5x>2 38. y = (9×2 – 6x + 2)ex3 4. h(x) = x tan 121×2 + 7 40. k(x) = x2 sec a1x b 1. ƒ(x) = 27 + x sec x 42. g(x) = tan 3x (x + 7)4 1. ƒ(u) = a sin u 1 + cos u b 2 1. g(t) = a1 + sin 3t 3 – 2t b -1 1. r = sin (u2) cos (2u) 46. r = sec2u tan a1 u b 1. q = sin a t 2t + 1 b 48. q = cotasin t t b 1. y = cos 1e-u22 50. y = u3e-2u cos 5u In Exercises 51–70, find dy>dt. 1. y = sin2 (pt – 2) 52. y = sec2 pt 2. y = (1 + cos 2t)-4 54. y = (1 + cot (t>2))-2 3. y = (t tan t)10 56. y = (t-3>4 sin t)4>3 4. y = ecos2 (pt-1) 58. y = (esin (t>2))3 5. y = a t2 t3 – 4t b 3 1. y = a3t – 4 5t + 2 b -5 1. y = sin (cos (2t – 5)) 62. y = cos a5 sin a t 3b b 1. y = a1 + tan4 a t 12b b 3 1. y = 1 6 11 + cos2 (7t)23 1. y = 21 + cos (t2) 66. y = 4 sin 121 + 1t2 2. y = tan2 (sin3 t) 68. y = cos4 (sec2 3t) 3. y = 3t (2t2 – 5)4 70. y = 43t + 32 + 21 – t Second Derivatives Find y_ in Exercises 71–78. 1. y = a1 + 1x b 3 1. y = 11 – 1×2-1 2. y = 1 9 cot (3x – 1) 74. y = 9 tan ax 3b 1. y = x (2x + 1)4 76. y = x2 (x3 – 1)5 2. 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. 1. ƒ(u) = u5 + 1, u = g(x) = 1x, x = 1 2. ƒ(u) = 1 – 1u , u = g(x) = 1 1 – x , x = -1 1. ƒ(u) = cot pu 10 , u = g(x) = 51x, x = 1 1. ƒ(u) = u + 1 cos2 u , u = g(x) = px, x = 1>4 1. ƒ(u) = 2u u2 + 1 , u = g(x) = 10×2 + x + 1, x = 0 1. ƒ(u) = au – 1 u + 1b 2 , u = g(x) = 1 x2 – 1, x = -1 1. Assume that ƒ_(3) = -1, g_(2) = 5, g(2) = 3, and y = ƒ(g(x)). What is y_ at x = 2? 1. If r = sin (ƒ(t)), ƒ(0) = p>3, and ƒ_(0) = 4, then what is dr>dt at t = 0? 1. 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. 1. 2ƒ(x), x = 2 b. ƒ(x) + g(x), x = 3 2. ƒ(x) # g(x), x = 3 d. ƒ(x)>g(x), x = 2 3. ƒ(g(x)), x = 2 f. 2ƒ(x), x = 2 4. 1>g2(x), x = 3 h. 2ƒ2(x) + g2(x), x = 2 5. 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. 1. 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 1. g(ƒ(x)), x = 0 f. (x11 + ƒ(x))-2, x = 1 2. ƒ(x + g(x)), x = 0 3. Find ds>dt when u = 3p>2 if s = cos u and du>dt = 5. 4. 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. 1. Find dy>dx if y = x by using the Chain Rule with y as a composite of 1. y = (u>5) + 7 and u = 5x – 35 2. y = 1 + (1>u) and u = 1>(x – 1). 3. Find dy>dx if y = x3>2 by using the Chain Rule with y as a composite of 1. y = u3 and u = 1x 2. y = 1u and u = x3. 3. Find the tangent to y = ((x – 1)>(x + 1))2 at x = 0. 4. Find the tangent to y = 2×2 – x + 7 at x = 2. 5. a. Find the tangent to the curve y = 2 tan (px>4) at x = 1. 6. 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. 1. Slopes on sine curves 2. 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. 1. 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. 1. 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. 1. 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. 1. 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.) 1. 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. 1. On what day is the temperature increasing the fastest? 2. 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) 1. 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. 1. 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. 1. 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. 1. 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). 1. 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 1. 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. 1. 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. 1. 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. 1. 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. 1. x1>4 = 21x 108. x3>4 = 2x1x COMPUTER EXPLORATIONS Trigonometric Polynomials 1. 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. 1. Graph dg>dt (where defined) over 3-p, p4. 2. Find dƒ>dt. 3. 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) 1. (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) 1. Graph dk>dt (where defined) over 3-p, p4. 2. Find dh>dt. 3. 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. 1. x2y + xy2 = 6 2. x3 + y3 = 18xy 2. 2xy + y2 = x + y 4. x3 – xy + y3 = 1 3. x2(x – y)2 = x2 – y2 6. (3xy + 7)2 = 6y 4. y2 = x – 1 x + 1 1. x3 = 2x – y x + 3y 1. x = sec y 10. xy = cot (xy) 2. x + tan (xy) = 0 12. x4 + sin y = x3y2 3. y sin a1y b = 1 – xy 14. x cos (2x + 3y) = y sin x 1. e2x = sin (x + 3y) 16. ex2y = 2x + 2y Find dr>du in Exercises 17–20. 1. u1>2 + r1>2 = 1 18. r – 22u = 3 2 u2>3 + 4 3 u3>4 1. sin (r u) = 1 2 1. cos r + cot u = er u Second Derivatives In Exercises 21–26, use implicit differentiation to find dy>dx and then d2y>dx2. 1. x2 + y2 = 1 22. x2>3 + y2>3 = 1 2. y2 = ex2 + 2x 24. y2 – 2x = 1 – 2y 3. 21y = x – y 26. xy + y2 = 1 4. If x3 + y3 = 16, find the value of d2y>dx2 at the point (2, 2). 5. 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. 1. y2 + x2 = y4 – 2x at (-2, 1) and (-2, -1) 2. (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. 1. x2 + xy – y2 = 1, (2, 3) 2. x2 + y2 = 25, (3, -4) 3. x2y2 = 9, (-1, 3) 4. y2 – 2x – 4y – 1 = 0, (-2, 1) 5. 6×2 + 3xy + 2y2 + 17y – 6 = 0, (-1, 0) 6. x2 – 23xy + 2y2 = 5, 123, 22 7. 2xy + p sin y = 2p, (1, p>2) 8. x sin 2y = y cos 2x, (p>4, p>2) 9. y = 2 sin (px – y), (1, 0) 10. x2 cos2 y – sin y = 0, (0, p) 11. 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? 1. Normals parallel to a line Find the normals to the curve xy + 2x – y = 0 that are parallel to the line 2x + y = 0. 1. 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 1. 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 1. 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 1. The folium of Descartes (See Figure 3.28.) 2. Find the slope of the folium of Descartes x3 + y3 – 9xy = 0 at the points (4, 2) and (2, 4). 1. At what point other than the origin does the folium have a horizontal tangent? 1. Find the coordinates of the point A in Figure 3.28 where the folium has a vertical tangent. Theory and Examples 1. Intersecting normal The line that is normal to the curve x2 + 2xy – 3y2 = 0 at (1, 1) intersects the curve at what other point? 1. 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. 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? 1. 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) 1. Verify that the following pairs of curves meet orthogonally. 2. x2 + y2 = 4, x2 = 3y2 3. x = 1 – y2, x = 1 3 y2 1. 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. 1. xy3 + x2y = 6 2. x3 + y2 = sin2 y 3. 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 . 1. Use the formula in Exercise 55 to find dy>dx if 2. 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. 1. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation. 1. Using implicit differentiation, find a formula for the derivative dy>dx and evaluate it at the given point P. 1. 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. 1. x3 – xy + y3 = 7, P (2, 1) 2. x5 + y3x + yx2 + y4 = 4, P (1, 1) 3. y2 + y = 2 + x 1 – x , P (0, 1) 1. y3 + cos xy = x2, P (1, 0) 2. x + tan a y xb = 2, P a1, p 4 b 1. xy3 + tan (x + y) = 1, P ap 4 , 0b 1. 2y2 + (xy)1>3 = x2 + 2, P (1, 1) 2. x21 + 2y + y = x2, P (1, 0) Exercises 3.8 Derivatives of Inverse Functions In Exercises 1–4: 1. Find ƒ -1(x). 2. Graph ƒ and ƒ -1 together. 3. 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). 1. ƒ(x) = 2x + 3, a = -1 2. ƒ(x) = (1>5)x + 7, a = -1 2. ƒ(x) = 5 – 4x, a = 1>2 4. ƒ(x) = 2×2, x Ú 0, a = 5 3. a. Show that ƒ(x) = x3 and g(x) = 1 3 x are inverses of one another. 1. 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. 1. Find the slopes of the tangents to the graphs of ƒ and g at (1, 1) and (-1, -1) (four tangents in all). 1. What lines are tangent to the curves at the origin? 2. a. Show that h(x) = x3>4 and k(x) = (4x)1>3 are inverses of one another. 1. 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. 1. Find the slopes of the tangents to the graphs at h and k at (2, 2) and (-2, -2). 1. What lines are tangent to the curves at the origin? 2. Let ƒ(x) = x3 – 3×2 – 1, x Ú 2. Find the value of dƒ -1>dx at the point x = -1 = ƒ(3). 1. Let ƒ(x) = x2 – 4x – 5, x 7 2. Find the value of dƒ -1>dx at the point x = 0 = ƒ(5). 1. 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. 1. 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. 1. y = ln 3x + x 12. y = 1 ln 3x 1. y = ln (t2) 14. y = ln (t3>2) + 2t 2. y = ln 3x 1. y = ln (sin x) 2. y = ln (u + 1) – eu 18. y = (cos u) ln (2u + 2) 3. y = ln x3 20. y = (ln x)3 4. y = t (ln t)2 22. y = t ln 2t 5. y = x4 4 ln x – x4 16 1. y = (x2 ln x)4 2. y = ln t t 26. y = t 2ln t 1. y = ln x 1 + ln x 1. y = x ln x 1 + ln x 1. y = ln (ln x) 30. y = ln (ln (ln x)) 2. y = u(sin (ln u) + cos (ln u)) 3. y = ln (sec u + tan u) 4. y = ln 1 x2x + 1 1. y = 1 2 ln 1 + x 1 – x 1. y = 1 + ln t 1 – ln t 1. y = 2ln 1t 2. y = ln (sec (ln u)) 38. y = ln a2sin u cos u 1 + 2 ln u b 1. 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. 1. y = 2x(x + 1) 42. y = 2(x2 + 1)(x – 1)2 2. y = A t t + 1 1. y = A 1 t(t + 1) 1. y = (sin u)2u + 3 46. y = (tan u)22u + 1 2. y = t(t + 1)(t + 2) 48. y = 1 t(t + 1)(t + 2) 1. y = u + 5 u cos u 1. y = u sin u 2sec u 1. y = x2x2 + 1 (x + 1)2>3 52. y = C (x + 1)10 (2x + 1)5 1. y = B 3 x(x – 2) x2 + 1 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. 1. y = ln (cos2 u) 56. y = ln (3ue-u) 2. y = ln (3te-t) 58. y = ln (2e-t sin t) 3. y = ln a eu 1 + eub 60. y = ln a 2u 1 + 2u b 1. y = e(cos t + ln t) 62. y = esin t (ln t2 + 1) In Exercises 63–66, find dy>dx. 1. ln y = ey sin x 64. ln xy = ex+y 2. xy = yx 66. tan y = ex + ln x In Exercises 67–88, find the derivative of y with respect to the given independent variable. 1. y = 2x 68. y = 3-x 2. y = 52s 70. y = 2(s2) 3. y = xp 72. y = t1-e 4. y = log2 5u 74. y = log3 (1 + u ln 3) 5. y = log4 x + log4 x2 76. y = log25 ex – log51x 6. y = log2 r # log4 r 78. y = log3 r # log9 r 7. y = log3 a ax + 1 x – 1b ln 3 b 80. y = log5 B a 7x 3x + 2b ln 5 1. y = u sin (log7 u) 82. y = log7 asin u cos u eu 2u b 1. y = log5 ex 84. y = log2 a x2e2 22x + 1 b 1. y = 3log2 t 86. y = 3 log8 (log2 t) 2. 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. 1. y = (x + 1)x 90. y = x(x+1) 2. y = (1t)t 92. y = t2t 3. y = (sin x)x 94. y = xsin x 4. y = xln x 96. y = (ln x)ln x Theory and Applications 1. 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.) 1. Show that limnSq a1 + xn b n = ex for any x 7 0. 1. If ƒ(x) = xn, n Ú 1, show from the definition of the derivative that ƒ_(0) = 0. 1. 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: 1. 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. 1. Solve the equation y = ƒ(x) for x as a function of y, and name the resulting inverse function g. 1. Find the equation for the tangent line to ƒ at the specified point (x0, ƒ(x0)). 1. 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. 1. 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. 1. y = 23x – 2, 2 3 … x … 4, x0 = 3 1. y = 3x + 2 2x – 11 , -2 … x … 2, x0 = 1>2 1. y = 4x x2 + 1 , -1 … x … 1, x0 = 1>2 1. y = x3 x2 + 1 , -1 … x … 1, x0 = 1>2 1. y = x3 – 3×2 – 1, 2 … x … 5, x0 = 27 10 1. y = 2 – x – x3, -2 … x … 2, x0 = 3 2 1. y = ex, -3 … x … 5, x0 = 1 2. 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. 1. y1>3 – 1 = (x + 2)3, -5 … x … 5, x0 = -3>2 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. 1. a. tan-1 1 b. tan-11-232 c. tan-1 a 1 23 b 1. a. tan-1(-1) b. tan-123 c. tan-1 a -1 23 b 1. a. sin-1 a-1 2 b b. sin-1 a 1 22 b c. sin-1 a-23 2 b 1. a. sin-1 a1 2b b. sin-1 a -1 22 b c. sin-1 a23 2 b 1. a. cos-1 a1 2b b. cos-1 a -1 22 b c. cos-1 a23 2 B 1. a. csc-1 22 b. csc-1 a -2 23 b c. csc-1 2 1. a. sec-11-222 b. sec-1 a 2 23 b c. sec-1(-2) 1. a. cot-1 (-1) b. cot-1 1232 c. cot-1 a -1 23 b Evaluations Find the values in Exercises 9–12. 1. sin acos-1 a22 2 b b 10. sec acos-1 1 2b 1. 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.) 1. lim xS1- sin-1 x 14. lim xS-1+ cos-1 x 1. lim xSq tan-1 x 16. lim xS-q tan-1 x 1. lim xSq sec-1 x 18. lim xS-q sec-1 x 1. 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. 1. y = cos-1 (x2) 22. y = cos-1 (1>x) 2. y = sin-122 t 24. y = sin-1 (1 – t) 3. y = sec-1 (2s + 1) 26. y = sec-1 5s 4. y = csc-1 (x2 + 1), x 7 0 5. y = csc-1 x 2 1. y = sec-1 1t , 0 6 t 6 1 30. y = sin-1 3 t2 1. y = cot-1 2t 32. y = cot-1 2t – 1 2. y = ln (tan-1 x) 34. y = tan-1 (ln x) 3. y = csc-1 (et) 36. y = cos-1 (e-t) 4. y = s21 – s2 + cos-1 s 38. y = 2s2 – 1 – sec-1 s 5. y = tan-12×2 – 1 + csc-1 x, x 7 1 6. y = cot-1 1x – tan-1 x 41. y = x sin-1 x + 21 – x2 1. y = ln (x2 + 4) – x tan-1 ax 2b Theory and Examples 1. 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 1. Two derivations of the identity sec_1 (_x) _ P _ sec_1 x 2. (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 1. (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. 1. a. tan-1 2 b. cos-1 2 2. a. csc-1 (1>2) b. csc-1 2 3. a. sec-1 0 b. sin-122 4. a. cot-1 (-1>2) b. cos-1 (-5) 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. 1. 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. 1. 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. 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. 1. 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. 1. 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. 1. a. y = tan-1 (tan x) b. y = tan (tan-1 x) 2. a. y = sin-1 (sin x) b. y = sin (sin-1 x) 3. a. y = cos-1 (cos x) b. y = cos (cos-1 x) T Use your graphing utility for Exercises 58–62. 1. Graph y = sec (sec-1 x) = sec (cos-1(1>x)). Explain what you see. 1. 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. 1. 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. 1. 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_. 1. 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 1. 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. 1. 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. 1. Assume that y = 5x and dx>dt = 2. Find dy>dt. 2. Assume that 2x + 3y = 12 and dy>dt = -2. Find dx>dt. 3. If y = x2 and dx>dt = 3, then what is dy>dt when x = -1? 4. If x = y3 – y and dy>dt = 5, then what is dx>dt when y = 2? 5. If x2 + y2 = 25 and dx>dt = -2, then what is dy>dt when x = 3 and y = -4? 1. If x2y3 = 4>27 and dy>dt = 1>2, then what is dx>dt when x = 2? 1. If L = 2×2 + y2, dx>dt = -1, and dy>dt = 3, find dL>dt when x = 5 and y = 12. 1. If r + s2 + y3 = 12, dr>dt = 4, and ds>dt = -3, find dy>dt when r = 3 and s = 1. 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 1. surface area change? 2. volume change? 3. 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? 1. 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. 1. How is dV>dt related to dh>dt if r is constant? 2. How is dV>dt related to dr>dt if h is constant? 3. How is dV>dt related to dr>dt and dh>dt if neither r nor h is constant? 1. 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. 1. How is dV>dt related to dh>dt if r is constant? 2. How is dV>dt related to dr>dt if h is constant? 3. How is dV>dt related to dr>dt and dh>dt if neither r nor h is constant? 1. 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 + _ 1. What is the value of dV>dt? 2. What is the value of dI>dt? 3. What equation relates dR>dt to dV>dt and dI>dt? 4. Find the rate at which R is changing when V = 12 volts and I = 2 amps. Is R increasing, or decreasing? 1. 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. 1. How are dP>dt, dR>dt, and dI>dt related if none of P, R, and I are constant? 1. How is dR>dt related to dI>dt if P is constant? 2. 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. 1. How is ds>dt related to dx>dt if y is constant? 2. How is ds>dt related to dx>dt and dy>dt if neither x nor y is constant? 1. How is dx>dt related to dy>dt if s is constant? 2. 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. 1. Assuming that x, y, and z are differentiable functions of t, how is ds>dt related to dx>dt, dy>dt, and dz>dt? 1. How is ds>dt related to dy>dt and dz>dt if x is constant? 2. How are dx>dt, dy>dt, and dz>dt related if s is constant? 3. 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. 1. How is dA>dt related to du>dt if a and b are constant? 2. How is dA>dt related to du>dt and da>dt if only b is constant? 3. How is dA>dt related to du>dt, da>dt, and db>dt if none of a, b, and u are constant? 1. 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? 1. 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? 1. 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. 1. 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. 1. How fast is the top of the ladder sliding down the wall then? 2. At what rate is the area of the triangle formed by the ladder, wall, and ground changing then? 1. 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 1. 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? 1. 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? 1. 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.? 1. 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. 1. 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. 1. How fast (centimeters per minute) is the water level falling when the water is 5 m deep? 1. How fast is the radius of the water’s surface changing then? Answer in centimeters per minute. 1. 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 1. At what rate is the water level changing when the water is 8 m deep? 1. What is the radius r of the water’s surface when the water is y m deep? 1. At what rate is the radius r changing when the water is 8 m deep? 1. 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. 1. 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? 1. 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. 1. How fast is the boat approaching the dock when 10 ft of rope are out? 1. At what rate is the angle u changing at this instant (see the figure)? 1. 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) 1. Making coffee Coffee is draining from a conical filter into a cylindrical coffeepot at the rate of 10 in3>min. 1. How fast is the level in the pot rising when the coffee in the cone is 5 in. deep? 1. 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? 1. 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? 1. 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? 1. 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)? 1. 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_ 1. 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.) 1. 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 1. 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? 1. 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. 1. 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. 1. At what rate is the player’s distance from third base changing when the player is 30 ft from first base? 1. At what rates are angles u1 and u2 (see the figure) changing at that time? 1. 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 1. 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? 1. 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? 1. 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. 1. ƒ(x) = x3 – 2x + 3, a = 2 2. ƒ(x) = 2×2 + 9, a = -4 3. ƒ(x) = x + 1x , a = 1 1. ƒ(x) = 23 x, a = -8 1. ƒ(x) = tan x, a = p 2. Common linear approximations at x _ 0 Find the linearizations of the following functions at x = 0. 1. 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. 1. ƒ(x) = x2 + 2x, a = 0.1 2. ƒ(x) = x-1, a = 0.9 3. ƒ(x) = 2×2 + 3x – 3, a = -0.9 4. ƒ(x) = 1 + x, a = 8.1 5. ƒ(x) = 23 x, a = 8.5 1. ƒ(x) = x x + 1 , a = 1.3 1. ƒ(x) = e-x, a = -0.1 2. ƒ(x) = sin-1 x, a = p>12 3. Show that the linearization of ƒ(x) = (1 + x)k at x = 0 is L(x) = 1 + kx. 1. Use the linear approximation (1 + x)k _ 1 + kx to find an approximation for the function ƒ(x) for values of x near zero. 1. ƒ(x) = (1 – x)6 b. ƒ(x) = 2 1 – x 1. ƒ(x) = 1 21 + x 1. ƒ(x) = 22 + x2 2. ƒ(x) = (4 + 3x)1>3 f. ƒ(x) = B 3 a1 – x 2 + xb 2 1. Faster than a calculator Use the approximation (1 + x)k _ 1 + kx to estimate the following. 1. (1.0002)50 b. 23 1.009 1. 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. 1. y = x3 – 32x 20. y = x21 – x2 2. y = 2x 1 + x2 22. y = 21x 3(1 + 1x) 1. 2y3>2 + xy – x = 0 24. xy2 – 4×3>2 – y = 0 2. y = sin (51x) 26. y = cos (x2) 3. y = 4 tan (x3>3) 28. y = sec (x2 – 1) 4. y = 3 csc 11 – 22×2 30. y = 2 cot a 1 1x b 1. y = e2x 32. y = xe-x 2. y = ln (1 + x2) 34. y = ln a x + 1 2x – 1 b 1. y = tan-1 (ex2) 36. y = cot-1 a 1 x2b + cos-1 2x 1. 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 1. the change _ƒ = ƒ(x0 + dx) – ƒ(x0); 2. the value of the estimate dƒ = ƒ_(x0) dx; and 3. 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 1. ƒ(x) = x2 + 2x, x0 = 1, dx = 0.1 2. ƒ(x) = 2×2 + 4x – 3, x0 = -1, dx = 0.1 3. ƒ(x) = x3 – x, x0 = 1, dx = 0.1 4. ƒ(x) = x4, x0 = 1, dx = 0.1 5. ƒ(x) = x-1, x0 = 0.5, dx = 0.1 6. 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. 1. The change in the volume V = (4>3)pr3 of a sphere when the radius changes from r0 to r0 + dr 1. The change in the volume V = x3 of a cube when the edge lengths change from x0 to x0 + dx 1. The change in the surface area S = 6×2 of a cube when the edge lengths change from x0 to x0 + dx 1. 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 1. 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 1. 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 1. The radius of a circle is increased from 2.00 to 2.02 m. 2. Estimate the resulting change in area. 3. Express the estimate as a percentage of the circle’s original area. 4. 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? 1. 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. 1. 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%? 1. 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 1. circumference? b. area? 2. 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 1. surface area? b. volume? 2. 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. 1. Tolerance 2. 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? 1. 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? 1. 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. 1. Estimate the allowable percentage error in measuring the diameter D of a sphere if the volume is to be calculated correctly to within 3%. 1. 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. 1. 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. 1. 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 ? 1. 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 1. 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. 1. With L held constant and g as the independent variable, calculate dT and use it to answer parts (b) and (c). 1. If g increases, will T increase or decrease? Will a pendulum clock speed up or slow down? Explain. 1. 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. 1. Quadratic approximations 2. Let Q(x) = b0 + b1(x – a) + b2(x – a)2 be a quadratic approximation to ƒ(x) at x = a with the properties: 1. i) Q(a) = ƒ(a) 2. ii) Q_(a) = ƒ_(a) iii) Q_(a) = ƒ_(a). Determine the coefficients b0, b1, and b2. 1. Find the quadratic approximation to ƒ(x) = 1>(1 – x) at x = 0. 1. 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. 1. 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 1. Find the quadratic approximation to h(x) = 21 + x at x = 0. Graph h and its quadratic approximation together. Comment on what you see. 1. What are the linearizations of ƒ, g, and h at the respective points in parts (b), (d), and (e)? 1. 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 1. E(a) = 0 The approximation error is zero at x = a. 2. 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 1. The linearization of 2x 2. Find the linearization of ƒ(x) = 2x at x = 0. Then round its coefficients to two decimal places. 1. Graph the linearization and function together for -3 … x … 3 and -1 … x … 1. 1. The linearization of log3 x 2. Find the linearization of ƒ(x) = log3 x at x = 3. Then round its coefficients to two decimal places. 1. 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: 1. Plot the function ƒ over I. 2. Find the linearization L of the function at the point a. 3. Plot ƒ and L together on a single graph. 4. Plot the absolute error _ ƒ(x) – L(x) _ over I and find its maximum value. 1. 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. 1. ƒ(x) = x3 + x2 – 2x, 3-1, 24, a = 1 2. ƒ(x) = x – 1 4×2 + 1 , c – 3 4 , 1 d , a = 1 2 1. ƒ(x) = x2>3(x – 2), 3-2, 34, a = 2 2. ƒ(x) = 2x – sin x, 30, 2p4, a = 2 3. ƒ(x) = x2x, 30, 24, a = 1 4. ƒ(x) = 2x sin-1 x, 30, 14, a = 1 2 Chapter 3 Questions to Guide Your Review 1. What is the derivative of a function ƒ? How is its domain related to the domain of ƒ? Give examples. 1. What role does the derivative play in defining slopes, tangents, and rates of change? 1. How can you sometimes graph the derivative of a function when all you have is a table of the function’s values? 1. What does it mean for a function to be differentiable on an open interval? On a closed interval? 1. How are derivatives and one-sided derivatives related? 2. Describe geometrically when a function typically does not have a derivative at a point. 1. How is a function’s differentiability at a point related to its continuity there, if at all? 1. What rules do you know for calculating derivatives? Give some examples. 1. Explain how the three formulas 2. d dx (xn) = nxn-1 b. d dx (cu) = c du dx 1. d dx (u1 + u2 + g+ un) = du1 dx + du2 dx + g+ dun dx enable us to differentiate any polynomial. 1. What formula do we need, in addition to the three listed in Question 9, to differentiate rational functions? 1. What is a second derivative? A third derivative? How many derivatives do the functions you know have? Give examples. 1. What is the derivative of the exponential function ex? How does the domain of the derivative compare with the domain of the function? 1. What is the relationship between a function’s average and instantaneous rates of change? Give an example. 1. 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. 1. How can derivatives arise in economics? 2. Give examples of still other applications of derivatives. 3. 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? 1. 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? 1. At what points are the six basic trigonometric functions continuous? How do you know? 1. What is the rule for calculating the derivative of a composite of two differentiable functions? How is such a derivative evaluated? Give examples. 1. 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. 1. What is implicit differentiation? When do you need it? Give examples. 1. 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? 1. 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? 1. What is the derivative of loga x? Are there any restrictions on a? 2. What is logarithmic differentiation? Give an example. 3. 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? 1. 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? 1. What are the derivatives of the inverse trigonometric functions? How do the domains of the derivatives compare with the domains of the functions? 1. How do related rates problems arise? Give examples. 2. Outline a strategy for solving related rates problems. Illustrate with an example. 1. 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. 1. 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. 1. y = x3 – 0.375×2 + 0.70x 2. y = 8 + 0.5×5 – 0.7×6 2. y = x5 – 5(x2 + 4) 4. y = x5 + 25x – 1 p + 5 1. y = (x + 3)2(x3 + 5×2) 6. y = (4x + 7)(2 – x)-2 2. y = (x3 + sin x + 3)4 8. y = a-3 – sec u 3 – u3 6 b 3 1. r = 1s 5 – 1s 1. r = 5 1s – 8 1. y = 2 cot2 x + csc2 x 12. y = 1 cos2 x – 3 tan x 1. r = tan4 (3 + 4t) 14. r = cot2 a6t b 1. r = (cos t + cot t)6 16. s = sec6 (8 – t2 + 6t3) 2. r = 23u tan u 18. r = 3u2tan u 3. r = tan 25u 20. r = cos 1u – 2u – 12 4. y = 1 3 x3 sec 3x 1. y = 52x cos 2x 2. y = x-3>2 csc (3x)3 24. y = 2x sec (x – 4)4 3. y = 6 tan x3 26. y = x3 cot 8x 4. y = x3 cos2 (4×2) 28. y = x-3 cos2 (x5) 5. s = a 6t t + 3b -3 1. s = -8 25(25t + 4)3 1. y = a 2x 2 – xb 3 1. y = a 32x 32x – 4 b 3 1. y = B x3 + x x3 34. y = 6x22x3 + 1×3 1. r = a cos u sin u + 1b 3 1. r = a2 – cos u 2 + sin u b 3 1. y = (3x + 2)23x + 2 38. y = 30 (5x – 3)1>5 (5x – 3)-1>6 2. y = 8 (9×3 + cos 2x)5>2 40. y = (2 + sin2 2x)-1>2 1. y = 16e-x>8 42. y = 25e25x 2. y = 1 5 xe5x – 1 25 e5x 44. y = x3e-3>x 1. y = ln (tan2 u) 46. y = ln (cos2 u) 2. y = log3 (x3>3) 48. y = log3 (8x + 5) 3. y = 9-x 50. y = 34x 4. y = 8×7.2 52. y = 25t-25 5. y = (x + 3)x+3 54. y = 5 (ln x)x>5 6. y = cos-1(2×21 – x2), 1 12 6 x 6 1 1. y = cos-1 a 1 2x b, x 7 1 1. y = ln sin-1 x 2. y = x sin-1 x + 21 – x2 3. y = x cot-1 x – 1 2 ln x 1. y = (1 – x2) tan-1 3x 2. y = t csc-1 t + 2t2 – 1, t 7 1 3. y = 52t + 1 tan-1 1t 4. y = csc-1 (sec u), 0 6 u 6 p>2 5. y = 21 – x2 esin-1 x Implicit Differentiation In Exercises 65–78, find dy>dx by implicit differentiation. 1. xy + 2x + 3y = 1 66. x2 + xy + y2 – 5x = 2 2. x3 + 4xy – 3y4>3 = 2x 68. 5×4>5 + 10y6>5 = 15 3. 1xy = 1 70. x2y2 = 1 4. y2 = x x + 1 1. y2 = A 1 + x 1 – x 1. ex+2y = 1 74. y2 = 2e-1>x 2. ln (x>y) = 1 76. x sin-1 y = 1 + x2 3. yetan-1 x = 2 78. xy = 22 In Exercises 79 and 80, find dp>dq. 1. p3 + 4pq – 3q2 = 2 80. q = (5p2 + 2p)-3>2 In Exercises 81 and 82, find dr>ds. 1. r cos 2s + sin2 s = p 82. 2rs – r – s + s2 = -3 2. Find d2y>dx2 by implicit differentiation: 3. x3 + y3 = 1 b. y2 = 1 – 2x 4. a. By differentiating x2 – y2 = 1 implicitly, show that dy>dx = x>y. 1. Then show that d2y>dx2 = -1>y3. Numerical Values of Derivatives 1. 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. 1. 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 1. g(ƒ(x)), x = 0 f. (x + ƒ(x))3>2, x = 1 2. ƒ(x + g(x)), x = 0 3. 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. 1. 1x ƒ(x), x = 1 b. 2ƒ(x), x = 0 2. 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 1. Find the value of dy>dt at t = 0 if y = 3 sin 2x and x = t2 + p. 2. Find the value of ds>du at u = 2 if s = t2 + 5t and t = (u2 + 2u)1>3. 1. Find the value of dw>ds at s = 0 if w = sin 1e1r2 and r = 3 sin (s + p>6). 1. Find the value of dr>dt at t = 0 if r = (u2 + 7)1>3 and u2t + u = 1. 1. If y3 + y = 2 cos x, find the value of d2y>dx2 at the point (0, 1). 2. 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. 1. ƒ(t) = 1 2t + 1 1. g(x) = 2×2 + 1 2. a. Graph the function ƒ(x) = e x2, -1 … x 6 0 -x2, 0 … x … 1. 1. Is ƒ continuous at x = 0? 2. Is ƒ differentiable at x = 0? Give reasons for your answers. 1. a. Graph the function ƒ(x) = e x, -1 … x 6 0 tan x, 0 … x … p>4. 1. Is ƒ continuous at x = 0? 2. Is ƒ differentiable at x = 0? Give reasons for your answers. 1. a. Graph the function ƒ(x) = e x, 0 … x … 1 2 – x, 1 6 x … 2. 1. Is ƒ continuous at x = 1? 2. Is ƒ differentiable at x = 1? Give reasons for your answers. 1. For what value or values of the constant m, if any, is ƒ(x) = e sin 2x, x … 0 mx, x 7 0 1. continuous at x = 0? 2. differentiable at x = 0? Give reasons for your answers. Slopes, Tangents, and Normals 1. 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. 1. Tangents with specified slope Are there any points on the curve y = x – e-x where the slope is 2? If so, find them. 1. Horizontal tangents Find the points on the curve y = 2×3 – 3×2 – 12x + 20 where the tangent is parallel to the x-axis. 1. Tangent intercepts Find the x- and y-intercepts of the line that is tangent to the curve y = x3 at the point (-2, -8). 1. Tangents perpendicular or parallel to lines Find the points on the curve y = 2×3 – 3×2 – 12x + 20 where the tangent is 1. perpendicular to the line y = 1 – (x>24). 2. parallel to the line y = 22 – 12x. 3. Intersecting tangents Show that the tangents to the curve y = (p sin x)>x at x = p and x = -p intersect at right angles. 1. 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. 1. 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. 1. Tangent parabola The parabola y = x2 + C is to be tangent to the line y = x. Find C. 1. 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). 1. 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)? 1. 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. 1. x2 + 2y2 = 9, (1, 2) 2. ex + y2 = 2, (0, 1) 3. xy + 2x – 5y = 2, (3, 2) 4. (y – x)2 = 2x + 4, (6, 2) 5. x + 1xy = 6, (4, 1) 6. x3>2 + 2y3>2 = 17, (1, 4) 7. Find the slope of the curve x3y3 + y2 = x + y at the points (1, 1) and (1, -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? 1. 120. x y _1 0 1 1 _1 _2 A 2 B x y 0 1 1 A B 2 2 3 4 1. Use the following information to graph the function y = ƒ(x) for -1 … x … 6. 1. i) The graph of ƒ is made of line segments joined end to end. 2. 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) 1. Repeat Exercise 121, supposing that the graph starts at (-1, 0) instead of (-1, 2). Trigonometric Limits Find the limits in Exercises 123–130. 1. lim xS0 sin x 2×2 – x 1. lim xS0 3x – tan 7x 2x 1. lim rS0 sin r tan 2r 1. lim uS0 sin (sin u) u 1. lim uS(p>2)- 4 tan2 u + tan u + 1 tan2 u + 5 1. lim uS0+ 1 – 2 cot2 u 5 cot2 u – 7 cot u – 8 1. lim xS0 x sin x 2 – 2 cos x 1. lim uS0 1 – cos u u2 Show how to extend the functions in Exercises 131 and 132 to be continuous at the origin. 1. 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. 1. y = 2(x2 + 1) 2cos 2x 1. y = 10A 3x + 4 2x – 4 1. y = a (t + 1)(t – 1) (t – 2)(t + 3)b 5 , t 7 2 1. y = 2u2u 2u2 + 1 1. y = (sin u)2u 138. y = (ln x)1>(ln x) Related Rates 1. 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. 1. How is dS>dt related to dr>dt if h is constant? 2. How is dS>dt related to dh>dt if r is constant? 3. How is dS>dt related to dr>dt and dh>dt if neither r nor h is constant? 1. How is dr>dt related to dh>dt if S is constant? 2. 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? 1. 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? 1. 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? 1. 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)? 1. 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. 1. Draining a tank Water drains from the conical tank shown in the accompanying figure at the rate of 5 ft3>min. 1. What is the relation between the variables h and r in the figure? 2. How fast is the water level dropping when h = 6 ft? r h Exit rate: 5 ft3_min 10_ 4_ 1. 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_ 1. 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. 1. How fast is the light moving along the shore when it reaches point A? 1. How many revolutions per minute is 0.6 rad>sec? 1 km A x u 1. 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 1. Find the linearizations of 2. tan x at x = -p>4 b. sec x at x = -p>4. Graph the curves and linearizations together. 1. 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. 1. Find the linearization of ƒ(x) = 21 + x + sin x – 0.5 at x = 0. 2. Find the linearization of ƒ(x) = 2>(1 – x) + 21 + x – 3.1 at x = 0. Differential Estimates of Change 1. 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. 1. Controlling error 2. 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%? 1. 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. 1. 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 1. the radius. b. the surface area. c. the volume. 2. 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 1. 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. 1. sin 2u = 2 sin u cos u 2. cos 2u = cos2 u – sin2 u 3. 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. 1. 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). 1. 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). 1. 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)? 1. 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. 1. Industrial production 2. 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. 1. 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? 1. 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? 1. 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.) 1. 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). 1. What is the particle’s starting position (t = 0)? 2. What are the points farthest to the left and right of the origin reached by the particle? 1. Find the particle’s velocity and acceleration at the points in part (b). 1. When does the particle first reach the origin? What are its velocity, speed, and acceleration then? 1. 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. 1. How long does it take the paper clip to reach its maximum height? With what velocity does it leave your hand? 1. 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? 1. 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? 1. 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. 1. Average and instantaneous velocity 2. 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. 1. What is the geometric significance of the result in part (a)? 2. 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 1. continuous at x = p. 2. differentiable at x = p. 3. Does the function ƒ(x) = • 1 – cos x x , x _ 0 0, x = 0 have a derivative at x = 0? Explain. 1. 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? 1. Discuss the geometry of the resulting graph of ƒ. 2. 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? 1. Discuss the geometry of the resulting graph of g. 2. Odd differentiable functions Is there anything special about the derivative of an odd differentiable function of x? Give reasons for your answer. 1. Even differentiable functions Is there anything special about the derivative of an even differentiable function of x? Give reasons for your answer. 1. 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. 1. (Continuation of Exercise 19.) Use the result of Exercise 19 to show that the following functions are differentiable at x = 0. 1. 0 x 0 sin x b. x2>3 sin x c. 23 x (1 – cos x) 1. h(x) = e x2 sin (1>x), x _ 0 0, x = 0 1. Suppose that a function ƒ satisfies the following conditions for all real values of x and y: 1. i) ƒ(x + y) = ƒ(x) # ƒ(y). 2. 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). 1. 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)! . 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 . 1. 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 1. the length of a clock pendulum whose period is T = 1 sec. 2. the change dT in T if the pendulum in part (a) is lengthened 0.01 ft. 1. 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 1. What can be said about the extreme values of a function that is continuous on a closed interval? 1. 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. 1. How do you find the absolute extrema of a continuous function on a closed interval? Give examples. 1. What are the hypotheses and conclusion of Rolle’s Theorem? Are the hypotheses really necessary? Explain. 1. What are the hypotheses and conclusion of the Mean Value Theorem? What physical interpretations might the theorem have? 1. State the Mean Value Theorem’s three corollaries. 2. What is the First Derivative Test for Local Extreme Values? Give examples of how it is applied. 1. How do you test a twice-differentiable function to determine where its graph is concave up or concave down? Give examples. 1. What is an inflection point? Give an example. What physical significance do inflection points sometimes have? 1. What is the Second Derivative Test for Local Extreme Values? Give examples of how it is applied. 1. What do the derivatives of a function tell you about the shape of its graph? 1. What is a cusp? Give examples. 2. List the steps you would take to graph a rational function. Illustrate with an example. 1. Outline a general strategy for solving max-min problems. Give examples. 1. Describe l’Hôpital’s Rule. How do you know when to use the rule and when to stop? Give an example. 1. How can you sometimes handle limits that lead to indeterminate forms q>q, q # 0, q – q, 1q, 00, and qq? Give examples. 1. 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? 1. Can a function have more than one antiderivative? If so, how are the antiderivatives related? Explain. 1. What is an indefinite integral? How do you evaluate one? What general formulas do you know for finding indefinite integrals? 1. What is an initial value problem? How do you solve one? Give an example. Chapter 4 Practice Exercises Extreme Values 1. Does ƒ(x) = x3 + 2x + tan x have any local maximum or minimum values? Give reasons for your answer. 1. Does g(x) = csc x + 2 cot x have any local maximum values? Give reasons for your answer. 1. 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. 1. 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. 1. 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. 1. 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. 1. ƒ(x) = x – 2 ln x, 1 … x … 3 2. ƒ(x) = (4>x) + ln x2, 1 … x … 4 3. 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. 1. 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. 1. How is this consistent with Theorem 2 in Section 4.1? Give reasons for your answer. 1. 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? 1. 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? 1. 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. 1. Graph ƒ over the interval -2.5 … x … 2.5. Where does the graph appear to have local extreme values or points of inflection? 1. 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 1. 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.) 1. (Continuation of Exercise 13.) 2. 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? 1. 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. 1. 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 1. a. Show that g(t) = sin2 t – 3t decreases on every interval in its domain. 1. How many solutions does the equation sin2 t – 3t = 5 have? Give reasons for your answer. 1. a. Show that y = tan u increases on every open interval in its domain. 1. 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? 1. a. Show that the equation x4 + 2×2 – 2 = 0 has exactly one solution on 30, 14 . 1. Find the solution to as many decimal places as you can. 2. a. Show that ƒ(x) = x>(x + 1) increases on every open interval in its domain. 1. Show that ƒ(x) = x3 + 2x has no local maximum or minimum values. 1. 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.) 1. 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. 1. 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. 1. 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. 1. 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 1. Estimate the open intervals on which the function y = ƒ(x) is 2. increasing. 3. decreasing. 4. 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. 1. y = x2 – (x3>6) 28. y = x3 – 3×2 + 3 2. y = -x3 + 6×2 – 9x + 3 3. y = (1>8)(x3 + 3×2 – 9x – 27) 4. y = x3(8 – x) 32. y = x2(2×2 – 9) 5. y = x – 3×2>3 34. y = x1>3(x – 4) 6. y = x23 – x 36. y = x24 – x2 7. y = (x – 3)2 ex 38. y = xe-x2 8. y = ln (x2 – 4x + 3) 40. y = ln (sin x) 9. 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. 1. y_ = 16 – x2 44. y_ = x2 – x – 6 2. y_ = 6x(x + 1)(x – 2) 46. y_ = x2(6 – 4x) 3. 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. 1. y = x2>3 + (x – 1)1>3 50. y = x2>3 + (x – 1)2>3 2. 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. 1. y = x + 1 x – 3 1. y = 2x x + 5 1. y = x2 + 1 x 56. y = x2 – x + 1 x 1. y = x3 + 2 2x 1. y = x4 – 1 x2 1. y = x2 – 4 x2 – 3 1. y = x2 x2 – 4 Using L’Hôpital’s Rule Use l’Hôpital’s Rule to find the limits in Exercises 61–72. 1. lim xS2 x3 – 5x + 2 x – 2 1. lim xS1 x5 – 1 x7 – 1 1. lim xS0 sin x x 64. lim xS0 sin x 2x + tan x 1. lim xS1 1 – x cos (px 2 ) 1. lim xS0 tan mx tan nx 1. lim xSp>2 sec 9x cos 5x 68. lim xS0+ 2x sec x 1. lim xSp>2 (sec x – tan x) 70. lim xS0 a 1 x5 – 1 x3b 1. lim xSq 12×4 – x2 + 5 – 2×4 + x22 1. lim xSq a x4 x3 – 1 – x4 x3 + 1 b Find the limits in Exercises 73–84. 1. lim xS0 15x – 1 x 74. lim uS0 5u – 1 u 1. lim xS0 5tan x – 1 ex – 1 1. lim xS0 5-tan x – 1 ex – 1 1. lim xS0 atan 3x – 2x 3x – sin2 x b 78. lim xS0 a7 – 7ex xex b 1. lim tS0+ t – ln (1 + 2t) t2 80. lim xS2 x2 – 4 2x + 2 – 23x – 2 1. lim tS0+ e4x x – 1x 1. lim yS0+ e-1>y ln y 1. lim xSq a1 + bx b kx 1. lim xSq a6 – 5x – 10 x3 b Optimization 1. The sum of two nonnegative numbers is 36. Find the numbers if 2. the difference of their square roots is to be as large as possible. 3. the sum of their square roots is to be as large as possible. 4. The sum of two nonnegative numbers is 20. Find the numbers 5. if the product of one number and the square root of the other is to be as large as possible. 1. if one number plus the square root of the other is to be as large as possible. 1. 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. 1. 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? 1. Find the height and radius of the largest right circular cylinder that can be put in a sphere of radius 23. 1. 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_ 1. 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? 1. Particle motion The positions of two particles on the s-axis are s1 = cos t and s2 = cos (t + p>4). 1. What is the farthest apart the particles ever get? 2. When do the particles collide? 3. 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. 1. 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 1. 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. 1. 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 1. d2r dt2 = 152t + 3 2t ; r_(1) = 8, r (1) = 0 1. d3r dt3 = -cos t; r_(0) = r_(0) = 0, r (0) = -1 Applications and Examples 1. 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 1. 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 1. 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 1. 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. 1. y = x ln 2x – x, c 1 2e , e 2 d 1. 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. 1. ƒ(x) = ex>2×4+1 132. g(x) = e23-2x-x2 2. 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. 1. y = (ln x)>2x b. y = e-x2 c. y = (1 + x)e-x 2. Graph ƒ(x) = x ln x. Does the function appear to have an absolute minimum value? Confirm your answer with calculus. 1. Graph ƒ(x) = (sin x)sin x over 30, 3p4. Explain what you see. 2. 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 1. What can you say about a function whose maximum and minimum values on an interval are equal? Give reasons for your answer. 1. 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. 1. 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. 1. Local extrema Use the sign pattern for the derivative 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. 1. Local extrema 2. 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? 1. 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? 1. 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. 1. 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 . 1. An inequality 2. Show that -1>2 … x>(1 + x2) … 1>2 for every value of x. 3. 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. 1. 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. 1. Extrema and inflection points Let h = ƒg be the product of two differentiable functions of x. 1. 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? 1. 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. 1. 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). 1. i) The values of a, b, and c are either 0 or 1. 2. ii) The graph of ƒ passes through the point (-1, 0). iii) The line y = 1 is an asymptote of the graph of ƒ. 1. 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 1. 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? 1. Proving the second derivative test The Second Derivative Test for Local Maxima and Minima (Section 4.4) says: 1. ƒ has a local maximum value at x = c if ƒ_(c) = 0 and ƒ_(c) 6 0 1. 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. 1. 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?) 1. 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 1. 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 1. 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 1. Evaluate the following limits. 2. lim xS0 2 sin 5x 3x 1. lim xS0 sin 5x cot 3x 1. lim xS0 x csc2 22x d. lim xSp>2 (sec x – tan x) 1. lim xS0 x – sin x x – tan x f. lim xS0 sin x2 x sin x 1. lim xS0 sec x – 1 x2 h. lim xS2 x3 – 8 x2 – 4 1. L’Hôpital’s Rule does not help with the following limits. Find them some other way. 1. lim xSq 2x + 5 2x + 5 1. lim xSq 2x x + 72x Theory and Examples 1. 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. 1. 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. 1. Graph y = (1>x) – 3. Where does the graph cross the x-axis? 1. Show that the recursion formula in this case is xn+1 = xn(2 – 3xn), so there is no need for division. 1. 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? 1. 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.) 1. 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. 1. 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 1. 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. 1. 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? 1. 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). 1. 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. 1. 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. 1. 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. 1. 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 1. the velocity y in terms of t. 2. the position s in terms of t. 3. 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. 1. Schwarz’s inequality 2. 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. 1. 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. 1. 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. 1. Show that the critical value of u for which dL>du equals zero is uc = cos-1 r4 R4 . 1. 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 1. How can you sometimes estimate quantities like distance traveled, area, and average value with finite sums? Why might you want to do so? 1. What is sigma notation? What advantage does it offer? Give examples. 1. What is a Riemann sum? Why might you want to consider such a sum? 1. What is the norm of a partition of a closed interval? 2. What is the definite integral of a function ƒ over a closed interval 3a, b4 ? When can you be sure it exists? 1. What is the relation between definite integrals and area? Describe some other interpretations of definite integrals. 1. What is the average value of an integrable function over a closed interval? Must the function assume its average value? Explain. 1. Describe the rules for working with definite integrals (Table 5.6). Give examples. 1. What is the Fundamental Theorem of Calculus? Why is it so important? Illustrate each part of the theorem with an example. 1. What is the Net Change Theorem? What does it say about the integral of velocity? The integral of marginal cost? 1. Discuss how the processes of integration and differentiation can be considered as “inverses” of each other. 1. How does the Fundamental Theorem provide a solution to the initial value problem dy>dx = ƒ(x), y(x0) = y0 , when ƒ is continuous? 1. How is integration by substitution related to the Chain Rule? 2. How can you sometimes evaluate indefinite integrals by substitution? Give examples. 1. How does the method of substitution work for definite integrals? Give examples. 1. 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 1. 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) 1. 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.) 1. Sketch a graph of the rocket’s height above ground as a function of time for 0 … t … 8. 1. 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? 1. 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) 1. Suppose that a 10 k=1 ak = -2 and a 10 k=1 bk = 25. Find the value of 1. a 10 k=1 ak 4 1. a 10 k=1 (bk – 3ak) 1. a 10 k=1 (ak + bk – 1) d. a 10 k=1 a5 2 – bkb 1. Suppose that a 20 k=1 ak = 0 and a 20 k=1 bk = 7. Find the values of 1. a 20 k=1 3ak b. a 20 k=1 (ak + bk) 1. 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. 1. lim }P}S0 a n k=1 (2ck – 1)-1>2 _xk , where P is a partition of 31, 54 1. lim }P}S0 a n k=1 ck(ck 2 – 1)1>3 _xk , where P is a partition of 31, 34 1. lim }P}S0 a n k=1 acosa ck 2 b b _xk , where P is a partition of 3-p, 04 1. lim }P}S0 a n k=1 (sin ck)(cos ck) _xk , where P is a partition of 30, p>24 1. 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 1. 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. 1. ƒ(x) = x2 – 4x + 3, 0 … x … 3 2. ƒ(x) = 1 – (x2>4), -2 … x … 3 3. ƒ(x) = 5 – 5×2>3, -1 … x … 8 4. ƒ(x) = 1 – 2x, 0 … x … 4 Find the areas of the regions enclosed by the curves and lines in Exercises 15–26. 1. y = x, y = 1>x2, x = 2 2. y = x, y = 1>2x, x = 2 3. 2x + 2y = 1, x = 0, y = 0 x y 1 0 1 “x + “y _ 1 1. x3 + 2y = 1, x = 0, y = 0, for 0 … x … 1 x y 0 1 1 x3 + “y _ 1, 0 _ x _ 1 1. x = 2y2, x = 0, y = 3 20. x = 4 – y2, x = 0 2. y2 = 4x, y = 4x – 2 3. y2 = 4x + 4, y = 4x – 16 4. y = sin x, y = x, 0 … x … p>4 5. y = 0 sin x 0 , y = 1, -p>2 … x … p>2 6. y = 2 sin x, y = sin 2x, 0 … x … p 7. y = 8 cos x, y = sec2 x, -p>3 … x … p>3 8. 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. 1. 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. 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. 1. Find the area of the region cut from the first quadrant by the curve x1>2 + y1>2 = a1>2. 1. Find the total area of the region enclosed by the curve x = y2>3 and the lines x = y and y = -1. 1. Find the total area of the region between the curves y = sin x and y = cos x for 0 … x … 3p>2. 1. Area Find the area between the curve y = 2(ln x)>x and the x-axis from x = 1 to x = e. 1. 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. 1. 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 1. 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. 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 1. Find the average value of ƒ(x) = mx + b 2. over 3-1, 14 b. over 3-k, k4 3. Find the average value of 4. y = 23x over 30, 34 b. y = 2ax over 30, a4 5. 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. 1. 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. 1. a. Verify that 1 ln x dx = x ln x – x + C. 2. Find the average value of ln x over 31, e4 . 3. Find the average value of ƒ(x) = 1>x on 31, 24 . 4. 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). 1. 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. 1. y = L x 2 22 + cos3 t dt 122. y = L 7×2 2 22 + cos3 t dt 1. y = L 1 x 6 3 + t4 dt 124. y = L 2 sec x 1 t2 + 1 dt 1. y = L 0 ln x2 ecos t dt 126. y = L e2x 1 ln (t2 + 1) dt 1. y = L sin-1 x 0 dt 21 – 2t2 1. y = L p>4 tan-1x e2t dt Theory and Examples 1. a. If L 1 0 7ƒ(x) dx = 7, does L 1 0 ƒ(x) dx = 1? 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. 1. 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 1. ƒ(x) … g(x) on the interval -2 … x … 5 2. 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.) 1. 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. 1. Find ƒ(4) if a. L x2 0 ƒ(t) dt = x cos px b. L ƒ(x) 0 t2 dt = x cos px. 1. Find ƒ(p/2) from the following information. 2. i) ƒ is positive and continuous. 3. 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. 1. 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). 1. 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.) 1. 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. 1. 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. 1. ƒ(x) = e x2>3, -8 … x 6 0 -4, 0 … x … 3 1. ƒ(x) = e 2-x, -4 … x 6 0 x2 – 4, 0 … x … 3 1. g(t) = e t, 0 … t 6 1 sin pt, 1 … t … 2 1. h(z) = e 21 – z, 0 … z 6 1 (7z – 6)-1>3, 1 … z … 2 1. ƒ(x) = c 1, -2 … x 6 -1 1 – x2, -1 … x 6 1 2, 1 … x … 2 1. h(r) = c r, -1 … r 6 0 1 – r2, 0 … r 6 1 1, 1 … r … 2 1. Find the average value of the function graphed in the accompanying figure. x y 0 1 2 1 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. 1. lim bS1-L b 0 dx 21 – x2 1. lim xSq 1x L x 0 tan-1 t dt 1. lim nSq a 1 n + 1 + 1 n + 2 + g + 1 2nb 1. lim nSq 1n 1e1>n + e2>n + g + e(n-1)>n + en>n2 Defining Functions Using the Fundamental Theorem 1. 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 1. Find g(1). b. Find g(3). c. Find g(-1). 2. Find all values of x on the open interval (-3, 4) at which g has a relative maximum. 1. Write an equation for the line tangent to the graph of g at x = -1. 2. Find the x-coordinate of each point of inflection of the graph of g on the open interval (-3, 4). 1. Find the range of g. 2. A differential equation Show that both of the following conditions are satisfied by y = sin x + 1 p x cos 2t dt + 1: 1. i) y_ = -sin x + 2 sin 2x 2. 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. 1. Find ƒ_(2) if ƒ(x) = eg(x) and g(x) = L x 2 t 1 + t4 dt. 1. 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 1. 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.) 1. Bound on an integral Let ƒ be a continuously differentiable function on 3a, b4 satisfying 1 b a ƒ(x) dx = 0. 1. 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. 1. 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. 1. 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. 1. ƒ(x) = L x 1>x 1t dt 26. ƒ(x) = L sin x cos x 1 1 – t2 dt 1. g(y) = L 22y 2y sin t2 dt 28. g(y) = L y2 2y e t t dt 1. y = L x2 x2>2 ln 2t dt 30. y = L 23 x 2x ln t dt 1. y = L ln x 0 sin e t dt 32. y = L e2x e42x ln t dt Theory and Examples 1. 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. 1. For what x 7 0 does x(xx) = (xx)x? Give reasons for your answer. 2. 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? 1. a. Find df > dx if ƒ(x) = L ex 1 2 ln t t dt. 1. Find ƒ(0). 2. 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 1. How do you define and calculate the volumes of solids by the method of slicing? Give an example. 1. How are the disk and washer methods for calculating volumes derived from the method of slicing? Give examples of volume calculations by these methods. 1. Describe the method of cylindrical shells. Give an example. 2. 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? 1. 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. 1. 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. 1. What is a center of mass? a centroid? 2. How do you locate the center of mass of a thin flat plate of material? Give an example. 1. 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. 1. 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. 1. 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. 1. 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. 1. 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 1. 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. 1. 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.) 1. 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. 1. 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. 1. 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. 1. 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. 1. 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. 1. 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. 1. 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. 1. 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.) 1. 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. 1. 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. 1. y = x1>2 – (1>3)x3>2, 1 … x … 4 2. x = y2>3, 1 … y … 8 3. y = x2 – (ln x)>8, 1 … x … 2 4. 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. 1. y = 22x + 1, 0 … x … 3; x@axis 2. y = x3>3, 0 … x … 1; x@axis 3. x = 24y – y2, 1 … y … 2; y@axis 4. x = 2y, 2 … y … 6; y@axis Work 1. 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.) 1. 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. 1. 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. 1. 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? 1. 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? 1. 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? 1. 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? 1. 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 1. Find the centroid of a thin, flat plate covering the region enclosed by the parabolas y = 2×2 and y = 3 – x2. 1. 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. 1. 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. 1. Find the centroid of a thin, flat plate covering the region enclosed by the parabola y2 = x and the line x = 2y. 1. 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.) 1. 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. 1. 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 1. 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). 1. 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). 1. 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? 1. a. Show that for 0 6 a … p>2, L a 0 21 + cos2 u du 7 2a2 + sin2 a. 1. Generalize the result in part (a). 2. 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. 1. 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 1. 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 1. 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 1. 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. 1. 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 1. 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? 1. 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? 1. 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 1. My – bdA if the plate lies to the right of the line, and 2. bdA – My if the plate lies to the left of the line. 3. 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 . 1. 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. 1. Find the limits of the coordinates of the centroid as a approaches b and discuss the meaning of the result. 1. 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 1. 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. 1. What integrals lead to logarithms? Give examples. 2. What are the integrals of tan x and cot x? sec x and csc x? 3. 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. 1. 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? 1. How do you solve separable first-order differential equations? 2. 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? 1. What are the six basic hyperbolic functions? Comment on their domains, ranges, and graphs. What are some of the identities relating them? 1. 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? 1. 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? 1. 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. 1. 3y = 2y+1 14. 4-y = 3y+2 2. 9e2y = x2 16. 3y = 3 ln x 3. ln ( y – 1) = x + ln y 18. ln (10 ln y) = ln 5x Theory and Applications 1. 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). 1. 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 -1 dx  ƒ(x) = 1 ƒ_(x) . 1. 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)? 1. 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.) 1. The functions ƒ(x) = ln 5x and g(x) = ln 3x differ by a constant. What constant? Give reasons for your answer. 1. a. If (ln x)>x = (ln 2)>2, must x = 2? 2. If (ln x)>x = -2 ln 2, must x = 1>2? Give reasons for your answers. 1. The quotient (log4 x)>(log2 x) has a constant value. What value? Give reasons for your answer. 1. logx (2) vs. log2 (x) How does ƒ(x) = logx (2) compare with g(x) = log2 (x)? Here is one way to find out. 1. Use the equation loga b = (ln b)>(ln a) to express ƒ(x) and g(x) in terms of natural logarithms. T 1. 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 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 1. x dy – 1y + 2y2 dx = 0, y(1) = 1 2. y-2 dx dy = ex e2x + 1 , y(0) = 1 1. What is the age of a sample of charcoal in which 90% of the carbon- 14 originally present has decayed? 1. 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? 1. 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. 1. lim tSq A(t) b. lim tSq V(t)>A(t) c. lim tS0 + V(t)>A(t) 1. Varying a logarithm’s base 2. Find lim loga 2 as aS 0+, 1-, 1+, and q. 3. Graph y = loga 2 as a function of a over the interval 0 6 a … 4. 1. 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. 1. Graph ƒ(x) = (sin x)sin x over 30, 3p4. Explain what you see. 2. Even-odd decompositions 3. 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. 1. 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. 1. What is the significance of the result in part (b)? T T T 1. Let g be a function that is differentiable throughout an open interval containing the origin. Suppose g has the following properties: 1. 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. 1. lim hS0 g(h) = 0 iii. lim hS0 g(h) h = 1 1. Show that g(0) = 0. 2. Show that g_(x) = 1 + 3g(x) 42. 3. Find g(x) by solving the differential equation in part (b). 4. 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. 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. 1. Find the volume of the solid. 2. 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. 1. 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. 1. Differentiate both sides of the equation in part (a) with respect to u to show that A_(u) = 1 2 . 1. 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 1. 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. 1. What integrals lead to logarithms? Give examples. 2. What are the integrals of tan x and cot x? sec x and csc x? 3. 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. 1. 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? 1. How do you solve separable first-order differential equations? 2. 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? 1. What are the six basic hyperbolic functions? Comment on their domains, ranges, and graphs. What are some of the identities relating them? 1. 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? 1. 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? 1. 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. 1. 3y = 2y+1 14. 4-y = 3y+2 2. 9e2y = x2 16. 3y = 3 ln x 3. ln ( y – 1) = x + ln y 18. ln (10 ln y) = ln 5x Theory and Applications 1. 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). 1. 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 -1 dx  ƒ(x) = 1 ƒ_(x) . 1. 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)? 1. 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.) 1. The functions ƒ(x) = ln 5x and g(x) = ln 3x differ by a constant. What constant? Give reasons for your answer. 1. a. If (ln x)>x = (ln 2)>2, must x = 2? 2. If (ln x)>x = -2 ln 2, must x = 1>2? Give reasons for your answers. 1. The quotient (log4 x)>(log2 x) has a constant value. What value? Give reasons for your answer. 1. logx (2) vs. log2 (x) How does ƒ(x) = logx (2) compare with g(x) = log2 (x)? Here is one way to find out. 1. Use the equation loga b = (ln b)>(ln a) to express ƒ(x) and g(x) in terms of natural logarithms. T 1. 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 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 1. x dy – 1y + 2y2 dx = 0, y(1) = 1 2. y-2 dx dy = ex e2x + 1 , y(0) = 1 1. What is the age of a sample of charcoal in which 90% of the carbon- 14 originally present has decayed? 1. 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? 1. 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. 1. lim tSq A(t) b. lim tSq V(t)>A(t) c. lim tS0 + V(t)>A(t) 1. Varying a logarithm’s base 2. Find lim loga 2 as aS 0+, 1-, 1+, and q. 3. Graph y = loga 2 as a function of a over the interval 0 6 a … 4. 1. 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. 1. Graph ƒ(x) = (sin x)sin x over 30, 3p4. Explain what you see. 2. Even-odd decompositions 3. 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. 1. 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. 1. What is the significance of the result in part (b)? T T T 1. Let g be a function that is differentiable throughout an open interval containing the origin. Suppose g has the following properties: 1. 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. 1. lim hS0 g(h) = 0 iii. lim hS0 g(h) h = 1 1. Show that g(0) = 0. 2. Show that g_(x) = 1 + 3g(x) 42. 3. Find g(x) by solving the differential equation in part (b). 4. 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. 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. 1. Find the volume of the solid. 2. Find the centroid of the region. Chapter 8 Techniques of Integration Chapter 8 Questions to Guide Your Review 1. What is the formula for integration by parts? Where does it come from? Why might you want to use it? 1. 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? 1. 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. 1. 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. 1. 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. 1. What restrictions can you place on the variables involved in the three basic trigonometric substitutions to make sure the substitutions are reversible (have inverses)? 1. What is the goal of the method of partial fractions? 2. 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) 1. is a product of distinct linear factors? 2. consists of a repeated linear factor? 3. contains an irreducible quadratic factor? What do you do if the degree of ƒ is not less than the degree of g? 1. How are integral tables typically used? What do you do if a particular integral you want to evaluate is not listed in the table? 1. What is a reduction formula? How are reduction formulas used? Give an example. 1. How would you compare the relative merits of Simpson’s Rule and the Trapezoidal Rule? 1. What is an improper integral of Type I? Type II? How are the values of various types of improper integrals defined? Give examples. 1. 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 1. 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.) 1. 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? 1. 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. 1. 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.) 1. 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). 1. 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. 1. 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 1. Use the Trapezoidal Rule to approximate the total fuel consumption during the hour. 1. If the automobile covered 60 mi in the hour, what was its fuel efficiency (in miles per gallon) for that portion of the trip? 1. 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 1. a. Show that 1 a 0 ƒ(x) dx = 1 a 0 ƒ(a – x) dx. 1. 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. 1. 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. 1. lim nSq a n k=1 ln A n 1 + kn 1. lim nSq a n-1 k=0 1 2n2 – k2 Applications 1. Finding arc length Find the length of the curve y = L x 0 2cos 2t dt, 0 … x … p>4. 1. Finding arc length Find the length of the graph of the function y = ln (1 – x2), 0 … x … 1>2. 1. 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. 1. 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. 1. 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. 1. 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 1. the x-axis. b. the line y = 1. 2. 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. 1. Show that ƒ is continuous at x = 0. 2. Find the volume of the solid. y 0 y _ x ln x x 1 2 1. 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. 1. 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. 1. Length of a curve Find the length of the curve y = ln x from x = 1 to x = e. 1. 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. 1. Length of a curve Find the length of the curve y = L x 1 32t – 1 dt , 1 … x … 16. 1. For what value or values of a does L q 1 a ax x2 + 1 – 1 2xb dx converge? Evaluate the corresponding integral(s). 1. For each x 7 0, let G(x) = 1 q 0 e-xt dt. Prove that xG(x) = 1 for each x 7 0. 1. 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. 1. 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. 1. If n is a nonnegative integer, _(n + 1) _ n! 2. Show that _(1) = 1. 3. 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) 1. Use mathematical induction to verify Equation (5) for every nonnegative integer n. 1. 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) 1. 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) 1. 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 1. What is an infinite sequence? What does it mean for such a sequence to converge? To diverge? Give examples. 1. What is a monotonic sequence? Under what circumstances does such a sequence have a limit? Give examples. 1. What theorems are available for calculating limits of sequences? Give examples. 1. What theorem sometimes enables us to use l’Hôpital’s Rule to calculate the limit of a sequence? Give an example. 1. What are the six commonly occurring limits in Theorem 5 that arise frequently when you work with sequences and series? 1. What is an infinite series? What does it mean for such a series to converge? To diverge? Give examples. 1. What is a geometric series? When does such a series converge? Diverge? When it does converge, what is its sum? Give examples. 1. Besides geometric series, what other convergent and divergent series do you know? 1. What is the nth-Term Test for Divergence? What is the idea behind the test? 1. What can be said about term-by-term sums and differences of convergent series? About constant multiples of convergent and divergent series? 1. 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? 1. How do you reindex a series? Why might you want to do this? 2. Under what circumstances will an infinite series of nonnegative terms converge? Diverge? Why study series of nonnegative terms? 1. What is the Integral Test? What is the reasoning behind it? Give an example of its use. 1. When do p-series converge? Diverge? How do you know? Give examples of convergent and divergent p-series. 1. What are the Direct Comparison Test and the Limit Comparison Test? What is the reasoning behind these tests? Give examples of their use. 1. What are the Ratio and Root Tests? Do they always give you the information you need to determine convergence or divergence? Give examples. 1. What is absolute convergence? Conditional convergence? How are the two related? 1. What is an alternating series? What theorem is available for determining the convergence of such a series? 1. 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? 1. What do you know about rearranging the terms of an absolutely convergent series? Of a conditionally convergent series? 1. What is a power series? How do you test a power series for convergence? What are the possible outcomes? 1. What are the basic facts about 2. sums, differences, and products of power series? 3. substitution of a function for x in a power series? 4. term-by-term differentiation of power series? 5. term-by-term integration of power series? Give examples. 1. 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. 1. What is a Maclaurin series? 2. Does a Taylor series always converge to its generating function? Explain. 1. What are Taylor polynomials? Of what use are they? 2. 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? 1. What is the binomial series? On what interval does it converge? How is it used? 1. How can you sometimes use power series to estimate the values of nonelementary definite integrals? To find limits? 1. 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. 1. an = 1 + (-1)n n 2. an = 1 – (-1)n 2n 1. an = 1 – 2n 2n 4. an = 1 + (0.9)n 1. an = sin np 2 1. an = sin np 2. an = ln (n2) n 8. an = ln (2n + 1) n 1. an = n + ln n n 10. an = ln (2n3 + 1) n 1. an = an – 5 n b n 1. an = a1 + 1n b -n 1. an = A n 3n n 14. an = a3n b 1>n 1. an = n(21>n – 1) 16. an = 2n 2n + 1 2. an = (n + 1)! n! 1. an = (-4)n n! Convergent Series Find the sums of the series in Exercises 19–24. 1. a q n=3 1 (2n – 3)(2n – 1) 1. a q n=2 -2 n(n + 1) 1. a q n=1 9 (3n – 1)(3n + 2) 1. a q n=3 -8 (4n – 3)(4n + 1) 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. 1. a q n=1 1 2n 1. a q n=1 -5 n 1. a q n=1 (-1)n 2n 1. a q n=1 1 2n3 1. a q n=1 (-1)n ln (n + 1) 1. a q n=2 1 n (ln n)2 1. a q n=1 ln n n3 32. a q n=3 ln n ln (ln n) 1. a q n=1 (-1)n n2n2 + 1 1. a q n=1 (-1)n 3n2 n3 + 1 1. a q n=1 n + 1 n! 1. a q n=1 (-1)n(n2 + 1) 2n2 + n – 1 1. a q n=1 (-3)n n! 1. a q n=1 2n 3n nn 1. a q n=1 1 2n(n + 1)(n + 2) 1. 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. 1. a q n=1 (x + 4)n n3n 42. a q n=1 (x – 1)2n-2 (2n – 1)! 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 1. a q n=1 xn nn 46. a q n=1 xn 2n 1. a q n=0 (n + 1)x2n-1 3n 48. a q n=0 (-1)n(x – 1)2n+1 2n + 1 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 – 1 4 + 1 16 – g + (-1)n 1 4n + g 1. 2 3 – 4 18 + 8 81 – g + (-1)n-1 2n n3n + g 1. p – p3 3! + p5 5! – g + (-1)n p2n+1 (2n + 1)! + g 1. 1 – p2 9 # 2! + p4 81 # 4! – g + (-1)n p2n 32n(2n)! + g 1. 1 + ln 2 + (ln 2)2 2! + g + (ln 2)n n! + g 1. 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 1 – 2x 1. 1 1 + x3 1. sin px 60. sin 2x 3 1. cos (x5>3) 62. cos x3 25 1. 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. 1. ƒ(x) = 23 + x2 at x = -1 2. ƒ(x) = 1>(1 – x) at x = 2 3. ƒ(x) = 1>(x + 1) at x = 3 4. 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: 1. Use power series to evaluate the limit. 2. Then use a grapher to support your calculation. 3. lim xS0 7 sin x e2x – 1 1. lim uS0 eu – e-u – 2u u – sin u 1. lim tS0 a 1 2 – 2 cos t – 1 t2b 76. lim hS0 (sin h)>h – cos h h2 1. lim zS0 1 – cos2 z ln (1 – z) + sin z 1. lim yS0 y2 cos y – cosh y Theory and Examples 1. 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. 1. 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. 1. Find the radius of convergence of the series a q n=1 2 # 5 # 8 # g# (3n – 1) 2 # 4 # 6 # g# (2n) xn. 1. 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. 1. 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. 1. Evaluate gqk =2 (1> (k2 – 1) ) by finding the limits as nS q of the series’ nth partial sum. 1. 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. 1. 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. 1. a. Find the Maclaurin series for the function x2>(1 + x). 2. Does the series converge at x = 1? Explain. 3. 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. 1. 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. 1. Prove that the sequence 5xn6 and the series gqk =1 (xk+1 – xk) both converge or both diverge. 1. Prove that gqn =1 (an>(1 + an)) converges if an 7 0 for all n and gqn =1 an converges. 1. Suppose that a1, a2, a3,c, an are positive numbers satisfying the following conditions: 1. i) a1 Ú a2 Ú a3 Ú g; 2. ii) the series a2 + a4 + a8 + a16 + g diverges. Show that the series a1 1 + a2 2 + a3 3 + g diverges. 1. 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. 1. a q n=1 1 (3n – 2)n+(1>2) 2. a q n=1 (tan-1 n)2 n2 + 1 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. 1. a1 = 1, an+1 = n(n + 1) (n + 2)(n + 3) an (Hint: Write out several terms, see which factors cancel, and then generalize.) 1. a1 = a2 = 7, an+1 = n (n – 1)(n + 1) an if n Ú 2 1. a1 = a2 = 1, an+1 = 1 1 + an if n Ú 2 1. 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. 1. cos x near x = 1 10. sin x near x = 6.3 2. ex near x = 0.4 12. ln x near x = 1.3 3. cos x near x = 69 14. tan-1 x near x = 2 Theory and Examples 1. 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? 1. 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. 1. Evaluate a q n=0 L n+1 n 1 1 + x2 dx. 1. Find all values of x for which a q n=1 nxn (n + 1)(2x + 1)n converges absolutely. 1. 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? 1. 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? 1. Use calculus to confirm your findings in parts (a) and (b). 2. Show that if gqn =1 an converges, then a q n=1 a 1 + sin (an) 2 b n converges. 1. 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. 1. How do you know that the functions sin x, ln x, and ex are not polynomials? Give reasons for your answer. 1. Find the value of a for which the limit lim xS0 sin (ax) – sin x – x x3 is finite and evaluate the limit. 1. Find values of a and b for which lim xS0 cos (ax) – b 2×2 = -1. 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). 1. (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. 1. If gqn =1 an converges, and if an _ 1 and an 7 0 for all n, 1. Show that gqn =1 an 2 converges. 1. Does gqn =1 an>(1 – an) converge? Explain. 1. (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).) 1. 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.) 1. 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. 1. Use part (a) to find the real solution greater than 1 of the equation x = a q n=1 n(n + 1) xn . 1. Quality control 2. Differentiate the series 1 1 – x = 1 + x + x2 + g+ x n + g to obtain a series for 1>(1 – x)2. 1. 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. 1. 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. 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.) 1. pk = 2-k b. pk = 5k-1 6k 1. pk = 1 k(k + 1) = 1 k – 1 k + 1 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. 1. Write Rn in closed from as a single fraction, and find R = limnSq Rn. 1. 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? 1. 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 1. Koont, COMAP, Inc., Lexington, MA.) 2. 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. 1. Verify the preceding equation for t0. 2. 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. 1. Given CH = 2 mg>mL, CL = 0.5 mg>mL, and k = 0.02 h-1, determine a scheme for administering the drug. 1. 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 1. 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. 1. Give some typical parametrizations for lines, circles, parabolas, ellipses, and hyperbolas. How might the parametrized curve differ from the graph of its Cartesian equation? 1. What is a cycloid? What are typical parametric equations for cycloids? What physical properties account for the importance of cycloids? 1. 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. 1. How can you sometimes find the area bounded by a parametrized curve and one of the coordinate axes? 1. 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. 1. What is the arc length function for a smooth parametrized curve? What is its arc length differential? 1. 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. 1. 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? 1. What consequence does the lack of uniqueness of polar coordinates have for graphing? Give an example. 1. 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. 1. 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. 1. 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. 1. 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. 1. x = t>2, y = t + 1; -q 6 t 6 q 2. x = 2t, y = 1 – 2t; t Ú 0 3. x = (1>2) tan t, y = (1>2) sec t; -p>2 6 t 6 p>2 4. x = -2 cos t, y = 2 sin t; 0 … t … p 5. x = -cos t, y = cos2 t; 0 … t … p 6. x = 4 cos t, y = 9 sin t; 0 … t … 2p Finding Parametric Equations and Tangent Lines 1. 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.) 1. 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. 1. x = (1>2) tan t, y = (1>2) sec t; t = p>3 2. x = 1 + 1>t2, y = 1 – 3>t; t = 2 3. Eliminate the parameter to express the curve in the form y = ƒ(x) . 4. x = 4t2, y = t3 – 1 5. x = cos t, y = tan t 6. Find parametric equations for the given curve. 7. Line through (1, -2) with slope 3 8. (x – 1)2 + ( y + 2)2 = 9 9. y = 4×2 – x 10. 9×2 + 4y2 = 36 Lengths of Curves Find the lengths of the curves in Exercises 13–19. 1. y = x1>2 – (1>3)x3>2, 1 … x … 4 2. x = y2>3, 1 … y … 8 3. y = (5>12)x6>5 – (5>8)x4>5, 1 … x … 32 4. x = (y3>12) + (1>y), 1 … y … 2 5. x = 5 cos t – cos 5t, y = 5 sin t – sin 5t, 0 … t … p>2 6. x = t3 – 6t2, y = t3 + 6t2, 0 … t … 1 7. x = 3 cos u, y = 3 sin u, 0 … u … 3p 2 1. 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. 1. x = t2>2, y = 2t, 0 … t … 25; x-axis 2. 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. 1. r cos au + p 3 b = 223 24. r cos au – 3p 4 b = 22 2 1. r = 2 sec u 26. r = -22 sec u 2. 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. 1. r = -4 sin u 30. r = 323 sin u 2. 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. 1. x2 + y2 + 5y = 0 34. x2 + y2 – 2y = 0 2. 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. 1. 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. 1. r = cos 2u b. r cos u = 1 c. r = 6 1 – 2 cos u 1. r = sin 2u e. r = u f. r2 = cos 2u 2. r = 1 + cos u h. r = 1 – sin u i. r = 2 1 – cos u 1. r2 = sin 2u k. r = -sin u l. r = 2 cos u + 1 2. Four-leaved rose 40. Spiral x y x y 1. Limaçon 42. Lemniscate x y x y 1. Circle 44. Cardioid x y x y 1. 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. 1. Enclosed by the limaçon r = 2 – cos u 2. Enclosed by one leaf of the three-leaved rose r = sin 3u 3. Inside the “figure eight” r = 1 + cos 2u and outside the circle r = 1 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. 1. r = -1 + cos u 2. r = 2 sin u + 2 cos u, 0 … u … p>2 3. r = 8 sin3 (u>3), 0 … u … p>4 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. 1. r = 2 1 + cos u 1. r = 8 2 + cos u 1. r = 6 1 – 2 cos u 1. 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. 1. e = 2, r cos u = 2 2. e = 1, r cos u = -4 3. e = 1>2, r sin u = 2 4. e = 1>3, r sin u = -6 Chapter 10 A dditional and Advanced Exercises Polar Coordinates 1. a. Find an equation in polar coordinates for the curve x = e2t cos t, y = e2t sin t; -q 6 t 6 q. 1. Find the length of the curve from t = 0 to t = 2p. 2. 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. 1. e = 2, r cos u = 2 4. e = 1, r cos u = -4 2. e = 1>2, r sin u = 2 6. e = 1>3, r sin u = -6 Theory and Examples 1. 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. 1. 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) 1. 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. 1. Find the value of tan c for the curve r = sin4 (u>4). 2. Find the angle between the radius vector to the curve r = 2a sin 3u and its tangent when u = p>6. 1. a. Graph the hyperbolic spiral ru = 1. What appears to happen to c as the spiral winds in around the origin? 1. Confirm your finding in part (a) analytically. Chapter 11 Vectors and the Geometry of Space Chapter 11 Questions to Guide Your Review 1. When do directed line segments in the plane represent the same vector? 1. How are vectors added and subtracted geometrically? Algebraically? 2. How do you find a vector’s magnitude and direction? 3. 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? 1. 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? 1. What geometric interpretation does the dot product have? Give examples. 1. What is the vector projection of a vector u onto a vector v? Give an example of a useful application of a vector projection. 1. 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? 1. What geometric or physical interpretations do cross products have? Give examples. 1. 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. 1. 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? 1. How do you find the distance from a point to a line in space? From a point to a plane? Give examples. 1. What are box products? What significance do they have? How are they evaluated? Give an example. 1. How do you find equations for spheres in space? Give examples. 2. How do you find the intersection of two lines in space? A line and a plane? Two planes? Give examples. 1. What is a cylinder? Give examples of equations that define cylinders in Cartesian coordinates. 1. 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. 1. 5u + 9v 2. u – v 2. -7u 4. 9v In Exercises 5–8, find the component form of the vector. 1. The vector obtained by rotating 80, 19 through an angle of 3p>4 radians 1. The unit vector that makes an angle of p>3 radian with the positive x-axis 1. The vector 2 units long in the direction 4i – j 2. 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. 1. 23i + 23j 10. -3i + 2j 2. Velocity vector v = (-5 sin t)i + (5 cos t)j when t = p>4. 3. 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. 1. 4i + 5j – 3k 14. i – 5j + k 2. Find a vector 8 units long in the direction of v = 3i + 2j – k. 3. 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. 1. 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. 1. u = i – j – k, v = 3i + 4j – 5k, w = 2i + 2j – k 2. u = j – k, v = k, w = i – j – k Lines, Planes, and Distances 1. 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. 1. 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. 1. (2, 2, 0); x = -t, y = t, z = -1 + t 2. (0, 4, 1); x = 2 + t, y = 2 + t, z = t 3. Parametrize the line that passes through the point (1, 2, 3) parallel to the vector v = -3i + 7k. 1. 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. 1. (6, 0, -6), x – y = 4 2. (3, 0, 10), 2x + 3y + z = 2 3. Find an equation for the plane that passes through the point (3, -2, 1) normal to the vector n = 2i + j + k. 1. 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. 1. P(1, -1, 2), Q(2, 1, 3), R(-1, 2, -1) 2. P(1, 0, 0), Q(0, 1, 0), R(0, 0, 1) 3. Find the points in which the line x = 1 + 2t, y = -1 – t, z = 3t meets the three coordinate planes. 1. 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. 1. Find the acute angle between the planes x = 7 and x + y + 22z = -3. 1. Find the acute angle between the planes x + y = 1 and y + z = 1. 2. Find parametric equations for the line in which the planes x + 2y + z = 1 and x – y + 2z = -8 intersect. 1. 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 . 1. The planes 3x + 6z = 1 and 2x + 2y – z = 3 intersect in a line. 2. Show that the planes are orthogonal. 3. Find equations for the line of intersection. 4. 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. 1. Is v = 2i – 4j + k related in any special way to the plane 2x + y = 5? Give reasons for your answer. 1. 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? 1. 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) . 1. Find the distance from the point (2, 2, 3) to the plane 2x + 3y + 5z = 0. 1. Find a vector parallel to the plane 2x – y – z = 4 and orthogonal to i + j + k. 1. 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. 1. 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. 1. 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. 1. Find the point in which the line through P(3, 2, 1) normal to the plane 2x – y + 2z = -2 meets the plane. 1. 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? 1. 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. 1. 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. 1. 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) . 1. 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. 1. 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)? 1. (2i – 3j + 3k) # ((x + 2)i + (y – 1)j + zk) = 0 2. x = 3 – t, y = -11t, z = 2 – 3t 3. (x + 2) + 11(y – 1) = 3z 4. (2i – 3j + 3k) * ((x + 2)i + ( y – 1)j + zk) = 0 5. (2i – j + 3k) * (-3i + k) # ((x + 2)i + ( y – 1)j + zk) = 0 1. 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) 1. the coordinates of D, 2. the cosine of the interior angle at B, 3. the vector projection of rBA onto rBC, 4. the area of the parallelogram, 5. an equation for the plane of the parallelogram, 6. the areas of the orthogonal projections of the parallelogram on the three coordinate planes. 1. 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. 1. (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. 1. x2 + y2 + z2 = 4 66. x2 + (y – 1)2 + z2 = 1 2. 4×2 + 4y2 + z2 = 4 68. 36×2 + 9y2 + 4z2 = 36 3. z = – (x2 + y2) 70. y = – (x2 + z2) 4. x2 + y2 = z2 72. x2 + z2 = y2 5. x2 + y2 – z2 = 4 74. 4y2 + z2 – 4×2 = 4 6. y2 – x2 – z2 = 1 76. z2 – x2 – y2 = 1 Chapter 11 Additional and Advanced Exercises 1. 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 1. 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? 1. 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. 1. 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) 1. 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.) 1. 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 1. 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) . 1. For a fixed b determine the value of a which minimizes the magnitude 0 T1 0 . 1. For a fixed a determine the value of b which minimizes the magnitude 0 T2 0 . 1. Determinants and planes 2. 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) . 1. 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 ? 1. 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. 1. 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 . 1. 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 . 1. 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. 1. 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 . 1. Find the distance between the planes 2x + 3y – z = 6 and 2x + 3y – z = 12. 1. 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. 1. Write equations for the planes that lie parallel to and 5 units away from the plane x – 2y + z = 3. 1. Prove that four points A, B, C, and D are coplanar (lie in a common plane) if and only if rAD # (rAB * rBC) = 0. 1. 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 1. 2i 2j 2k 2. i – j + k 2i + j – 2k -i + 2j – k 3. 2i + j 2i – j + k i + 2k 4. i + j – 2k -i – k 2i + 4j – 2k 5. Cross and dot products Show that if u, v, w, and r are any vectors, then 1. u * (v * w) + v * (w * u) + w * (u * v) = 0 2. u * v = (u # v * i)i + (u # v * j)j + (u # v * k)k 3. (u * v) # (w * r) =  u # w v # w u # r v # r  . 1. Cross and dot products Prove or disprove the formula u * (u * (u * v)) # w = – 0 u 0 2 u # v * w. 1. By forming the cross product of two appropriate vectors, derive the trigonometric identity sin (A – B) = sin A cos B – cos A sin B. 1. 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 .) 1. 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 1. State the rules for differentiating and integrating vector functions. Give examples. 1. 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. 1. What is special about the derivatives of vector functions of constant length? Give an example. 1. 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. 1. 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? 1. How do you measure distance along a smooth curve in space from a preselected base point? Give an example. 1. What is a differentiable curve’s unit tangent vector? Give an example. 1. 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? 1. What is a plane curve’s principal normal vector? When is it defined? Which way does it point? Give an example. 1. How do you define N and k for curves in space? 2. What is a curve’s binormal vector? Give an example. 3. 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? 1. 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. 1. Find the point on the curve y = ex where the curvature is greatest. 2. 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? 1. 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). 1. 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. 1. 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. 1. r(t) = (4 cos t)i + 122 sin t2j, t = 0 and p>4 2. r(t) = 123 sec t2i + 123 tan t2j, t = 0 3. 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. 1. 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? 1. 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. 1. Sketch the curve traced by P during the interval 0 … t … 3. 2. Find v and a at t = 0, 1, 2, and 3 and add these vectors to your sketch. 1. 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 1. 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? 1. 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? 1. 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. 1. 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. 1. 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? 1. How high did the javelin go? Motion in Space Find the lengths of the curves in Exercises 15 and 16. 1. r(t) = (2 cos t)i + (2 sin t)j + t2k, 0 … t … p>4 2. 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. 1. r(t) = 4 9 (1 + t)3>2 i + 4 9 (1 – t)3>2 j + 1 3 t k, t = 0 1. r(t) = (et sin 2t)i + (et cos 2t)j + 2et k, t = 0 1. r(t) = t i + 1 2 e2t j, t = ln 2 1. 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. 1. r(t) = (2 + 3t + 3t2)i + (4t + 4t2)j – (6 cos t)k 2. r(t) = (2 + t)i + (t + 2t2)j + (1 + t2)k 3. Find T, N, B, and k, as functions of t if r(t) = (sin t)i + 122 cos t2j + (sin t)k. 1. 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? 1. 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. 1. 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). 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. 1. 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 1. 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. 1. 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

1. 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.

1. Find the angular velocity du>dt when u = 2p.
2. Express the particle’s u@ and z-coordinates as functions of t.
3. 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

1. Suppose the curve in Exercise 1 is replaced by the conical helix

r = au, z = bu shown in the accompanying figure.

1. Express the angular velocity du>dt as a function of u.
2. 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

1. 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.

1. 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.

1. Show that this particular equation has a solution between

x = 0 and x = 2.

method to find the solution to as many places as you can.

1. 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 .

1. Express x # and y # in terms of r # and ru #

by evaluating the dot

products v # i and v # j.

1. Express r # and ru #

in terms of x # and y # by evaluating the dot

products v # ur and v # uu .

1. Express the curvature of a twice-differentiable curve r = ƒ(u) in

the polar coordinate plane in terms of ƒ and its derivatives.

1. 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.

1. Find the beetle’s acceleration and velocity in polar form when

it is halfway to (1 in. from) the origin.

1. 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?

1. 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

1. What is a real-valued function of two independent variables?

Three independent variables? Give examples.

1. 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.

1. 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?

1. 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?

1. 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.

1. What can be said about algebraic combinations and composites of

continuous functions?

1. Explain the two-path test for nonexistence of limits.
2. How are the partial derivatives 0ƒ>0x and 0ƒ>0y of a function

ƒ(x, y) defined? How are they interpreted and calculated?

1. 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.

1. 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.

1. What does it mean for a function ƒ(x, y) to be differentiable?

What does the Increment Theorem say about differentiability?

1. 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?

1. 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?

1. 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.

1. 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.

1. 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.

1. How can you use directional derivatives to estimate change?
2. 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?

1. What can you say about the accuracy of linear approximations of

functions of two (three) independent variables?

1. 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.

1. How do you define local maxima, local minima, and saddle points

for a differentiable function ƒ(x, y)? Give examples.

1. 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.

1. 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.

1. 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.

1. ƒ(x, y) = 9×2 + y2 2. ƒ(x, y) = ex+y
2. 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.

1. ƒ(x, y, z) = x2 + y2 – z 6. g(x, y, z) = x2 + 4y2 + 9z2
2. 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.

1. lim

(x,y)S(p, ln 2)

ey cos x 10. lim

(x,y)S(0,0)

2 + y

x + cos y

1. lim

(x,y)S(1,1)

x – y

x2 – y2 12. lim

(x,y)S(1,1)

x3y3 – 1

xy – 1

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.

1. lim

(x,y)S(0,0)

y

x2 – y

1. lim

(x,y)S(0,0)

x2 + y2

xy

y_x2 xy_0

1. 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?

1. 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.

1. ƒ(r, u) = r sin u – r cos u
2. ƒ(x, y) = 1

2

ln (x2 – y2) + sin-1

y

x

1. ƒ(R1, R2, R3) = 1

R1

+ 1

R2

+ 1

R3

1. ƒ(x, y, z) = cos (4px – y + 5z)
2. P(n, R, T, V ) = nRT

V

(the ideal gas law)

1. 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.

1. g(x, y) = y + xy
2. ƒ(x, y) = ey – x cos y
3. g(x, y) = y – xy – 8y3 + ln (y2 – 1)
4. g(x, y) = x2 – 9xy – sin y + 5ex

Chain Rule Calculations

1. Find dw> dt at t = 0 if w = sin (xy + p), x = et, and y =

ln (t + 1).

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.

1. Find 0w>0r and 0w>0s when r = p and s = 0 if w =

sin (2x – y), x = r + sin s, y = rs.

1. Find 0w>0u and 0w>0y when u = y = 0 if w =

ln21 + x2 – tan-1 x and x = 2eu cos y.

1. 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.

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. 1 – x – y2 – sin xy = 0, P(0, 1)
2. 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.

1. ƒ(x, y) = cos x cos y, P0(p>4, p>4), v = 3i + 4j
2. ƒ(x, y) = x2e-2y, P0(1, 0), v = i + j
3. ƒ(x, y, z) = ln (2x + 3y + 6z), P0(-1, -1, 1),

v = 2i + 3j + 6k

1. ƒ(x, y, z) = x2 + 3xy – z2 + 2y + z + 4, P0(0, 0, 0),

v = i + j + k

1. 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.

1. 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)?

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).

1. Find ƒx(1, 2) and ƒy(1, 2).
2. Find the derivative of ƒ at (1, 2) in the direction toward the

point (4, 6).

1. Which of the following statements are true if ƒ(x, y) is differentiable

1. 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.

1. The derivative of ƒ at (x0 , y0) in the direction of u is a vector.
2. The directional derivative of ƒ at (x0 , y0) has its greatest value

in the direction of _ƒ.

1. 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.

1. x2 + y + z2 = 0; (0, -1, {1), (0, 0, 0)
2. 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 .

1. x2 – y – 5z = 0, P0(2, -1, 1)
2. 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.

1. z = ln (x2 + y2), (0, 1, 0)
2. 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 .

1. 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.

1. Surfaces: x2 + 2y + 2z = 4, y = 1

Point: (1, 1, 1 > 2)

1. 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.

1. ƒ(x, y) = sin x cos y, P0(p>4, p>4)

R:  x – p

4  … 0.1,  y – p

4  … 0.1

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.

1. ƒ(x, y, z) = xy + 2yz – 3xz at (1, 0, 0) and (1, 1, 0)
2. 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

1. 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?

1. 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?

1. 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?

1. 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?

1. Error in estimating a product Let y = uy and z = u + y,

where u and y are positive independent variables.

1. 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?

1. Show that the percentage error in the calculated value of z is

less than the percentage error in the value of y.

1. 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.

1. ƒ(x, y) = x2 – xy + y2 + 2x + 2y – 4
2. ƒ(x, y) = 5×2 + 4xy – 2y2 + 4x – 4y
3. ƒ(x, y) = 2×3 + 3xy + 2y3
4. ƒ(x, y) = x3 + y3 – 3xy + 15
5. ƒ(x, y) = x3 + y3 + 3×2 – 3y2
6. 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.

1. ƒ(x, y) = x2 + xy + y2 – 3x + 3y

R: The triangular region cut from the first quadrant by the line

x + y = 4

1. 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

1. ƒ(x, y) = y2 – xy – 3y + 2x

R: The square region enclosed by the lines x = {2 and y = {2

1. 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

1. 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

1. 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

1. ƒ(x, y) = x3 + y3 + 3×2 – 3y2

R: The square region enclosed by the lines x = {1 and y = {1

1. ƒ(x, y) = x3 + 3xy + y3 + 1

R: The square region enclosed by the lines x = {1 and y = {1

Lagrange Multipliers

1. Extrema on a circle Find the extreme values of ƒ(x, y) =

x3 + y2 on the circle x2 + y2 = 1.

1. Extrema on a circle Find the extreme values of ƒ(x, y) = xy

on the circle x2 + y2 = 1.

1. Extrema in a disk Find the extreme values of ƒ(x, y) =

x2 + 3y2 + 2y on the unit disk x2 + y2 … 1.

1. Extrema in a disk Find the extreme values of ƒ(x, y) =

x2 + y2 – 3x – xy on the disk x2 + y2 … 9.

1. Extrema on a sphere Find the extreme values of ƒ(x, y, z) =

x – y + z on the unit sphere x2 + y2 + z2 = 1.

1. Minimum distance to origin Find the points on the surface

x2 – zy = 4 closest to the origin.

1. 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?

1. 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.

1. 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.

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

1. 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.

1. 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.

1. If a and b are constants, w = u3 + tanh u + cos u, and u =

ax + by, show that

a 0w

0y = b 0w

0x .

1. 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

1. 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.

1. 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

0x =

0y =

02ƒ

0x2 =

02ƒ

0y2 = 1

at that point.

1. 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.

1. 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.

1. 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.

1. 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.

1. 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.

1. Tangent plane and normal line
2. Sketch the surface x2 – y2 + z2 = 4.
3. Find a vector normal to the surface at (2, -3, 3). Add the

1. Find equations for the tangent plane and normal line at

(2, -3, 3).

Chapter 13 A dditional and Advanced Exercises

Partial Derivatives

1. 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

1. 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.

1. 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.

1. 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.

1. 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

1. x

0x + y

0y = nƒ(x, y)

1. x2 a02ƒ

0x2b + 2xya 02ƒ

0x0yb + y2 a02ƒ

0y2b = n(n – 1)ƒ.

1. Surface in polar coordinates Let

ƒ(r, u) = •

sin 6r

6r

, r _ 0

1, r = 0,

where r and u are polar coordinates. Find

1. lim

rS0

ƒ(r, u) b. ƒr(0, 0) c. ƒu(r, u), r _ 0.

z _ f (r, u)

1. Properties of position vectors Let r = xi + yj + zk and let

r = 0 r 0 .

1. Show that _r = r>r.
2. Show that _(rn) = nrn-2r.
3. Find a function whose gradient equals r.
4. Show that r # dr = r dr.
5. Show that _(A # r) = A for any constant vector A.
6. 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.

1. 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).

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

1. 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).

1. 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).

1. 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.

1. 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

1. 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.)

1. 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

1. Finding functions from partial derivatives Suppose that ƒ

and g are functions of x and y such that

0y =

0g

0x and

0x =

0g

0y ,

and suppose that

0x = 0, ƒ(1, 2) = g(1, 2) = 5, and ƒ(0, 0) = 4.

Find ƒ(x, y) and g(x, y).

1. 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.

1. 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).

1. 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

1. 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.

1. 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?

1. Which direction tangential to S at the point (1, 1, 8) will make

the rate of change of temperature a maximum?

1. 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

1. Find all solutions of the one-dimensional heat equation of the

form w = ert sin px, where r is a constant.

1. 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

1. Define the double integral of a function of two variables over a

bounded region in the coordinate plane.

1. How are double integrals evaluated as iterated integrals? Does the

order of integration matter? How are the limits of integration

determined? Give examples.

1. How are double integrals used to calculate areas and average values.

Give examples.

1. 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.

1. Define the triple integral of a function ƒ(x, y, z) over a bounded

region in space.

1. How are triple integrals in rectangular coordinates evaluated?

How are the limits of integration determined? Give an example.

1. How are double and triple integrals in rectangular coordinates

used to calculate volumes, average values, masses, moments, and

centers of mass? Give examples.

1. 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?

1. How are triple integrals in cylindrical and spherical coordinates

evaluated? How are the limits of integration found? Give

examples.

1. How are substitutions in double integrals pictured as transformations

of two-dimensional regions? Give a sample calculation.

1. 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

1. 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.

1. 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.

1. 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.

1. 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.

1. The square bounded by the lines x = 1, y = 1 in the first quadrant
2. 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

1. 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.

1. Integrate ƒ(x, y) = 1> (1 + x2 + y2)2 over
2. Triangular region The triangle with vertices (0, 0), (1, 0),

and 11, 232.

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

1. 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

1. 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

1. 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.

1. Average value Find the average value of r over the solid

sphere r … a (spherical coordinates).

Cylindrical and Spherical Coordinates

1. 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.

1. 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

1. 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

1. 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.

1. 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.

1. 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.

1. Describe the solid by giving equations for the surfaces that

form its boundary.

1. Convert the integral to cylindrical coordinates but do not

evaluate the integral.

1. 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

1. 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).

1. 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.

1. 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

1. 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.

1. Centroid Find the centroid of the region between the parabola

x + y2 – 2y = 0 and the line x + 2y = 0 in the xy-plane.

1. 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.

1. 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

1. x = {2, y = {1 in the xy-plane
2. 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 .)

1. 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.

1. 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.

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.

1. 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.

1. Centroid Find the centroid of the region in the polar coordinate

plane defined by the inequalities 0 … r … 3, -p>3 … u … p>3.

1. 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.

1. 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.

1. Sketch the region and show the centroid in your sketch.
2. 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-?

1. Sketch the region for a = 5p>6 and show the centroid in

Substitutions

1. 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.

1. 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.)

Volumes

1. 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.

1. 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.

1. 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.

1. 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.

1. 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.

1. 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

1. 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.

1. Find the radius of the hole and the radius of the sphere.
2. Evaluate the integral.
3. Sphere and cylinder Find the volume of material cut from the

solid sphere r2 + z2 … 9 by the cylinder r = 3 sin u.

1. Two paraboloids Find the volume of the region enclosed by

the surfaces z = x2 + y2 and z = (x2 + y2 + 1)>2.

1. 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

1. 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.)

1. 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.

1. Rewrite the Cartesian integral with the order of integration

reversed.

1. 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.

1. 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

1. 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?

1. 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.

1. 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.

1. 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

1. 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.

1. 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).

1. 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)

1. 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

1. 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.

1. 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.

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.

1. 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

1. (2)
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

.

1. Substitute y = 2t in Equation (2) to show that

_(1>2) = 2I = 2p.

1. 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.

1. 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?

1. 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.

1. 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.”)

1. What would be the smallest tilt of the satellite dish so that it

holds no water?

1. 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.

1. 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

1. What are line integrals of scalar functions? How are they evaluated?

Give examples.

1. How can you use line integrals to find the centers of mass of

springs or wires? Explain.

1. What is a vector field? What is the line integral of a vector field?

What is a gradient field? Give examples.

1. 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.

1. What is the Fundamental Theorem of line integrals? Explain how

it relates to the Fundamental Theorem of Calculus.

1. Specify three properties that are special about conservative fields.

How can you tell when a field is conservative?

1. What is special about path independent fields?
2. What is a potential function? Show by example how to find a

potential function for a conservative field.

1. What is a differential form? What does it mean for such a form to

be exact? How do you test for exactness? Give examples.

1. What is Green’s Theorem? Discuss how the two forms of Green’s

Theorem extend the Net Change Theorem in Chapter 5.

1. 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.

1. How do you integrate a scalar function over a parametrized surface?

Of surfaces that are defined implicitly or in explicit form?

Give examples.

1. 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.

1. What is the curl of a vector field? How can you interpret it?
2. What is Stokes’ Theorem? Explain how it generalizes Green’s

Theorem to three dimensions.

1. What is the divergence of a vector field? How can you interpret it?
2. What is the Divergence Theorem? Explain how it generalizes

Green’s Theorem to three dimensions.

1. 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

1. 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

1. 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.

1. Integrate ƒ(x, y, z) = 2×2 + z2 over the circle

r(t) = (a cos t)j + (a sin t)k, 0 … t … 2p.

1. 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

1. 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.

1. 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

1. 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.

1. Area of a parabolic cap Find the area of the cap cut from the

paraboloid y2 + z2 = 3x by the plane x = 1.

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.

1. 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.

1. 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

1. 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

1. Parabolic cylinder cut by planes Integrate
2. g(x, y, z) =

yz

24y2 + 1

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.

1. 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.

1. 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.)

1. Spherical band The portion of the sphere x2 + y2 + z2 = 36

between the planes z = -3 and z = 323

1. Parabolic cap The portion of the paraboloid z = – (x2 + y2) >2

above the plane z = -2

1. Cone The cone z = 1 + 2×2 + y2, z … 3
2. 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

1. Portion of paraboloid The portion of the paraboloid y =

2(x2 + z2), y … 2, that lies above the xy-plane

1. Portion of hemisphere The portion of the hemisphere

x2 + y2 + z2 = 10, y Ú 0, in the first octant

1. Surface area Find the area of the surface

r(u, y) = (u + y)i + (u – y)j + yk,

0 … u … 1, 0 … y … 1.

1. Surface integral Integrate ƒ(x, y, z) = xy – z2 over the surface

in Exercise 25.

1. 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

1. 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?

1. F = x i + y j + z k
2. F = (x i + y j + zk )>(x2 + y2 + z2)3>2
3. F = xeyi + yezj + zexk
4. F = (i + zj + yk)>(x + yz)

Find potential functions for the fields in Exercises 33 and 34.

1. F = 2i + (2y + z)j + (y + 1)k
2. 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.

1. F = 2xy i + j + x2 k 36. F = 2xy i + x2 j + k
2. 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:

1. By using the parametrization of the curve to evaluate the work

integral.

1. By evaluating a potential function for F.
2. Flow along different paths Find the flow of the field F =

_(x2zey)

1. 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.

1. 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.

1. 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

1. 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

1. 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.

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.

1. 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).

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.

1. 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 .

1. 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.

1. 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.

1. 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.

1. Square F = (2xy + x)i + (xy – y)j

C: The square bounded by x = 0, x = 1, y = 0, y = 1

1. Triangle F = (y – 6×2)i + (x + y2)j

C: The triangle made by the lines y = 0, y = x, and x = 1

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.

1. 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.

1. 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.

1. Cube F = 2xyi + 2yzj + 2xzk

D: The cube cut from the first octant by the planes x = 1, y = 1,

z = 1

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

1. 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

1. 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

1. 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.

1. 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.

1. Cylindrical can Use the Divergence Theorem to find the flux of

F = xy2i + x2yj + yk outward through the surface of the region

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.

1. The limaçon x = 2 cos t – cos 2t, y = 2 sin t – sin 2t,

0 … t … 2p

y

x

0 1

1. The deltoid x = 2 cos t + cos 2t, y = 2 sin t – sin 2t,

0 … t … 2p

y

x

0 3

1. The eight curve x = (1>2) sin 2t, y = sin t, 0 … t … p (one loop)

y

x

1

_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

1. 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.

1. 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.

1. 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.

1. 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.

1. 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.

1. 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.

1. 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?

1. 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.

1. 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.

1. 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.

1. Find the total work done by evaluating the line integral in part (a).
2. 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.

1. 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.

1. 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.

1. Find the total work done by evaluating the surface integral in

part (a).

1. 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.

1. 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.

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.

1. 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.

1. Use the Divergence Theorem to find the magnitude of the

buoyant force in part (b).

1. 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.

1. 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.

1. 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.

1. Prove or disprove that if _ # F = 0 and _ * F = 0, then F = 0.
2. 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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