Calculus Concepts And Contexts 4th Edition by James Stewart – Test Bank

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Calculus Concepts And Contexts 4th Edition by James Stewart – Test Bank

 

Section 1.2: Mathematical Models: A Catalog of Essential Functions

 

  1. Classify the function f(x) =
a. Power function e. Algebraic function
b. Root function f. Trigonometric function
c. Polynomial function g. Exponential function
d. Rational function h. Logarithmic function

 

 

ANS:  D                    PTS:   1

 

  1. Classify the function f(x) =
a. Power function e. Algebraic function
b. Root function f. Trigonometric function
c. Polynomial function g. Exponential function
d. Rational function h. Logarithmic function

 

 

ANS:  C                    PTS:   1

 

  1. Classify the function f(x) = sin (5)  sin (3) x.
a. Power function e. Algebraic function
b. Root function f. Trigonometric function
c. Polynomial function g. Exponential function
d. Rational function h. Logarithmic function

 

 

ANS:  C                    PTS:   1

 

  1. The following time-of-day and temperature (F°) were gathered during a gorgeous midsummer day in Fargo, North Dakota:
Time of Day Temperature
18 74
17 73
16 73
15 72
14 70
13 70
12 68
11 66
10 63
9 62
8 59
7 58

Source: National Weather Service; www.weather.gov

 

(a)  Make a scatter plot of these data.

 

(b)  Fit a linear model to the data.

 

(c)  Fit an exponential model to the data.

 

(d)  Fit a quadratic model to the data.

 

(e)  Use your equations to make a table showing the predicted temperature for each model, rounded to the nearest degree.

 

(f)  The actual temperature at 8:00 p.m. (20 hours) was 70° F. Which model was closest? Which model was second-closest?

 

(g)  All of the models give values that are too high for each of the times after 6:00 PM. What is one possible explanation for this?

 

ANS:

 

(b)  y = 1.561x  47.68

(c)  y = 49.89802 e

(d)  y =

(e)  Linear: 79

Exponential: 80

Quadratic: 75

(f)  Closest: quadratic. Second-closest: linear

(g)  Answers may vary, but only one explanation is that the data only reflect the part of the day when the air is warming and do not take into account cooling that takes place later in the day into evening. The only model that begins to reflect this is the quadratic model.

 

PTS:   1

 

  1. Consider the data below:
t 1 2 3 4 5 6
y 2.4 19 64 152 295 510

 

(a)  Fit both an exponential curve and a third-degree polynomial to the data.

 

(b)  Which of the models appears to be a better fit? Defend your choice.

 

ANS:

(a)

 

(b)  A third degree polynomial, for example, , appears to be a better fit.

 

PTS:   1

 

  1. The following table contains United States population data for the years 1981–1990, as well as estimates based on the 1990 census.

 

Year U.S. Population (millions)   Year U.S. Population (millions)
1981 229.5   1991 252.2
1982 231.6   1992 255.0
1983 233.8   1993 257.8
1984 235.8   1994 260.3
1985 237.9   1995 262.8
1986 240.1   1996 265.2
1987 242.3   1997 267.8
1988 244.4   1998 270.2
1989 246.8   1999 272.7
1990 249.5   2000 275.1

Source: U.S. Census Bureau website

 

(a)  Make a scatter plot for the data and use your scatter plot to determine a mathematical model of the U.S. population.

 

(b)  Use your model to predict the U.S. population in 2003.

 

ANS:

(a)

 

A linear model seems appropriate. Taking t = 0 in 1981, we obtain the model P(t) = 2.4455t + 228.5.

 

(b)  P (22)  282.3

 

PTS:   1

 

  1. The following table contains United States population data for the years 1790–2000 at intervals of 10 years.

 

Year Years since 1790 U.S. population (millions)   Year Years since 1790 U.S. population (millions)
1790 0 3.9   1900 110 76.0
1800 10 5.2   1910 120 92.0
1810 20 7.2   1920 130 105.7
1820 30 9.6   1930 140 122.8
1830 40 12.9   1940 150 131.7
1840 50 17.1   1950 160 150.7
1850 60 23.2   1960 170 178.5
1860 70 31.4   1970 180 202.5
1870 80 39.8   1980 190 225.5
1880 90 50.2   1990 200 248.7
1890 100 62.9   2000 210 281.4

 

(a)  Make a scatter plot for the data and use your scatter plot to determine a mathematical model for the U.S. population.

 

(b)  Use your model to predict the U.S. population in 2005.

 

ANS:

(a)

 

Answers will vary, but a quadratic or cubic model is most appropriate.

Linear model: P(t) = 1.28545t  40.47668;

quadratic model: P(t) = 0.006666t0.1144t  5.9;

cubic model: P(t) = (6.6365 10) t0.004575t0.057155t  3.7;

exponential model: P(t) = 6.04852453  1.020407795

 

(b)  Linear model: P(215)  235.9;

quadratic model: P(215)  289.4;

cubic model: P(215) 293.4;

exponential model: P(215)  465.6

 

PTS:   1

 

  1. Refer to your models from Problems 6 and 7 above. Why do the two data sets produce such

different models?

 

ANS:

Problem 6 covers a much shorter time span, so its data exhibit local linearity, while Problem 7 shows nonlinear population growth over a longer time span.

 

PTS:   1

 

  1. The following are the winning times for the Olympic Men’s 110 Meter Hurdles:

 

Year Time   Year Time   Year Time
1896 17.6   1932 14.6   1976 13.3
1900 15.4   1936 14.2   1980 13.39
1904 16   1948 13.9   1984 13.2
1906 16.2   1952 13.7   1988 12.98
1908 15   1956 13.5   1992 13.12
1912 15.1   1960 13.8   1996 12.95
1920 14.8   1964 13.6   2000 13
1924 15   1968 13.3   2004 12.91
1928 14.8   1972 13.24    

 

(a)  Make a scatter plot of these data.

 

(b)  Fit a linear model to the data.

 

(c)  Fit an exponential model to the data

 

(d)  Fit a quadratic model to the data.

 

(e)  Use your equations to make a table showing the predicted winning time for each model for the 2008 Olympics, rounded to the nearest hundredth of a second.

 

(f)  The actual time for the 2008 Olympics was 12.93 seconds. Which model was closest? Which model was second-closest?

 

ANS:

(a)

 

(b)  y = -0.0320057x + 76.595

(c)  y = 1053.09176(0.997791842)

(d)  y = 0.000322x  1.2872778x  1299.573

(e)  Linear: 12.33

Exponential: 12.44

Quadratic: 13.04

(f)  Closest: quadratic. Second-closest: exponential

 

PTS:   1

Section 2.2: The Limit of a Function

 

Use the graph below for the following questions:

 

 

  1. For the function whose graph is given above, determine
a. –3 e. 1
b. –2 f. 2
c. –1 g. 3
d. 0 h. Does not exist

 

 

ANS:  B                    PTS:   1

 

  1. For the function whose graph is given above, determine
a. –3 e. 1
b. –2 f. 2
c. –1 g. 3
d. 0 h. Does not exist

 

 

ANS:  H                    PTS:   1

 

  1. For the function whose graph is given above, determine
a. –3 e. 1
b. –2 f. 2
c. –1 g. 3
d. 0 h. Does not exist

 

 

ANS:  C                    PTS:   1

 

  1. For the function whose graph is given above, determine
a. –3 e. 1
b. –2 f. 2
c. –1 g. 3
d. 0 h. Does not exist

 

 

ANS:  C                    PTS:   1

 

Use the graph below for the following questions:

 

 

  1. For the function whose graph is given above, determine f (–2).
a. –3 e. 1
b. –2 f. 2
c. –1 g. 3
d. 0 h. Does not exist

 

 

ANS:  H                    PTS:   1

 

  1. For the function whose graph is given above, determine
a. –3 e. 1
b. –2 f. 2
c. –1 g. 3
d. 0 h. Does not exist

 

 

ANS:  C                    PTS:   1

 

  1. For the function whose graph is given above, determine
a. –3 e. 1
b. –2 f. 2
c. –1 g. 3
d. 0 h. Does not exist

 

 

ANS:  H                    PTS:   1

 

  1. For the function whose graph is given above, determine
a. –3 e. 1
b. –2 f. 2
c. –1 g. 3
d. 0 h. Does not exist

 

 

ANS:  A                    PTS:   1

 

  1. Use the given graph to find the indicated quantities:

 

 

(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
(j)  f (–1) (k)  f (0) (l)  f (1)
(m)  f (2)    

 

 

ANS:

(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
(j)  f (–1) = 0 (k)  f (0) » –1.7 (l)  f (1) = 1
(m)  f (2) = –1    

 

 

PTS:   1

 

  1. Use the graph of f below to determine the value of each of the following quantities, if it exists. If it does not exist, explain why.

 

 

(a) (b) (c) (d)  f (–2) (e)  f (2)

 

 

ANS:

(a)  The limit does not exist because the left- and right-hand limits are different.

(b) (c)
(d)  f (–2) = –1 (e)  f (2) = 2

 

 

PTS:   1

 

  1. Use the given graph to find the indicated quantities:

 

 

(a) (b) (c)  f (3)
(d) (e) (f)  f (–2)

 

 

ANS:

(a)  –2 (b)  Does not exist (c)  1
(d)  –1 (e)  –1 (f)  Undefined

 

 

PTS:   1

 

  1. (a)  Explain in your own words what is meant by

 

(b)  Is it possible for this statement to be true yet for f –(2) = 5? Explain.

 

ANS:

(a)  (Answers will vary.)  means that the values of f can be made as close as desired to 3 by taking values of x close enough to –2, but not equal to –2.

(b)  Yes, it is possible for , but  The limit refers only to how the function behaves when x is close to –2. It does not tell us anything about the value of the function at x = –2.

 

PTS:   1

 

  1. Sketch the graph of a function f on [–5, 5] that satisfies all of the following conditions:

 

f (–4) = 2, f (–3) = –1, f(–2) = 2, f(1) = –3, f(2) = 1, f(3) = 0, f(4) = –3, , , and

 

ANS:

(Answers will vary)

 

 

PTS:   1

 

  1. Consider the function  Make an appropriate table of values in order to determine the indicated limits:

 

(a) (b)

 

(c)  Does  exist? If so, what is its value? If not, explain.

 

ANS:

 

x f(x)   x f(x)
–2.9 –79  
–2.99 –799   –3.000001 8000001
–2.999 –7999   –3.00001 800001
–2.9999 –79999   –3.0001 80001
–2.99999 –799999   –3.001 8001
–2.999999 –7999999   –3.01 801
  –3.1 81

 

Using the table values, the limits appear to be:

 

(a) (b)

(c)  Since the right-hand limit and the left-hand limit have different values, the limit does not exist at = –3.

 

PTS:   1

 

  1. Use a table of values to estimate the value of each of the following limits, to 4 decimal places.

 

(a) (b) (c)

 

 

ANS:

(a)  1.0986

(b)  1.6667

(c)  2.7183

 

PTS:   1

 

  1. A cellular phone company has a roaming charge of 32 cents for every minute or fraction of a minute when you are out of your zone.

 

(a)  Sketch a graph of the \out-of-your-zone” costs, C, of cellular phone usage as a function of the length of the call, t, for .

 

(b)  Evaluate:

 

(i)            (ii)

 

(c)  Explain the significance of the left limit (i) and the right limit (ii) to the cell phone user.

 

(d)  For what values of t does C (t) not have a limit? Justify your answer.

 

ANS:

(a)

 

(b)  (i)  64 cents

(ii) 96 cents

 

(c)  The fact that  shows that there is an abrupt change in the cost of cellular phone usage at t = 2.

 

(d)  For  does not exist, since

 

PTS:   1

 

  1. If f (x) = 2x, how close to 3 does x have to be to ensure that f (x) is within 0.1 of 8?

 

ANS:

Answers will vary. One reasonable answer is that f (x) is within 0.1 of 8 when x is within 0.017 of 3, that is, when 2.983 < x < 3.017.

 

PTS:   1

 

  1. Determine  by producing an appropriate table.

 

ANS:

 

x f(x)   x
1.001 1716.924   0.99 99.995683
1.0001 1051.654   0.999 632.304575
1.00001 1005.012   0.9999 951.671108
1.000001 1000.500   0.99999 995.021352
1.0000001 1000.049   0.999999 999.499236
      0.9999999 999.950052

 

From the tables, it appears that

 

PTS:   1

Section 3.5: Implicit Differentiation

 

  1. If  find the value of  at the point (3, 4).
a. e. 0
b. f. 1
c. g.
d. h.

 

 

ANS:  H

 

  1. If  find the value of  at the point ().
a. e.
b. f.
c. g.
d. h. 4

 

 

ANS:  D

 

  1. If  find the value of  at the point (4, 1).
a. e. 1
b. 0 f. 3
c. 2 g.
d. h. 4

 

 

ANS:  G

 

  1. Find the y-intercept of the tangent to the ellipse  at the point ().
a. 3 e.
b. f.
c. g.
d. h.

 

 

ANS:  H

 

  1. Find the slope of the tangent to the curve  at the point .
a. e. 1
b. f. 2
c. g. 4
d. 0 h. 8

 

 

ANS:  B

 

  1. Let   If
a. 0 e.
b. 1 f.
c. 2 g.
d. h. 8

 

 

ANS:  G

 

  1. Find the slope of the tangent line to the curve  at the point (3, 3).
a. e. 1
b. f. 2
c. g. 3
d. 0 h. 4

 

 

ANS:  C

 

  1. Find the equation of the line normal to the curve defined by the equation  at the point (2, ).
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  A

 

  1. What is the slope of the tangent line to the curve  at the point (0, 1)?
a. e. 1
b. f. 2
c. g. 3
d. 0 h. 4

 

 

ANS:  C

 

  1. If  , find the value of  at the point (, 4)?
a. e. 1
b. f. 2
c. g. 4
d. 0 h. 8

 

 

ANS:  B

 

  1. If , find an expression for .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  E

 

  1. If , find an expression for .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  A

 

  1. Find  if .

 

ANS:

 

  1. Find  if .

 

ANS:

 

  1. Find  if .

 

ANS:

 

  1. Find an equation of the tangent line to the curve  at (1, 1).

 

ANS:

3x + 5y – 8 = 0

 

  1. Find an equation of the tangent line to the curve  at (3, 2).

 

ANS:

6x – 5y – 8 = 0

 

  1. The curve  has two tangents at x = 1.  What are their equations?

 

ANS:

6x – 5y + 9 = 0, 4x – 5y – 14 = 0

 

  1. There are two lines passing through the point (1, 0) tangent to the parabola .  Find their equations.

 

ANS:

y = x + 1, y = x – 1

 

  1. Find the point(s) where the curve  has a horizontal tangent.

 

ANS:

 

  1. If  find the value of  at the point (5, 4).

 

ANS:

 

  1. If , find an expression for .

 

ANS:

 

  1. If sin y = x, find the value of  at the point .

 

ANS:

 

  1. If sin y = cos x, find the value of  at the point .

 

ANS:

 

  1. Find an equation of the tangent line to the curve  at the point (1, 2).

 

ANS:

The slope is , so an equation of the line is .

 

  1. If , find an expression for .

 

ANS:

 

  1. Use implicit differentiation to find .

 

ANS:

 

  1. Show that the curves  and  are orthogonal.

 

ANS:

, 6xy + 2y = 0,

 

  1. Show that the curves  and  are orthogonal.

 

ANS:

, 3x2y – y3 – x= 4,

 

  1. Lake bottoms are frequently mapped using contour lines, which are curves joining points of

the same depth. The path of steepest descent is orthogonal to the contour lines. Given the

contour map below, sketch the path of steepest descent from starting positions A and B to

the deepest point C.

 

 

ANS:

Section 4.7: Newton’s Method

 

  1. Use Newton’s method with the initial approximation  to find , the second approximation to a root of the equation .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  B                    PTS:   1

 

  1. If Newton’s method is used to solve  with first approximation , what is the second approximation, ?
a. –0.500 e. –0.600
b. –0.525 f. –0.625
c. –0.550 g. –0.650
d. –0.575 h. –0.675

 

 

ANS:  F                    PTS:   1

 

  1. Use Newton’s method to find the root of  that lies between 0 and 1.
a. 0.316 e. 0.474
b. 0.333 f. 0.500
c. 0.158 g. 0.079
d. 0.167 h. 0.084

 

 

ANS:  B                    PTS:   1

 

  1. Use Newton’s method to approximate the root of  that lies between  and .
a. 0.473 e. –0.473
b. 0.372 f. –0.372
c. 0.563 g. –0.563
d. 1 h. None of these

 

 

ANS:  E                    PTS:   1

 

  1. Use Newton’s method to approximate the root of  that lies between  and .
a. 1.673 e. 2.473
b. 1.7693 f. 1.75
c. 1.77 g. 1.8693
d. 1 h. None of these

 

 

ANS:  B                    PTS:   1

 

  1. Given , use Newton’s method to find the iterative formula for .
a. e.
b. f.
c. g.
d. h. None of these

 

 

ANS:  G                    PTS:   1

 

  1. Given , use Newton’s method to find the iterative formula for .
a. e.
b. f.
c. g.
d. h. None of these

 

 

ANS:  C                    PTS:   1

 

  1. Use Newton’s method to approximate the root of  that lies between  and .

 

ANS:

–0.486

 

PTS:   1

 

  1. Use Newton’s method to approximate the root of  that lies between  and .

 

ANS:

1.728

 

PTS:   1

 

  1. Given , use Newton’s method to find the iterative formula for .

 

ANS:

 

PTS:   1

 

  1. Given , use Newton’s method to find the iterative formula for .

 

ANS:

 

PTS:   1

 

  1. Given , use Newton’s method to find the iterative formula for .

 

ANS:

 

PTS:   1

 

  1. If Newton’s method is used to find the cube root of a number a with first approximation , find an expression for .

 

ANS:

 

PTS:   1

 

  1. Sketch the graph of  on the interval . Suppose that Newton’s method is used to approximate the positive root of f with initial approximation .

 

(a) On your sketch, draw the tangent lines that you would use to find  and , and estimate the numerical values of  and .

 

(b) To approximate the negative root of f, use  as the starting approximation. As before, draw the tangent lines that you would use to find  and , and estimate the numerical values of  and .

 

(c) Suppose that you had used  as the starting point for approximating the negative root. Discuss what would happen.

 

ANS:

(a) (b) (c)
Newton’s method would fail to find the negative root if  were chosen. (It would converge slowly to the positive root.)

 

 

PTS:   1

 

  1. Sketch the graph of  on . For each initial approximation given below, determine graphically what happens if Newton’s Method is used to approximate the roots of .

 

(a)

 

(b)

 

(c)

 

(d)

 

(e)

 

ANS:

(a) (b) (c)
Using  Newton’s Method will approximate the root between –3 and –2. Using  Newton’s Method will approximate the root between –1 and 0. Using  Newton’s Method will approximate the root between 0 and 1.
(d) (e)
Using  Newton’s Method will approximate the root between 0 and 1. Using  Newton’s Method will converge very slowly to the root between –3 and –2, because  is close to 0.

 

 

PTS:   1

 

  1. Use Newton’s method to find  correct to four decimal places.

 

ANS:

9.43398

 

PTS:   1

 

  1. Find, correct to six decimal places, the root of .

 

ANS:

0.5109734

 

PTS:   1

 

  1. Sketch the graph of . Clearly the only x-intercept is zero. However, Newton’s method fails to converge here. Explain this failure.

 

ANS:

 

PTS:   1

 

  1. Use Newton’s Method to approximate all real roots of .

 

ANS:

–3.192258 or 2.19258

 

PTS:   1

 

  1. (a)  Explain why Newton’s Method is unable to find a root for  if  or .

 

(b)  Using ,use Newton’s Method to approximate a root of this equation to four decimal places.

 

ANS:

(a)  Since (b)  –2.19582

 

 

PTS:   1

Section 6.1: More About Areas

 

  1. Find the area of the region bounded by the curves  and
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  A

 

  1. Find the area of the region bounded by the curves  and
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  G

 

  1. The area of the region bounded by  and  between and  is
a. e.
b. f.
c. g.
d. 0 h.

 

 

ANS:  A

 

  1. Find the area of the region bounded by the curves  and
a. e.
b. f.
c. g. 4
d. h. 2

 

 

ANS:  E

 

  1. Find the area of the region bounded by the curves  and
a. e.
b. f.
c. g.
d. 20 h.

 

 

ANS:  F

 

  1. The area of the region bounded by  and the y-axis is
a. e.
b. f. 1
c. g. 2
d. 0 h.

 

 

ANS:  B

 

  1. Find the area of the region bounded by the parabola  and the line .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  B

 

  1. Find the area of the region bounded by the curve , and the x-axis.
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  D

 

  1. Find the area of the region bounded by , and the x-axis.
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  E

 

  1. Find the area of the region bounded by , and the x-axis.
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  C

 

  1. Find the area of the region bounded by the curves and .

 

ANS:

 

  1. Find the area of the region bounded by the curves  and .

 

ANS:

 

  1. Find the area of the region bounded by the curves  and the x-axis.

 

ANS:

 

  1. Find the area of the region bounded by the curves  and .

 

ANS:

 

  1. Find the area of the region bounded by the curves  and .

 

ANS:

 

  1. Let R be the region bounded by: , the tangent to at , and the x-axis.

Find the area of R integrating

 

(a)  with respect to x.

 

(b)  with respect to y.

 

ANS:

(a)

(b)

 

  1. Find the area of the region bounded by the curves  and .

 

ANS:

About 10.43

 

  1. Using the help of a graphing calculator, find the area of the region bounded by the curves  and .

 

ANS:

 

  1. Find the area of the region bounded by the curves

 

ANS:

 

  1. Find the area of the shaded region:

 

ANS:

 

  1. Find the area of the shaded region:

 

ANS:

 

  1. Find the area of the shaded region:

 

ANS:

 

  1. Find the area of the shaded region:

 

ANS:

9

 

  1. Find the area of the region bounded by

 

ANS:

 

  1. Find the area of the region bounded by

 

ANS:

8

 

  1. A particle is moving in a straight line and its velocity is given by  where t is measure in seconds and v in meters per second. Find the distance traveled by the particle during the time interval .

 

ANS:

28 m

 

  1. A stone is thrown straight up from the top of a tower that is 80 ft tall with initial velocity 64 ft/s. What is the total distance traveled by the stone when it hits the ground?

 

ANS:

208 feet

 

  1. Express the area of the given region as a definite integral. Do not evaluate.

 

ANS:

OR

 

  1. Express the area of the given region as a definite integral. Do not evaluate.

 

ANS:

OR

Section 7.6: Predator-Prey Systems

 

  1. Suppose that we model populations of aphids and ladybugs with the system of differential equations:

Find the equilibrium solution.

a. e.
b. f.
c. g.
d. h.

 

 

ANS:  A

 

  1. Suppose that we model populations of aphids and ladybugs with the system of differential equations:

Find the expression for .

a. e.
b. f.
c. g.
d. h.

 

 

ANS:  D

 

  1. Suppose that we model populations of predators and preys (in millions) with the system of differential equations:

Find the equilibrium solution.

a. e.
b. f.
c. g.
d. h.

 

 

ANS:  E

 

  1. Suppose that we model populations (in millions) of predators and preys with the system of differential equations:

Find the expression for .

a. e.
b. f.
c. g.
d. h.

 

 

ANS:  F

 

  1. A predator-prey system is modeled by the system of differential equations , , where a, b, c, and d are positive constants.

 

(a)  Which variable, x or y, represents the predator? Defend your choice.

 

(b)  Show that the given system of differential equations has the two equilibrium solutions  and .

 

(c)  Explain the significance of each of the equilibrium solutions.

 

ANS:

(a)   represents the predator. In the absence of prey, the predators will die out.

(b)  Solve  to get these solutions.

(c)   and  implies that there is neither predator nor prey. The population of both predator and prey are not changing if

 

  1. Consider the predator-prey system , where x and y are in millions of creatures and t represents time in years.

 

(a)  Find equilibrium solutions for this system.

 

(b)  Explain why it is reasonable to approximate this predator-prey system as , if the initial conditions are x(0) = 0.001 and y(0) = 0.002.

 

(c)  Describe what this approximate system tells about the rate of change of each of the specie populations x(t) and y(t) near (0, 0).

 

(d)  Find the solution for the approximate system given in part (b).

 

(e)  Sketch x (t) and y (t) as determined in part (d) on the same coordinate plane.

 

(f)  Sketch a phase trajectory through (0.001; 0.002) for the predator-prey system. Describe in words what happens to each population of species and the interaction between them.

 

ANS:

(a)   or

(b)  Near ,

(c)  Near , the prey population increases exponentially, and the predator population decreases exponentially.

(d)

(e)

(f)

As the population of predator decreases, the population of prey increases.

 

  1. Consider the following predator-prey system where x and y are in millions of creatures and t represents time in years:

 

(a)  Show that (4, 2) is the nonzero equilibrium solution.

 

(b)  Find an expression for .

 

(c)  The direction field for the differential equation is given below:

 

 

(i)  Locate (4, 2) on the graph.

(ii)  Sketch a rough phase trajectory through P indicated in the graph.

 

(d)  With the aid of the phase trajectory, answer the following questions:

 

(i)  For the region  and 0 < y < 2, is x (t) increasing or decreasing? Is y (t) increasing or decreasing? Describe in words how the two species interact with one another.

(ii)  For the region x > 4 and 0 < y < 2, is x (t) increasing or decreasing? Is y (t) increasing or decreasing? Describe in words how the two species interact with one another.

(iii)  For the region x > 4 and y > 2, is x (t) increasing or decreasing? Is y (t) increasing or decreasing? Describe in words how the two species interact with one another.

(iv)  For the region 0 < x < 4 and y > 2, is x (t) increasing or decreasing? Is y (t) increasing or decreasing? Describe in words how the two species interact with one another.

 

(e) Suggest a pair of species which might interact in the manner described by this system.

 

ANS:

(a)  Solve the system of equations

(b)

(c)

 

(d)  (i)

In this region the number of predator decreases because of the lack of prey, whereas the prey population can increase due to lack of predators.

(ii)

In this region the prey population has increased so much that the predator population can also increase.

(iii)

In this region the predator population has increased so much that the prey population is in decline.

(iv)

In this region, due to the lack of prey, both predator and prey are in decline.

(e)  Wolves and rabbits.

 

  1. A phase portrait of a predator-prey system is given below in which F represents the population of foxes (in thousands) and R the population of rabbits (in thousands).

 

 

(a)  Referring to the graph, what is a reasonable non-zero equilibrium solution for the system?

 

(b)  Write down a possible system of differential equations which could have been used to produce the given graph.

 

(c)  Describe how each population changes as time passes, using the initial condition P indicated on the graph.

 

(d)  Use your description in part (c) to make a rough sketch of the graph of R and F as functions of time.

 

ANS:

(a)  (4, 2)

(b)    (Answers may vary.)

(c)  Initially, the numbers of both species increase. At a certain point, the rabbit population begins a steep decline, followed closely by the fox population. Then the rabbit population begins to increase, again followed by the fox population, and the cycle begins anew.

 

(d)

 

  1. The population of two species is modeled by the system of equations .

 

(a)  Find an expression for .

 

(b)  A possible direction field for the differential equation in part (a) is given below:

 

 

 

Use this graph to sketch a phase portrait with each of P, Q, R, and S as an initial condition. Describe the behavior of the trajectories near the nonzero equilibrium solutions.

 

(c)  Graph x and y as function of t. What happens to the population of the two species as the time t increases without bound?

 

ANS:

(a)

(b)

(c)

The predator and prey populations rise and fall in cycles.

 

  1. In each of the given systems, x and y are populations of two different species which are solutions to the differential equations. For each system, describe how the species interact with one another (for example, do they compete for the same resources, or cooperate for mutual benefit?) and suggest a pair of species that might interact in a manner consistent with the given system of equations.

 

(a) (d)
(b) (e)
(c) (f)

 

 

ANS:

(a)  Predator-prey system  for example, robins and worms.

(b)  Compete for the same resource  for example, cheetahs and lions compete for wildebeest.

(c)  Cooperate for mutual benefit  for example, clownfish and anemone.

(d)  Cooperate for mutual benefit  for example, clownfish and anemone.

(e)  Predator-prey system  for example, whales and krills.

(f)  No interaction  for example, whales and tigers.

Section 9.4: The Cross Product

 

  1. Find the cross product .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  A                    PTS:   1

 

  1. Find , given that , , and the angle between a and b is .
a. e.
b. f.
c. 10 g.
d. 5 h. –5

 

 

ANS:  D                    PTS:   1

 

  1. Find the cross product ; where a = – i + 2j + 4k and b = 7i + 3j.
a. 15i 8j + 10k e. –7i – 23j + 33k
b. 3i + 11j – 9k f. –10i + 27j + 13k
c. 7i + 13j + k g. –12i + 28j – 17k
d. 14i + 5j – 12k h. 18i + 32j + 7k

 

 

ANS:  G                    PTS:   1

 

  1. Find the length of the cross product of the vectors  and .
a. e.
b. f. 2
c. g.
d. h.

 

 

ANS:  H                    PTS:   1

 

  1. Find the area of the triangle whose vertices are A(1,–1, 2), B(4, 0, 1), and C(2,–1, 2).
a. e.
b. f.
c. g. 3
d. h.

 

 

ANS:  C                    PTS:   1

 

  1. Find a unit vector orthogonal to both of the vectors  and .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  D                    PTS:   1

 

  1. Find a unit vector orthogonal to the plane through the points (1, 0, 0), (0, 1, 0), and (0, 2, 2).
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  H                    PTS:   1

 

  1. Given three points P(1, –1, 0), Q(0, 1, 2) and R(–1, –1, 1), find the distance from the point P to the line passing through the points Q and R.
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  C                    PTS:   1

 

  1. Given three points P(1, –1, 0), Q(0, 1, 2) and R(–1, –1, 1), find the distance from the point Q to the line passing through the points P and R.
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  B                    PTS:   1

 

  1. Given three points P(1, –1, 0), Q(0, 1, 2) and R(–1, –1, 1), find the distance from the point R to the line passing through the points Q and P.
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  D                    PTS:   1

 

  1. Find the volume of the parallelepiped determined by the vectors , , and .
a. 1 e. 5
b. 2 f. 6
c. 3 g. 7
d. 4 h. 8

 

 

ANS:  A                    PTS:   1

 

  1. Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS, where P (–211), Q(–102), R(052), and S(130).
a. 19 e. 20
b. 22 f. 40
c. 38 g. 44
d. 80 h. 72

 

 

ANS:  A                    PTS:   1

 

  1. Find the torque vector  of the force F =  acting on a rigid body at the point given by the position vector r = .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  C                    PTS:   1

 

  1. Find the cross product , where a = i + 2j + 3k and b = 2i – j + k.

 

ANS:

5i + 5j – 5k

 

PTS:   1

 

  1. Consider the points P = (1, 2, 3), Q = (2, –1, 0) , and R = (–1, 4, 1). Find the area of the triangle PQR.

 

ANS:

 

PTS:   1

 

  1. Let a and b be vectors such that  = 2 and  = 3. Assume that the angle between a and b is . Find .

 

ANS:

 

PTS:   1

 

  1. Find , where a and b are given in the figure.

 

ANS:

 

PTS:   1

 

  1. Let a and b be vectors. Under what conditions is ? When is ?

 

ANS:

when (1) a = b, (2) a = 0 or b = 0; (3) a and b are parallel; in all cases ;  always.

 

PTS:   1

 

  1. Use the property of the cross product that  to derive a formula for the distance d from a point P to a line l. Use this formula to find the distance from the origin to the line through (21, –4) and (33, –2).

 

ANS:

Let u be a vector from l to P, v be a vector parallel to l and  be the angle between u and v. Note that d = . But . So .

 

PTS:   1

 

  1. Given the vectors u = 2i + j – k and w = i + j + 4k, find a vector of length 2 which is orthogonal to both u and w.

 

ANS:

 

PTS:   1

 

  1. Find , where a and b are given in the figure.

 

ANS:

 

PTS:   1

 

  1. Let . Find ), if it exists.

 

ANS:

0

 

PTS:   1

 

  1. Let . Find the area A of the parallelogram determined by u and v, if it exists.

 

ANS:

 

PTS:   1

 

  1. Find the distance from the point P(1, 1, 1) to the line passing through the points Q(201) and R(032).

 

ANS:

 

PTS:   1

 

  1. Give a counterexample showing that the statement “If , then b = c” is not always true.

 

ANS:

For example, if a = i, b = 2i, and c = 3i, then , but .

 

PTS:   1

 

  1. For vectors a and b, given that , which of the following statements is true?
a. a and b are parallel d. a and b are perpendicular
b. a and b are not parallel e. a and b are not perpendicular
c. a = b  

 

 

ANS:  D                    PTS:   1

 

  1. Use vectors to determine whether the points P(000), Q(500), R(266), and S(766) are coplanar.

 

ANS:

Yes

 

PTS:   1

 

  1. If vector a = 3i + 4j – 2k; and vector b = 2i j + 5k, find a vector perpendicular to both vectors.

 

ANS:

Any scalar multiple of 18i 19j – 11k is perpendicular

 

PTS:   1

 

  1. Find the volume of the parallelepiped spanned by the vectors 2i + 3j + 5k, 3 j + 4k, and –i – 2j + 3k.

 

ANS:

64

 

PTS:   1

 

  1. Find the volume of the parallelepiped below given P = (1, –32), Q = (3, –13), R = (21, –4), and S = (–121).

 

ANS:

91

 

PTS:   1

 

  1. Find the area of quadrilateral ABCD. Note that ABCD is not a parallelogram.

 

ANS:

 

PTS:   1

 

  1. Suppose that , and let . Which of the following statements is true?
a. c is perpendicular to both a and b c. c is perpendicular to a
b. c is perpendicular to neither a nor b d. c is perpendicular to b

 

 

ANS:  C                    PTS:   1

 

  1. Compute  if ,, and .

 

ANS:

21

 

PTS:   1

 

  1. If , , and , find a value for z which guarantees that a, b, and c are coplanar.

 

ANS:

 

PTS:   1

 

  1. Find the distance from the point P(–102) to the plane passing through the points A(–211), B(052), and C(130).

 

ANS:

 

PTS:   1

 

  1. If we know that , which of the following statements is true?
a. b and c are parallel d. a is parallel to b – c
b. e. a is a unit vector
c. b = c  

 

 

ANS:  D                    PTS:   1

 

  1. If vectors a, b, and c are mutually orthogonal, find .

 

ANS:

 

PTS:   1

 

  1. Explain why there is no vector a such that

 

ANS:

which implies  and  are not orthogonal.

 

PTS:   1

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