Calculus for the Life Sciences Marvin L. Bittinger Test Bank.pdf (19)

Type:

Test Bank

Resource:

Calculus for the Life Sciences

Edition: Author(s):

1st Edition Marvin L. Bittinger Neal Brand John Quintanilla

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CONTENTS

I. CHAPTER TESTS, FORMS A-F and FINAL EXAMINATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

There are six DIFFERENT test forms for each chapter and the final examination. All six test forms are modeled after the Chapter Tests in the text.

Included at the end of each chapter test and the final exams are both synthesis and technology-based exercises.

The synthesis exercises are meant to be more challenging and they generally require students to go beyond the immediate objectives of the chapter. Synthesis exercises are separated from the main body of the test by a solid line. The technology-based exercises follow the synthesis exercises and are separated by a bold line. Both the synthesis and the technology-based exercises have been placed at the end of the tests to make it easy to omit them if the instructor wishes to do so.

Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Final Examinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

II. ANSWER KEYS FOR TESTS, FORMS A-F, AND FINAL EXAMINATION . . . . . . . . . . . . . . . . . . . . . 283

III. ANSWERS TO THE EVEN-NUMBERED EXERCISE SETS IN THE TEST . . . . . . . . . . . .. . . . . . . . . . 369

If the instructor wants students to have all of the answers for the exercises in the exercise sets of the test, these answers can be easily duplicated.

IV. ANSWERS TO THE EXTENDED LIFE CONNECTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

These answers are provided for the Extended Life Science Connections that occur at the end of each chapter in the text. The Extended Life Science Connections are designed to consider interesting applications of calculus in the life sciences. These problems make use of curve fitting, regression, modeling and problem-solving using technology; and allow for possible group or collaborative learning.

Special thanks to Joanne Ha for technical and production support. Special thanks to Joe Vetere for producing numerous graphs required for this manual and test bank.

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

1.Heart Attacks and CholesterolThe . following graph relates the annual heart attach rate per 10,000 men and their blood cholesterol level. (Copyright 1989, CSPI, Adapted from Nutrition Action Healthletter)

Chapter 1, Form A

Use the graph to answer the following:

(a) What is the annual heart attack rate per 10,000 men for those whose blood cholesterol level is 275 mg/dl?

1. (a) ____________________

(b) What is the blood cholesterol level for men with a heart attack rate of 100 attacks per 10,000 individuals?

1. (b) ____________________

2.A function is given byf ( x) = x3 − 4. Find

(a) f (−2).

2. (a) ____________________

(b)fxh ( +).

2. (b) ____________________

1 3. What are the slope and the y-intercept y =of− x − 5?

2 2 4. Find an equation of the line with slope , containing the point 3 (3, −6 ).

3. ______________________

4.______________________

the(−points 6,−8 ) . (3, 5. Find the slope of the line containing 5.______________________ −2 ) and

2

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form A

Find the average rate of change.

6.

7.

6. ___________________________

7.___________________________ 8. Ohm’s Law. The electrical current I, in amperes, in a circuit is directly proportional to the voltage V. When 45 volts are applied, the current is 15 amperes. Find an equation of variation expressing I as a function of V.

8.___________________________

9. For the following graph of function, f, determine (a)f(1); (b) the domain; (c) all x-values such that f(x)=3; and (d) the range.

9.(a) _________________________

9.(b) _________________________

9.(c) _________________________

9.(d) _________________________

10. Solve for x : x2 + 4 x − 8 =0.

10. ___________________________

1 11. Convert to rational exponents: .

5 x4

11 . ___________________________

12. Convert to radical notationy:− 23 .

12. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form A

2 13. Graph: f ( x) = x

3

13.

5

y

4 3 2 1 −5

−4

−3

−2

−1

x 1

−1 −2

2

3

4

5

−3 −4 −5

14. Graph: f ( x) =

x 2− 16

14.

x− 4

10

y

8 6 4 2 −10

−8

−6

−4

−2

−2

x 2

4

6

8

10

−4 −6 −8 −10

Determine the following exactly.

15. cos(−5π6)

15. ______________________

16. sin(5π6)

16. ______________________

17. tan(π2)

17. ______________________ 18. Solve for the missing side, x.

x 50

4.27

18. ______________________

4

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form A

Find all values of t that satisfy the equation.

19. 4sin2t=3

19. ___________________________

20. 2sin2t−sint−1=0

20. ___________________________

For each of the following functions, find the amplitude, period, and midline.

Also, find the maximum and minimum

21. y = 3sin (2t ) − 4

21. ___________________________

22. y = cos(2t ) − 2

22. ___________________________

23. Determine if the graph should be modeled by either y = a sinbt + k or y=acosbt+k, Then find a, b, and k. y 1 −π

−3π/ 4 −π/2

−π/4

−

t π/4

π/2

3π/4

π

1 − 2 − 3 − 4

23. ____________________________

− 5 − 6 −

7 ⎧ x 2 1, for x ≥ 1 24. Graph: f ( x) = ⎪ + − ⎨ x − 2, for x < −1 8 ⎪ ⎩

24.

5

y

4 3 2 1 −5

−4

−3

−2

−1

−1 −2 −3 −4 −5

x 1

2

3

4

5

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form A

25.

5

Milk Consumption. The number of gallons of lowfat milk consumed by the average American in a year for several years is recorded in the following table. (Copyright 1990, CSPI. Adapted from Nutrition ) .

Action Healthletter Year, x

Number of Gallons, g

1970 1975 1980 1985 1990 1995

4.5 7 9 11 12.5 12

(a) Using the data points (1975, 7) and (1995, 12) find a linear function that fits the data.

25. (a) _____________________

(b) Use the linear function to predict the number of gallons of lowfat milk the average American drank in 1988.

25. (b) _____________________ ___________________________________________________________ 26. Solve for x: 2+

1 x

= −2x.

26. ______________________

27. Use your grapher to graph x) = x3 − 4:xf2 (+ 2 x − 3.

Then sketch the graph below.

27.

28. Find the zeros of the function: f(x)=3x2−9−3.

28. ______________________

29.Milk Consumption. Use the data in Question 25. (a) Use the REGRESSION feature to fit a linear function to the data.

29. (a) ____________________

(b) Use the linear function to predict the lowfat milk consumption of the average American 1988.

29. (b) ____________________

CALCULUS FOR THE LIFE SCIENCES

Name: ______________________

Chapter 1, Form B 1. Medicine. The following graph relates the number of milligrams of ibuprofen in the bloodstream to the number of hours that have elapsed since 400 mg of the medication was swallowed.

Use the graph to answer the following:

(a) How much ibuprofen is in the bloodstream three hours after swallowing 400 mg?

1. (a) _________________________

(b) For what lengths of time is there approximately 140 mg of ibuprofen in the bloodstream?

1. (b) _________________________

2.A function is given byf ( x) = 2 x3 + 4. Find

(a) f (−2).

2. (a) _________________________

(b)fxa ( +).

2. (b) _________________________

3. What are the slope and the y-intercept y =−4 of x + 5 ?

3. ___________________________

2 4. Find an equation of the line with slope , containing the point 5 2 (5, −).

4.___________________________

(8,5) the points (4, 5. Find the slope of the line containing 5.___________________________ . −2 ) and

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form B

7

Find the average rate of change.

6.

7.

6. _____________________

7.______________________ 8. Hooke’s Law. Hooke’s Law for Springs, states that the distance d, a spring is stretched or compressed is directly proportional to the force F, in newtons. When 25 newtons are applied, the spring stretched 0.5 meter. Find an equation of variation expressing F as a function of d.

8.______________________

9. For the following graph of function, f, determine (a)f(−1); (the b) domain; (c) all x-values such that f(x)=0; and (the d) range.

9.(a) _____________________

9.(b) _____________________

9.(c) _____________________

9.(d) _____________________

10. Solve for x : x2 − 2 x − 4 = 0.

10. ______________________

1 11. Convert to rational exponents: .

3 y2

11 . ______________________

4 12. Convert to radical notation:3 p− .

12. ______________________

8

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form B

11.

Graph: f(x)=3

11.

x

5

y

4 3 2 1 −5

−4

−3

−2

−1

x 1

−1 −2

2

3

4

5

−3 −4 −5

14.Graph:f ( x) =

x2− x − 6

14.

x+ 2

10

y

8 6 4 2 −10

−8

−6

−4

−2

−2

x 2

4

6

8

10

−4 −6 −8 −10

Determine the following exactly.

15. sin(3π4) 16.

15. __________________________

cos(5π6)

17.

16. __________________________

tan(−π2) 17. __________________________ 18. Solve for the missing x.

side,

40

5.21 x

18. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form B

9

Find all values of t that satisfy the equation.

19. 4cos2t=3

19. ______________________

20. 2cos2t+3cost+1=0

20. ______________________

For each of the following functions, find the amplitude, period, and midline.

Also, find the maximum and minimum

21. y=

1 cos (3 2 πt)+3

21. ______________________

t 22. y = 2 sin⎛⎜ ⎞⎟ − 2 ⎝3⎠

22. ______________________

23. Determine if the graph should be modeled by either y = a sinbt + k or y=acosbt+k, Then find a, b, and k. y 3 2 1

t −4

−3

−2

−1

1

2

3

4

−1 −2

⎧ x 2 4, for x ≥ 0 24. Graph: f ( x) = ⎪ − ⎨ x + 1, for x < 0 ⎪ ⎩

23. ______________________

24.

5

y

4 3 2 1 −5

−4

−3

−2

−1

−1 −2 −3 −4 −5

x 1

2

3

4

5

10

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form B

25. Computer Prices. The following table lists the average price, in dollars, of a personal computer at Jaytech in several recent years.

Year, x

Average price, p

1993

1630

1996

1540

1999

1270

2002

990

2005

640

(a) Using the data points (1993, 1630) and (1999, 1270) find a linear function that fits the data.

25. (a) __________________________

(b) Use the linear function to estimate the average price of a personal computer in 2001.

25. (b) __________________________

________________________________________________________________

26. Solve for x: 9x+

8 x

=12.

26. ___________________________

27. Use your grapher to graph x) = 5 x3 −:6fx(2 + x − 2.

Then sketch the graph below.

27.

)= x2−: 4−f5(−x6.

28. Find the zeros of the function

28. ___________________________

29. Computer Prices. Use the data in Question 25. (a) Use the REGRESSION feature to fit a linear function to the data.

29. (a) _________________________

(b) Use the linear function to estimate the average price of a personal computer in 1997.

29. (b) ________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 1, Form C 1.Memory. The following graph relates the average number of words participants in a psychology experiment were able to memorize in a given amount of time.

Use the graph to answer the following:

(a) Estimate the number of words memorized after 8 min.

1. (a) ____________________

(b) After how many minutes is the number of words memorized approximately ten?

1. (b) ____________________

2.A function is given byf ( x) = 3 x2 − 5. Find

).(a) f (−7

2. (a) ____________________

( + ) (b)fx1.

2. (b) ____________________

2 3. What are the slope and the y-intercept of 3 x+8? y=−

5 4.Find an equation of the line with slope−, containing the point 8

(4,0).

3. ______________________

4.______________________

(2,9) and (−5,−6) 5. Find the slope of the line containing the points 5. . ______________________

12

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form C

Find the average rate of change.

6.

7.

6. ___________________________

7.___________________________ 8. Weight of the Moon. The weight M of an object on the moon is directly proportional to its weight E on earth. A person who weighs 192 lb on Earth weighs 32 lb on the moon. Find an equation of variation expressing E as a function of M.

8.___________________________

9. For the following graph of function, f, determine (a)f(−2); (b) domain; (c) all x-values such that f(x)=1; and the (d) range. the

9.(a) _________________________

9.(b) _________________________

9.(c) _________________________

9.(d) _________________________

10. Solve for x : x2 + 8 x +13 = 0.

10. ___________________________

5 11. Convert to rational exponents: .

3 x

11 . ___________________________

12. Convert to radical notation a−:67 .

12. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form C 13

13.

Graph: f ( x) =

6

( x−3)

13.

5

y

4 3 2 1 −5

−4

−3

−2

−1

x

−1 −2

1

2

3

4

5

2

4

6

8

10

−3 −4 −5

14.

Graph:f ( x) =

x2 − 3 x −10 x +2

14.

10

y

8 6 4 2 −10

−8

−6

−4

−2

−2

x

−4 −6 −8 −10

Determine the following exactly.

15. cos(7π6)

15. ______________________

16. sin(π2)

16. ______________________

17. tan(3π2)

17. ______________________ 18. Solve for the missing side, x.

7.93 35

x

18. ______________________

14

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form C

Find all values of t that satisfy the equation.

19. 2sin2t=1

19. ___________________________

20. cos2t+cost=0

20. ___________________________

For each of the following functions, find the amplitude, period, and midline.

Also, find the maximum and minimum. t 21. y = 2 cos⎛⎜ ⎞⎟ − 5 ⎝2⎠

21. ___________________________

1 sin ( 3t 3) +2 2

22. ___________________________

22. y =

23. Determine if the graph should be modeled by either y = a sinbt + k or y=acosbt+k, Then find a, b, and k. y

1 −3

−2

−1

t 1

2

3

− 1 − 2 − 3

23. ___________________________

− 4 − 5 −

2 ⎧− x − 3, forx ≥ −2 24. Graph:f ( x) =⎪⎨6 ⎪⎩x−1, for x < −2

24.

5

y

4 3 2 1 −5

−4

−3

−2

−1

−1 −2 −3 −4 −5

x 1

2

3

4

5

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form C 15

25. Total Revenue. The following table lists the revenue in thousands of dollars, of a small bike shop for various years.

Year, x

Total Revenue, R (in thousands)

1997

85

1998

94

1999

97

2001

100

2003

105

(a) Using the data points (1997, 85) and (1999, 97) find a linear function that fits the data.

25. (a) _____________________

(b) Use the linear function to predict the revenue of the bike shop in 2005.

25. (b) _____________________

___________________________________________________________

26. Solve for x: 4x+

11 x

=12.

26. ______________________

27. Use your grapher to graph x) = 2 x3 −:5fx(2 + x − 4.

Then sketch the graph below.

27.

28. Find the zeros of the function f :( x) =

x2 − 1 − 3 − 5.

28. ______________________

29. Total Revenue. Use the data in Question 25. (a) Use the REGRESSION feature to fit a linear function to the data.

29. (a) ____________________

(b) Use the linear function to predict the revenue of the bike shop in 2007.

29. (b) ____________________

CALCULUS FOR THE LIFE SCIENCES

Name: ______________________

Chapter 1, Form D 1.Height of a Projectile.The following graph relates the height H, in feet, of a projectile with an initial velocity of 64 ft/sec to the number of secondst after a launch.

Use the graph to answer the following:

(a) What is the height of the projectile 3 sec after launch?

1. (a) _________________________

(b) How many seconds after launch is the projectile 28 ft above ground?

1. (b) _________________________

2.A function is given byf ( x) = 2 x2 + 3. Find

(a) f (−1).

2. (a) _________________________

(b)fa( 3). −

2. (b) _________________________

3. What are the slope and the y-intercept y =of5 x − 4?

3. ___________________________

5 4. Find an equation of the line with slope , containing the point 8

(6, − 2).

5. Find the slope of the line containing the points (− 3,9)and (5, −3 ) .

4.___________________________

5.___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form D 17

Find the average rate of change.

6.

7.

6. _____________________

7.______________________ 8. Charles’s Law. Charles’s Law states that the volume V, occupied by a gas (at a constant pressure) is directly proportional to its absolute temperature T. An experiment with balloon shows that the volume of the balloon is 0.80 liter at 270 K. Find an equation of variation expressing T as a function of V.

8.______________________

9. For the following graph of function, f, determine (a)f(1); (b) the domain; (c) all x-values such that f(x)=2; and (d) the range.

9.(a) _____________________

9.(b) _____________________

9.(c) _____________________

9.(d) _____________________

10. Solve for x : x2 + 8 x +13 = 0.

10. ______________________

5 11. Convert to rational exponents: .

3 n

11 . ______________________

18

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form D

12. Convert to radical notationy:− 47 .

13.

Graph: f(x=) −3

12. ___________________________

13.

x− 4

5

y

4 3 2 1 −5

−4

−3

−2

−1

x 1

−1 −2

2

3

4

5

8

10

−3 −4 −5

14.Graph:f ( x) =

x2− 9

14.

x+ 3

10

y

8 6 4 2 −10

−8

−6

−4

−2

−2

x 2

4

6

−4 −6 −8 −10

Determine the following exactly.

15. sin(−5π4)

15. __________________________

16. cos(−2π3)

16. __________________________

17. tan(3π2) 17. __________________________ 18. Solve for the missing side, x.

42

x

6.24

18. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form D 19

Find all values of t that satisfy the equation.

19. 3tan2t=1

19. ______________________

20. 2sin2t+sint−1=0

20. ______________________

For each of the following functions, find the amplitude, period, and midline.

Also, find the maximum and minimum

21. y = 4sinπt ( )+1

21. ______________________

22. y = 3cos(3t ) − 1

22. ______________________

23. Determine if the graph should be modeled by either y = a sinbt + k or y=acosbt+k, Then find a, b, and k.

3

y

2

1

t −4

−3

−2

−1

1

2

3

4

−1

⎧ x 2 4, for x ≥ 1 24. Graph:f ( x) = ⎪ − ⎨ x + 3, for x < 1 ⎪ ⎩

23. ______________________

24.

5

y

4 3 2 1 −5

−4

−3

−2

−1

−1 −2 −3 −4 −5

x 1

2

3

4

5

20

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form D

25.

Average Income. The following table shows the average yearly income, in dollars, of individuals based on years of schooling.

Years of Schooling, x

Average Income, I

8

18,000

10

20,000

12

26,000

14

30,000

(a) Using the data points (8, 18,000) and (14, 30,000) find a linear function that fits the data.

25. (a) __________________________

(b) Use the linear function to predict average income for an individual with 16 years of schooling.

25. (b) __________________________

________________________________________________________________

26. Solve for x: x + 1 = −

1

x

26. ___________________________

.

27. Use your grapher to graph:( x f ) = x3 − 2 x3 − 2 x + 3.

Then sketch the graph below.

27.

28. Find the zeros of the function f :( x) = 3 x2 + 1 − 4.

28. ___________________________

29. Average Income. Use the data in Question 25. (a) Use the REGRESSION feature to fit a linear function to the data.

29. (a) _________________________

(b) Use the linear function to estimate the average income of an individual with 13 years of schooling.

29. (b) ________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 1, Form E 1.Medical Concentration.The concentration C, in parts per million, of a certain antibiotic in the bloodstream after t hours is illustrated in the following graph.

Use the graph to answer the following:

(a) Find the approximate concentration after 10 hr.

1. (a) ____________________

(b) For lengths of time is the concentration approximately 8 parts per million?

1. (b) ____________________

2.A function is given byf ( x) = 4 x2 + x . Find

(a) f (−4).

2. (a) ____________________

(b)fx( 2). −

2. (b) ____________________

3. What are the slope and the y-intercept y =of−2x−6?

1 4.Find an equation of the line with slope −, containing the point 4

(2, − 8).

3. ______________________

4.______________________

(−______________________ 1,1) and (2,−8) 5. Find the slope of the line containing the points 5. .

22

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form E

Find the average rate of change.

6.

7.

6. __________________________

7.___________________________

8. Pressure and Depth. The pressure P, in lbft2, a liquid exerts at a given point on a submarine is directly proportional to the depth d, in feet, of the point below the surface of the liquid. When a given point is 3 ft below 8.___________________________ the surface of the liquid, the pressure is 187.5 lbft2.

Find Forequation the following graph of function , f, determine (a)f(3); of variation expressing P as a function 9. an of d . (b) the domain; (c) all x-values such that f(x)=2; and (d) the range.

9.(a) _________________________

9.(b) _________________________

9.(c) _________________________

9.(d) _________________________

10. Solve for x : x2 + 4 x − 3 =0.

10. ___________________________

2 11. Convert to rational exponents: .

3 p2

11 . ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form E 23

12. Convert to radical notationy:−12 .

13. Graph: f(x)=

4 x−4

12. ______________________

13.

5

y

4 3 2 1 −5

−4

−3

−2

−1

x 1

−1 −2

2

3

4

5

−3 −4 −5

14. Graph: f ( x) =

x2+ 5x + 6 x+ 2

14.

10

y

8 6 4 2 −10

−8

−6

−4

−2

−2

x 2

4

6

8

10

−4 −6 −8 −10

Determine the following exactly.

15. cos(−4π3)

15. ______________________

16. sin(π6)

16. ______________________

17. tan(−π)

17. ______________________

18. Solve for the missing side, x.

57

3.21

x

18. ______________________

24

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form E

Find all values of t that satisfy the equation.

19. 4sin2t=1

19. ___________________________

20. 2cos2t+3cost+1=0

20. ___________________________

For each of the following functions, find the amplitude, period, and midline.

Also, find the maximum and minimum t 21. y = 3cos⎛⎜ 3 ⎞⎟ − 2 ⎝ ⎠

21. ___________________________

22. y = 2 cos πt() +3

22. ___________________________

y = a sinbt + k 23. Determine if the graph should be modeled by either or y=acosbt+k, Then find a, b, and k.

6

y

5 4 3 2 1

t −4π

−3π

−2π

−π

π

2π

3π

4π

−

23. ___________________________

1 − 2

3, forx ≥ 0 ⎧ 24. Graph:f ( x) =⎪x⎨ − ⎪⎩−2 x, for x < 0 2

24.

5

y

4 3 2 1 −5

−4

−3

−2

−1

−1 −2 −3 −4 −5

x 1

2

3

4

5

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form E 25

25. Test Grades. The following table shows the study time for a particular chapter in a calculus class and the corresponding test grade for that chapter.

Study Time (in min), t 40

Test Grade (in percent), g 77

60

83

120

85

200

91

300

95

(a) Using the data points (60, 83) and (300, 95) to find a linear function that fits the data.

25. (a) _____________________

(b) Use the linear function to predict the test grade for a student who studies 240 min.

25. (b) _____________________ ___________________________________________________________ 26. Solve for x: 6x+

2 x

26. ______________________

= −3.

27. Use your grapher to graph x) = 2 x3 −:3fx(2 + x + 2.

Then sketch the graph below.

27.

28. Find the zeros of the function f :( x) = 2 2 − x2 − 2.

28. ______________________

29. Test Grades. Use the data in Question 25. (a) Use the REGRESSION feature to fit a linear function to the data.

29. (a) ____________________

(b) Use the linear function to estimate the test grade of a student who studies 30 minutes.

29. (b) ____________________

CALCULUS FOR THE LIFE SCIENCES

Name: ______________________

Chapter 1, Form F 1.Memory. The following graph relates the average number of words participants in a psychology experiment were able to memorize in a given amount of time.

Use the graph to answer the following:

(a) Estimate the number of words memorized after 10 min.

1. (a) _________________________

(b) After how many minutes is the number of words memorized approximately six?

1. (b) _________________________

2.A function is given byf ( x) = 3 x2 − x . Find

(a) f (−4).

2. (a) _________________________

(b)fxa ( +).

2. (b) _________________________

3. What are the slope and the y-intercept y =−2 of x + 3 ?

3. ___________________________

2 4.Find an equation of the line with slope − , containing the point 3

(5, − 3).

4.___________________________

5.Find the slope of the line containing the points . 5.___________________________ (− 4,−2 ) and (3,10)

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form F 27

Find the average rate of change.

6.

7.

6. _____________________

7.______________________ 8. Muscle Weight and Body Weight. The weight M, of muscles in a human is directly proportional to the body weight W. A person who weighs 155 lb has 68 lb of muscles. Find an equation of variation expressing W as a function of M.

8.______________________

9. For the following graph of function, f, determine (a)f(−2); (the b) domain; (c) all x-values such that f(x)=4; and (the d) range.

9.(a) _____________________

9.(b) _____________________

9.(c) _____________________

9.(d) _____________________

10. Solve for x : x2 − 6 x + 4 = 0.

10. ______________________

5 11. Convert to rational exponents: . x

11 . ______________________

28

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form F

12. Convert to radical notationm :− 54 .

13. Graph : f ( x = )−

2

12. ___________________________

13.

x

5

y

4 3 2 1 −5

−4

−3

−2

−1

x 1

−1 −2

2

3

4

5

−3 −4 −5

14. Graph: f ( x) =

x 2 + 2 x −8

14.

x+ 4

10

y

8 6 4 2 −10

−8

−6

−4

−2

−2

x 2

4

6

8

10

−4 −6 −8 −10

Determine the following exactly.

15. sin(2π3)

15. __________________________

16. cos(11π6)

16. __________________________

tan(0)

17. __________________________

17.

18. Solve for the missing side, x.

x

7.93 35

18. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form F 29

Find all values of t that satisfy the equation.

19. 4cos2t=1

19. ______________________

20. 2sin2x+3sinx+1=0

20. ______________________

For each of the following functions, find the amplitude, period, and midline.

Also, find the maximum and minimum 1 t)−3 21. y= 2 sin (3

21. ______________________

t 22. y = sin⎛⎜ ⎞⎟ − 1 ⎝2⎠

22. ______________________

23. Determine if the graph should be modeled by either y = a sinbt + k or y=acosbt+k, Then find a, b, and k.

6

y

5 4 3 2 1 −2π

−3 π/ 2 −π

−π/2

−1

t π/2

π

3π/2

2π

−2 −3

23. ______________________

−4

⎧ x 2 1, for x > 0 24. Graph: f ( x) = ⎪ + ⎨ x − 2, for x ≤ 0 ⎪ ⎩

24.

5

y

4 3 2 1 −5

−4

−3

−2

−1

−1 −2 −3 −4 −5

x 1

2

3

4

5

30

CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form F

25. Insurance Rates. The following table shows the comparison of the cost, in dollars, of a $100,000 life insurance policy for female nonsmokers at certain ages Age, a

Cost, c

31

170

32

172

33

176

34

178

35

182

(a) Using the data points (32, 172) and (35, 182) find a linear function that fits the data.

25. (a) __________________________

(b) Use the linear function to predict life insurance cost for a female nonsmoker of age 38.

25. (b) __________________________ ________________________________________________________________ 26. Solve for x: 25x+

12 =30. x

26. ___________________________

27. Use your grapher to graph x) = − 3x+:6fx(− 3.

Then sketch the graph below.

27.

f :( x) = 3 10 − x2 − 4.

28. Find the zeros of the function

28. ___________________________

29. Insurance Rates. Use the data in Question 25. (a) Use the REGRESSION feature to fit a linear function to the data.

29. (a) _________________________

(b) Use the linear function to estimate the life insurance cost for a female nonsmoker of age 42.

29. (b) ________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 2, Form A f Limits Graphically. Consider the following graph of function for Questions 1 – 6.

Find the limit, if it exists.

1. limf

(x )

1.______________________

2. lim

f(x)

2.______________________

3. limf ( x )

3.______________________

4. limf ( x

4. _____________________

x→−3

x→−2 x→1

x→ 2

5.

limf x→ 4

6.limf

x→5

)

5.______________________

(x

6.______________________

)

Determine whether the function is continuous.

7.

(x

8.

7. ______________________

)

8. ______________________

32

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form A

For the function in Question 7, answer the following.

9. Findlimf(x).

9.___________________________

(1.)

10. ___________________________

11. Is f continuous 1at?

11. ___________________________

12. Findlim f (x) .

12. ___________________________

x→1

10. Findf

x→ 4

).

(4

13. ___________________________

14. Is f continuous at4?

14. ___________________________

13. Findf

Find the limit if it exists.

15. lim(2 − x + 3x4)

15. ___________________________

x+2

16. ___________________________

x→ 2

16.

lim

x→−2+ x2 − 4

8 17. lim x→0x

17. ___________________________

18. Find a simplified difference quotient for

f(x)=4x2−3.

18. __________________________

19. Find an equation of the tangent line to the graph of

y=2xx + )(at 3 the point(1,5).

19. ___________________________

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form A 33 y = x3 − 3 x2 at which 20. Find the points on the graph of the tangent line is horizontal.

20. ______________________

Find dydx .

y =3 x

21. ______________________

8 x3

22. ______________________

23. y = x 4 5

23. ______________________

21.

22. y = −

24. y

=−5

x2 + 0.4x+16

24. ______________________

25.

25. y = cot (3)x

______________________

Differentiate.

26.

x

f (x)= 4 − x

27. y =

26. ______________________

(3x4x−+ 46)

−5

2

27. ______________________

28. f ( x) = x 2x3 + 1

28. ______________________

x 29. f ( x) =cos⎛⎜ ⎞⎟ ⎝2⎠

29. ______________________

30. f ( x) =cot( x) csc (4)x

30. ______________________

31. f ( x) =

32. f ( x) =

sin2 ( x)

tan ( x) −x 2

cos (x )() tan 3

31. ______________________

32. ______________________

34

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form A

2 33. f ( x) = x +4+

cos x (2 − x )

33. ___________________________

d3y . dx3

34. ___________________________

34. For y = 4x 6 − 9 x 3, find

35. Given that an object is at position ( )st= 4 t + 2sin(t)at any timet, find each of the following.

a) vt(. )

35. (a) _________________________

b) a (t ) .

35. (b) _________________________

5π c) The position, velocity and acceleration when = 6t .

35. (c) _________________________

36. Recreation. The number of games N in a season in which each team plays every other team once is given

by N n2−n . a)

=1( ) 2

Find the rate of change of number of games with respect to the number of teams.

36. (a) _________________________

b) How many games are played if there are 15 teams?

36. (b) _________________________

at=15 teams . c) Find the rate of change n

36. (c) _________________________

37. Find ( f g )( x) and ( g f )(x), given that f ( x) = x − 3 x2 and g ( x=)x3 .

37. ___________________________

38. Differentiatey = (5− 4 x)43(5 + 4 x)23.

38. ___________________________

39. Findlim

x3 − 8

.

39. ___________________________

t .

40. Findlim t→0 cos(t )

40. ___________________________

x→ 2

x− 2

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form A 35 41. Using a grapher, graph fand fover the given interval. ′ Then estimate points at which the tangent line to fis horizontal. Sketch the graphs. f(x)=4x3−25x2+32x+ 4 x;[0,5]

41. ______________________

42. Find the following limit using tables on a grapher. Start with Step =0.1 and then go to 0.01,0.001,and0.0001.

When you think you know the limit, graph and use the TRACE feature to further verify your assertion.

lim x→ 0

1 6 − 4 x− 4 x

42 . ______________________

CALCULUS FOR THE LIFE SCIENCES

Name: ______________________

Chapter 2, Form B f Limits Graphically. Consider the following graph of function for Questions 1 – 6.

Find the limit, if it exists.

(x)

1.___________________________

2. lim f ( x )

2.___________________________

1. limf x→−5

x→−3

3. limf ( x )

3.___________________________

4. limf ( x )

4. __________________________

x→ 0

x→3

limf x→5

(x)

5.___________________________

6.limf

(x)

6.___________________________

5.

x→6

Determine whether the function is continuous.

7.

8.

7. ___________________________

8. ___________________________

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form B 37 For the function in Question 7, answer the following.

9. Findlimf(x). x→−2

9.______________________

(− 2).

10. ______________________

11. Is f continuous at −2 ?

11. ______________________

12. Findlim f (x) .

12. ______________________

10. Findf

x→ 2

).

(2

13. ______________________

14. Is f continuous at2?

14. ______________________

13. Findf

Find the limit if it exists.

15. lim(x4−2x2+3) x→3

16.

lim

7+x

x→−7 − 49− x2

15. ______________________

16. ______________________

15 17. lim x→0x

17. ______________________

18. Find a simplified difference quotient for

18. ______________________

f(x)=3x2−6.

19. Find an equation of the tangent line to the graph of

y = 3 xx + )(at 8 the point(2,10).

19. ______________________

38

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form B

y = x3 − 2 x2 at which 20. Find the points on the graph of the tangent line is horizontal.

20. ___________________________

Find dydx .

y = 45 x

21. ___________________________

11 22. y = − x 2

22. ___________________________

23. y = x 3 5

23. ___________________________

21.

24. y

=−

0.6x2 + 4.1x− 100

25. y = x 2 sinx()

24. ___________________________

25. ___________________________

Differentiate.

26.

f (x)=

27. y =

2x x +2

26. ___________________________

(6 x 2x++2x ) 5

4 6 −

28. f ( x) = x 2x3 − 2

(

)=sin( x)

27. ___________________________

28. ___________________________

29. f x

29. ___________________________

cscx() 30. f ( x) =tan (2) x

30. ___________________________

31. f ( x) =

32. f ( x) =

cos2 ( x)

sec ( x) −x 2

sin(sec x )) (

31. ___________________________

32. ___________________________

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form B 39

cos x (2 x + 1 )

33. ______________________

d3y . dx3

34. ______________________

33. f ( x) = 2 x+5−

34. For y = 5x4 − x 2 +6, find

35. Given that an object is at position ( )st= 2 t − 5cos(t)at any timet, find each of the following.

a) vt(. )

35. (a) _____________________

b) a (t ) .

35. (b) _____________________

π c) The position, velocity and acceleration when = t.

3

35. (c) _____________________

36. Surface Area. The surface area Sof an 8-in.high can is given by S= 2πr2+16πr, where ris the length of the radius..

a) Find the rate of change of the surface area with respect to the length of the radius.

36. (a) _____________________

b) What is the surface area of a can that is 8-in. high and has a radius of 2in.?

36. (b) _____________________

c) Find the rate of change rat= 2in.

36. (c) _____________________

37. Find ( f g )(x)and ( g f)(x), given that f ( x) =x2+ 4 x and g ( x ) = x .

37. ______________________

38. Differentiate y = 2( − 3 x)23(5 + x)12.

38. ______________________

27 3 39. Findlim 3 − x . x→3 −x

39. ______________________

cot ( ) +cos(t) . t t→0 t

40. Findlim

40. ______________________

40

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form B

41. Using a grapher, graph fand fover the given interval. ′

Then estimate points at which the tangent line to fis horizontal. Sketch the graphs.

f(x)=6x3−20x2+10+x3 x;[0,5]

41. ___________________________

Insert page 44 rectangle

42. Find the following limit using tables on a grapher. Start with Step =0.1 and then go to 0.01,0.001,and0.0001.

When you think you know the limit, graph and use the TRACE feature to further verify your assertion.

lim x→ 0

x2 + 1 −1 x

42 . ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 2, Form C f Limits Graphically. Consider the following graph of function for Questions 1 – 6.

Find the limit, if it exists.

1. limf

(x)

1.______________________

2. lim

f(x)

2.______________________

x→−5

x→−3

3. limf ( x )

3.______________________

4. lim f ( x )

4. _____________________

x→ 0

x→−2

5. limf ( x

5.______________________

x→5

6.limf

x→6

)

6.______________________

(x

Determine whether the function is continuous.

7.

)

8.

7. ______________________

8. ______________________

42

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form C

For the function in Question 7, answer the following.

9. Findlimf(x). x→ 2

9. ___________________________

(2 ) .

10. ___________________________

2?

11. Is f continuous at

11. ___________________________

10. Findf

12. Findlimf

x→3

( x) .

12. ___________________________

(3).

13. ___________________________

14. Is f continuous at3?

14. ___________________________

13. Findf

Find the limit if it exists.

15. lim(3−x2+ 10x) x1+ →−

x2 + 3 x −18 x→−6 x +6

15. ___________________________

lim

16. ___________________________

3 17. lim− x→0 x

17. ___________________________

16.

18. Find a simplified difference quotient for

f(x)=2x2+5x.

18. __________________________

19. Find an equation of the tangent line to the graph of

y =4 x +( 10the point(5,18). )−at x

19. ___________________________

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form C 43 y = 2 x3 − 3 x2 at which 20. Find the points on the graph of the tangent line is horizontal.

20. ______________________

Find dydx .

y =6 x

21. ______________________

7 22. y = − x7

22. ______________________

23. y = x 2 5

23. ______________________

21.

24. y

=−3

25. y =

x2 + 10.1x − 6

x cos x )( 2

24. ______________________

25. ______________________

Differentiate.

26.

f (x)=

3x 1− x

26. ______________________

(

)

27. ______________________

28. f ( x) = x 3x4 − 3

28. ______________________

27. y = 6 x 2 − 10 x +1

−4

( 29. f x) = cos(3x )

(

)

)

30. f x = sin(4x cos (3) x

31. f ( x) =

cos(x) sin2(x)−x

( (2))x tan 32. f ( x) = cos

29. ______________________

30. ______________________

31. ______________________

32. ______________________

44

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form C

2 33. f ( x) = x −3+

sinx (x − 3 )

34. For y = 3x 6 − 4 x 3, find

d3y dx3

.

33. ___________________________

34. ___________________________

35. Given that an object is at position ( st ) = sin(3) t− 2tat any timet, find each of the following.

a) vt(. )

35. (a) _________________________

b) a (t ) .

35. (b) _________________________

5π c) The position, velocity and acceleration when = 3 t.

35. (c) _________________________

36. Electing Officers.For a club consisting of npeople, the number of ways in which a president, vice-president, and treasurer can be elected is given by P=n3−3n2+2n.

a) Find the rate of change in the number of ways to elect the 3 officers with respect to the number of club members.

36. (a) _________________________

b) How many ways are there to elect the 3 officers when there are 20 club members?

36. (b) _________________________

at= 20. c) Find the rate of change n

36. (c) _________________________

37. Find ( f g )(x) and ( g f )(x), given that x 2 and g ( x)= x 2 + 5 . f ( x) = 3 −

37. ___________________________

x (10+ x)16.

38. Differentiatey = (6 − 3 )13

38. ___________________________

x3 + 8 . x→−2 x + 2

39. ___________________________

1 − cos(t) . t→0 t

40. ___________________________

39. Findlim

40. Findlim

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form C 45

41. Using a grapher, graph fand fover the given interval. ′

Then estimate points at which the tangent line to fis horizontal. Sketch the graphs.

f(x)=3x5−15x2+5x−;[ 3,3]

41. ______________________

42. Find the following limit using tables on a grapher. Start with Step =0.1 and then go to 0.01,0.001,and0.0001.

When you think you know the limit, graph and use the TRACE feature to further verify your assertion.

lim x→6

2x + 4 − 4 x −6

42 . ______________________

CALCULUS FOR THE LIFE SCIENCES

Name: ______________________

Chapter 2, Form D f Limits Graphically. Consider the following graph of function for Questions 1 – 6.

Find the limit, if it exists.

(x)

1.___________________________

2. lim f ( x )

2.___________________________

1. limf x→−5

x→−2

3. limf ( x )

3.___________________________

4. limf ( x )

4. __________________________

x→ 0

x→3

limf

(x)

5.___________________________

6. lim f ( x )

6.___________________________

5.

x→ 4

x→−4

Determine whether the function is continuous.

7.

8.

7. ___________________________

8. ___________________________

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form D 47 For the function in Question 7, answer the following.

9. Findlimf(x). x→−4

10. Findf

(− 4).

9.______________________

10. ______________________

4?

11. Is f continuous at

11. ______________________

12. Findlim f (x) .

12. ______________________

x→ 0

).

(0

13. ______________________

14. Is f continuous at0?

14. ______________________

13. Findf

Find the limit if it exists.

15. lim(2x3−2x2+5x) x→− 1

16.

17.

lim

9+ x

x→−9+ 81− x 2

6 lim + 6 x→−6 x

18. Find a simplified difference quotient for

f(x)=−2x2+5.

15. ______________________

16. ______________________

17. ______________________

18. ______________________

19. Find an equation of the tangent line to the graph of

y = − xx+) at ( 4 the point (2,0).

19. ______________________

48

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form D

y = 6 x3 + 9 x2 at which 20. Find the points on the graph of the tangent line is horizontal.

20. ___________________________

Find dydx .

21.

y = 133 x

21. ___________________________

4 22. y = x8

22. ___________________________

23. y = x 3 8

23. ___________________________

24. y

=−6

x2 − 5 x + 4.9

24. ___________________________

25.

25. y = sec (2) x

___________________________

Differentiate.

26.

f (x)=

27. y =

5x 3−x

26. ___________________________

(4 x 3x−+25) 2

−4

27. ___________________________

28. f ( x) = x 3x2 − 1

28. ___________________________

x 29. f ( x) =sin⎛⎜ 4 ⎞⎟ ⎝ ⎠

29. ___________________________

tan (4) x 30. f ( x) =cos (2) x

30. ___________________________

31. f ( x) =

32. f ( x) =

cot2 ( x)

sin ( x) −x 2

cos(sin x )) (

31. ___________________________

32. ___________________________

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form D 49

sin( )

33. ______________________

d3y . dx3

34. ______________________

x 4−x 33. f ( x) =1− x2 − 2

34. For y = 2 x 6 − 10 x , find

35. Given that an object is at position ( )st=

cos ()2t

any timet, find each of the following.

2

at

a) vt(. )

35. (a) _____________________

b) a (t ) .

35. (b) _____________________

7π c) The position, velocity and acceleration when = 6t .

35. (c) _____________________

36. Volume of an ice cream coneThe . volume of an ice cream cone with a height of 4in. is given by

V=4πr3.

3

a) Find the rate of change of volume with respect to to the length of the radius.

36. (a) _____________________

b) What is the volume when the radius is 0.5 in.?

36. (b) _____________________

r in. c) Find the rate of change when = 0.5

36. (c) _____________________

37. Find ( f g )(x) andg( f )(x), given that f ( x) =x2+ 2 x2 and g(x) = x.

37. ______________________

38. Differentiatey = (4− 3 x)65(1 + x)25.

38. ______________________

39. Findlim

x3 − 125

.

39. ______________________

cos(t)− t . t→0 t

40. ______________________

x→5

40. Findlim

x−5

50

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form D

41. Using a grapher, graph fand f′over the given interval.

Then estimate points at which the tangent line to fis horizontal. Sketch the graphs.

f(x)=2x5−5x2− x + 2;[−3,3]

41. ___________________________

42. Find the following limit using tables on a grapher. Start with Step =0.1 and then go to 0.01,0.001,and0.0001.

When you think you know the limit, graph and use the TRACE feature to further verify your assertion.

lim

x→12

x−3 −3 x− 12

42 . ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 2, Form E f Limits Graphically. Consider the following graph of function for Questions 1 – 6.

Find the limit, if it exists.

(x )

1.______________________

2. limf ( x

2.______________________

3. limf )

3.______________________

4. limf ( x

4. _____________________

1. limf x→−3

x→ 4

x→ 2

x→ 6

lim

f)(

5.______________________

6. lim x( x )

6.______________________

5.

x→−2

x→−5

f)(

Determine whether the function is continuous.

7.

x)

8.

7. ______________________

8. ______________________

52

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form E

For the function in Question 7, answer the following.

9. Findlimf(x). x→−3

9.___________________________

(− 3).

10. ___________________________

− 3?

11. Is f continuous at

11. ___________________________

10. Findf

12. Findlim

x→−5

f (x) .

12. ___________________________

(−5) .

13. ___________________________

− 5?

14. Is f continuous at

14. ___________________________

13. Findf

Find the limit if it exists.

15. lim(−x3−4 3x2+) x→− 3

15. ___________________________

x −6 16. lim 2 x− x→6− x− 2 24

16. ___________________________

4 17. lim + 3 x→−3 x

17. ___________________________

18. Find a simplified difference quotient for

18. __________________________

f(x)=−x2+5x.

19. Find an equation of the tangent line to the graph of

y=−2x+(6 x)at the point(2,−1).

19. ___________________________

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form E 53 y = x3 − 2 x2 at which 20. Find the points on the graph of the tangent line is horizontal.

20. ______________________

Find dydx .

y = 53 x

21. ______________________

120 22. y = x5

22. ______________________

23. y = x 73

23. ______________________

1 24. y =−5 x2 + x − 16 2

24. ______________________

tan x )(

25. ______________________

21.

25. y =

3

Differentiate.

26.

7x

f (x)= 2 −x

(

26. ______________________

)

27. y = 3 x 3 − 5x2 + 8

−3

28. f ( x) = x 3x4 − 3

27. ______________________

28. ______________________

( 29. f x) = cos(2x )

29. ______________________

(

)

30. f x = cot(2x )secx()

31. f ( x) =

32. f ( x) =

cos2 ( x)

tan ( x) −x 2

sin(xsec )) ( 2

30. ______________________

31. ______________________

32. ______________________

54

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form E

cos x (x + 5 )

33. ___________________________

d3y . dx3

34. ___________________________

33. f ( x) =3 x + 4 +

34. For y = 280x − 3 x5 , find

35. Given that an object is at position ( )st=

sin(3) t

any timet, find each of the following.

3

at

a) vt(. )

35. (a) _________________________

b) a (t ) .

35. (b) _________________________

π c) The position, velocity and acceleration when = t.

9

35. (c) _________________________

36. Daily accidents.The average number of daily accidents involving drivers of age ais approximated

by P =0.4 a2−40 a +1039. a) Find the rate of change of the average number of accidents with respect to age.

36. (a) _________________________

b) How many daily accidents involve a 20-year old driver?

36. (b) _________________________

a = 20. c) Find the rate of change when

36. (c) _________________________

37. Find ( f g )(x) andg ( f )(x), given that f ( x) = x − x2 and g ( x)= 4 x 2 .

37. ___________________________

38. Differentiatey = (8− 2 x)32(4 + x)12.

38. ___________________________

39. Findlim

x3 + 64

x→−4

40. Findlim

t→0

x+ 4

sin(t) . t

.

39. ___________________________

40. ___________________________

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form E 55 41. Using a grapher, graph fand fover the given interval. ′ Then estimate points at which the tangent line to fis horizontal. Sketch the graphs.

f(x)=2x5+4x2−7x;[ −5,5]

41. ______________________

42. Find the following limit using tables on a grapher. Start with Step =0.1 and then go to 0.01,0.001,and0.0001.

When you think you know the limit, graph and use the TRACE feature to further verify your assertion.

lim x→5

6x − 5 − 5 x −5

42 . ______________________

CALCULUS FOR THE LIFE SCIENCES

Name: ______________________

Chapter 2, Form F f Limits Graphically. Consider the following graph of function for Questions 1 – 6.

Find the limit, if it exists.

(x )

1.___________________________

f(x)

2.___________________________

1. limf x→−3

2.

lim

x→−4.5

3. limf ( x

3.___________________________

4. limf )

4. __________________________

5. limf ( x

5.___________________________

x→ 0

x→3

x→5

6.limf

x→ 2

)

6.___________________________

(x

Determine whether the function is continuous.

7.

)

8.

7. ___________________________

(x )

8. ___________________________

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form F 57 For the function in Question 7, answer the following.

9. Findlimf(x). x→−2

(− 2).

10. Findf

9.______________________

10. ______________________

2?

11. Is f continuous at

11. ______________________

12. Findlim f (x) .

12. ______________________

x→ 2

).

(2

13. ______________________

14. Is f continuous at2?

14. ______________________

13. Findf

Find the limit if it exists.

15. lim(−x3+6x−6) x→ 4

16.

x +2 x− x→−2− x − 7 18 lim

2

5 17. lim x→ 2 x− 2

15. ______________________

16. ______________________

17. ______________________

18. Find a simplified difference quotient for

f(x)=−4x2+1.

18. ______________________

19. Find an equation of the tangent line to the graph of

y=−4x+(6 x)at the point(3,10).

19. ______________________

58

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form F

y = 3 x3 −9xat which 20. Find the points on the graph of the tangent line is horizontal.

20. ___________________________

Find dydx .

21.

y =8 x

21. ___________________________

3 22. y = x 4

22. ___________________________

23. y = x 27

23. ___________________________

24. y

=−4

x2 − 3.1x+ 40

24. ___________________________

25.

25. y = csc (4)x

___________________________

Differentiate.

26.

f (x)=

27. y =

4x 8−x

26. ___________________________

16 x + − 3x) (2x3

−4

2

28. f ( x) = x 2 x6 + 3

(

29. f x)

sin (5)x tan (3 x

=

) sec (2) x

30. f ( x

27. ___________________________

28. ___________________________

29. ___________________________

30. ___________________________

) 31. f ( x)==

32. f ( x) =

sin2 ( x)

cot ( x) −x 2

cot(xsin )) ( 2

31. ___________________________

32. ___________________________

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form F 59

sinx (7 − x )

33. ______________________

d3y . dx3

34. ______________________

33. f ( x) = 7− x 2 −

34. For y = 4x 6 − 3 x 2, find

35. Given that an object is at position ( )st=

sin ( )2t

any timet, find each of the following.

c)

4

at

a) vt(. )

35. (a) _____________________

b) a (t ) .

35. (b) _____________________

7π The position, velocity and acceleration when = 12 t .

35. (c) _____________________

36. Spheres in a pyramid.The number of spheres in a triangular pyramid with x layers is given by N=1x3+1x2+1x.

6

2

3

a) Find the rate of change of the number of spheres with respect to the number of layers.

36. (a) _____________________

b) How many spheres are there in a pyramid 10 layers deep?

36. (b) _____________________

x = 10. c) Find the rate of change when

36. (c) _____________________

37. Find ( f g )(x ) and ( g f )(x), given that 2 − f ( x) = 2 x 5 and g(x)= x .

37. ______________________

38. Differentiatey = 5( − 2 x)14( x + 6 )34.

38. ______________________

1 + x3 . x→−1 1 + x

39. Findlim

40. Findlim

t→0

sin ( )− t t t

39. ______________________

.

40. ______________________

60

CALCULUS FOR LIFE SCIENCES, Chapter 2, Form F

41. Using a grapher, graph fand fover the given interval. ′

Then estimate points at which the tangent line to fis horizontal. Sketch the graphs.

f ( x )=2 x 3 − 6 x 2+2+x2 x ;[0,5]

41. ___________________________

42. Find the following limit using tables on a grapher. Start with Step =0.1 and then go to 0.01,0.001,and0.0001.

When you think you know the limit, graph and use the TRACE feature to further verify your assertion.

lim x→ 2

7x + 2 − 4 2− x

42 . ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Find the relative extrema and points of inflection of the function. List your answers in terms of ordered pairs. Then sketch a graph of the function.

1. f ( x) = x2 − 2 x − 8

10

2. f ( x) = 4 x4 − 12 x 2 + 9

y

10

8

8

6

6

4

−8

−6

−4

−2

3. f ( x) = ( x − 2

2

x 2

−2

4

6

8

10

−10

−8

−6

−4

−2

−4

−6

−6

−8

−8

−10

−10

) 1 2 + 3 y

10

8

8

6

6

−6

−4

−2

2

4

6

8

10

−10

−8

−6

−4

−2

−4

−6

−6

−8

−8

−10

−10

3

y

10

8

8

6

6

−4

−2

−2

2. ___________________

__________________ 3. __ x 2

4

6

8

10

4. ___________________

y

__________________ 5. __

4

2 −6

10

6. f ( x) = x 16 − x2

4

−8

8

y

−2

−4

5. f ( x) = ( x + 1 ) + 3

−10

6

2

x

−2

10

4

4

2 −8

2

40 4. f ( x) = x 2+5

4

−10

x

−2

−4

10

__________________ 1. __

y

4

2 −10

Chapter 3, Form A

2

x 2

4

6

8

10

−10

−8

−6

−4

−2

−2

−4

−4

−6

−6

−8

−8

−10

−10

x 2

4

6

8

10

6. ___________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 3, Form A

62

Find the relative extrema of the function on [0,2π]. List your answers in terms of ordered pairs. Then sketch a graph of the function.

7. f ( x)=

1.0

8. f (x)=sin(2 x)+2 x

cosx 4 + 3cosx

y

10

_______________________ 1. __

y

0 8

0.7 5

6

x

0.5 π/2

−00.2

π

2π

3π/2

4

50.2−

2

5 0 0.5

x

− 0.7 5

2. ________________________

π/2

π

3π/2

2π

−

1.0 0

Sketch a graph of the function.

9.

3 x+2

f(x)=

4 10. f ( x) = − 2 − x 1

10

y

10

8

8

6

6

4

4

2 −10

−8

−6

−4

11. f ( x) =

−2

2

x 2

−2

4

6

8

10

−10

−8

−6

−4

−2

−4

−6

−6

−8

−8

−10

−10

x2− 9

12. f ( x) =

x

y

10

8

8

6

6

−6

−4

−2

−2

4

6

8

10

y

4

2 −8

2

x+ 4 5 x−

4

−10

x

−2

−4

10

y

2

x 2

4

6

8

10

−10

−8

−6

−4

−2

−2

−4

−4

−6

−6

−8

−8

−10

−10

x 2

4

6

8

10

CALCULUS FOR THE LIFE SCIENCES, Chapter 3, Form A 63 Find absolute maximum and minimum values of the function, if they exist, over the indicated interval. Where no interval is specified, use the real line. List your answers in terms of ordered pairs.

13. f ( x) = x(x− 3)

14.

13. ______________________

( )

f x = 4 x3 +3 x 2 − 6x − 5; [−2, 1]

14. ______________________

( ) 15. f x = 2 x2 + 5.4x− 6

15. ______________________

16. f ( x) = sin2x + 2 sin x; [0,2 π]

16. ______________________

( )

( )

;(2−, π 2 ) 17. f x = 3 tan2x +2 tanxπ

17. ______________________

; 2, (− π 2) 18. f x = cos2x + tan 2 xπ

18. ______________________

54 19. f ( x) =x2 + x ; (0,∞)

19. ______________________

20. Of all numbers whose difference is 12, find the two that have the minimum product.

20. ______________________

21. Minimize Q = x2 + y2 , where x+ y = 4.

21. ______________________

22. From a thin piece of cardboard 48 in. by 48 in., square corners are cut out so that the sides can be folded up to make a box.

What dimensions will yield a box of maximum volume? What is the maximum volume?

22. ______________________

23. Find the linearization tof( x) =

x x +1

at a =3.

cosx at a =π 2.

24. Find the linearization tof( x) = 1 +

23. ______________________

24. ______________________

64

CALCULUS FOR THE LIFE SCIENCES, Chapter 3, Form A

48.

25. Use linearization to approximate

25. ___________________________

26. Use Newton’s method and x1 =1 to find an approximate

solution of 3x3+2x=7.

26. ___________________________

27. Use Newton’s method and x1 =1 to find an approximate solution of sinx=x−cosx .

27. ___________________________

dydx . Then 28. Differentiate the following implicitly to find find the slope of the curve at the given point.

x2+y3=17;(4,1) 29. A board 10 ft long leans against a vertical wall. If the lower end is being moved away from the wall at a rate of 0.3 ft/sec, how fast is the upper end coming down when the lower end is 8 ft from the wall?

28. ___________________________

29. ___________________________

________________________________________________________________ 30. Find the absolute maximum and minimum values of the function, if they exist, over the indicated interval.

f(x)= 4

x2 +x

3

;[0, )∞

30. ___________________________

31. Use a grapher to estimate the relative extrema of the function.

f(x)=4x3−30x2+40x+3x

31. ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 3, Form B

Find the relative extrema of the function. List your answers in terms of ordered pairs. Then sketch a graph of the function.

1. f ( x) = x2 + 4 x − 5

10

2. f ( x) = 3 x4 − 122 + 4

y

10

8

8

6

6

4

4

2 −10

−8

−6

−4

−2

3. f ( x) = ( x − 2

2

x 2

−2

4

6

8

10

−10

−8

−6

−4

−2

−4

−6

−6

−8

−8

−10

−10

) 4 2 + 3 y

10

8

8

6

6

−6

−4

−2

2

4

6

8

10

−10

−8

−6

−4

−2

−4

−6

−6

−8

−8

−10

−10

3

y

10

8

8

6

6

−4

−2

−2

2. ___________________

__________________ 3. __ x 2

4

6

8

10

4. ___________________

y

__________________ 5. __

4

2 −6

10

6. f ( x) = x 16 − x2

4

−8

8

y

−2

−4

5. f ( x) = ( x + 3 ) − 4

−10

6

2

x

−2

10

4

4

2 −8

2

3 4. f ( x) = 2− x +1

4

−10

x

−2

−4

10

__________________ 1. __

y

2

x 2

4

6

8

10

−10

−8

−6

−4

−2

−2

−4

−4

−6

−6

−8

−8

−10

−10

x 2

4

6

8

10

6. ___________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 3, Form B

66

Find the relative extrema and points of inflection of the function on [0,2π].

List your answers in terms of ordered pairs. Then sketch a graph of the function.

7. f ( x) =

1.0

8. f ( x)=sin2x−2cosx

1 4 + 3cosx

y

2.0

0

0

0.7

1.5

5 π/2

−00.2

π

0 −0.

50.2−

0.5 50

5 0 0.5

0 −1.

− 0.7

00

5

−

−1.

1.0 0

50

Sketch a graph of the function.

9.

00

10

π/2

π

2π

3π/2

2 10. f ( x) = 2− x −4

y

10

8

8

6

6

4

−8

−6

−4

11. f ( x) =

−2

2

x 2

−2

4

6

8

10

−10

−8

−6

−4

−2

−4

−6

−6

−8

−8

−10

−10

x2−1

12. f ( x) =

x

y

10

8

8

6

6

−6

−4

−2

−2

4

6

8

10

y

4

2 −8

2

x −1 4 x+

4

−10

x

−2

−4

10

y

4

2 −10

2. ________________________

−2.

5 x−3

f(x)=

x

1.0

2π

3π/2

_______________________ 1. __

0

x

0.5

y

2

x 2

4

6

8

10

−10

−8

−6

−4

−2

−2

−4

−4

−6

−6

−8

−8

−10

−10

x 2

4

6

8

10

CALCULUS FOR THE LIFE SCIENCES, Chapter 3, Form B 67 Find absolute maximum and minimum values of the function, if they exist, over the indicated interval. Where no interval is specified, use the real line. List your answers in terms of ordered pairs.

13. f ( x) = x(6− x)

14.

( )

f x =2 x 3 − 5 x 2 − 4 x +

13. ______________________

3; [−2, 1]

14. ______________________

( ) 15. f x =−3x2+6.6x+4

15. ______________________

16. f ( x) = cos2 x + 2cos x;[0, π] 2

16. ______________________

( )

( )

(0, x 17. f x = 3 cot 2 x − 2 cot ;π)

17. ______________________

2πx); ( 18. f x = sin2 x − cot0,

18. ______________________

19. f ( x)=x2+2;(0,∞) x

19. ______________________

20. Of all numbers whose difference is 9, find the two that have the minimum product.

20. ______________________

21. Minimize Q = x2 + 2 y2 , where x+ y = 6.

21. ______________________

22. From a thin piece of cardboard 72 in. by 72 in., square corners are cut out so that the sides can be folded up to make a box.

What dimensions will yield a box of maximum volume? What is the maximum volume?

22. ______________________

23. Find the linearization tof( x) =x x − 1 at a =5.

23. ______________________

( 2 24. Find the linearization tof x) = 1( −sinx) at a =π .

24. ______________________

3 26.

25. Use linearization to approximate

25. ______________________

68

CALCULUS FOR THE LIFE SCIENCES, Chapter 3, Form B

1=− 26. Use Newton’s method and x1to find an approximate

26. ___________________________

solution of x−2x3=5.

27. Use Newton’s method and x1 =1to find an approximate solution of cosx=x−sinx .

27. ___________________________

dydx . Then 28. Differentiate the following implicitly to find find the slope of the curve at the given point.

2x2+y3=6;(−1,2) 29. A board 26 ft long leans against a vertical wall. If the lower end is being moved away from the wall at a rate of 0.1 ft/sec, how fast is the upper end coming down when the lower end is 24 ft from the wall?

28. ___________________________

29. ___________________________

________________________________________________________________ 30. Find the absolute maximum and minimum values of the function, if they exist, over the indicated interval.

2 x2 ;[0,∞

f(x)= 8

+x

3

)

30. ___________________________

31. Use a grapher to estimate the relative extrema of the function.

f ( x )=4 x 3 − 28 x 2+40 xx+2

31. ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 3, Form C

Find the relative extrema of the function. List your answers in terms of ordered pairs. Then sketch a graph of the function.

1. f ( x) = x2 − 2 x − 3

10

2. f ( x) = 4 x4 − 4 x2 + 1

y

10

8

8

6

6

4

4

2 −10

−8

−6

−4

−2

3. f ( x) = ( x − 4

2

x 2

−2

4

6

8

10

−10

−8

−6

−4

−2

−4

−6

−6

−8

−8

−10

−10

) 1 2 + 3

4. f ( x) =

−6

−4

−2

10 8

6

6

2

−2

4

6

8

10

−10

−8

−6

−4

−2

−4 −6

−8

−8

−10

−10

x 2

4

6

8

10

4. ___________________

6. f ( x) = x 49 − 9 x2

y

10

8

8

6

6

−2

y

__________________ 5. __

4

2 −2

2. ___________________

__________________ 3. __

−2

−6

4

−4

10

y

2

−4

10

−6

8

4

x

3

−8

6

x +4

y

5. f ( x) = ( x − 1 )

−10

4

8

8

2 −8

2

2

4

−10

x

−2

−4

10

__________________ 1. __

y

2

x 2

4

6

8

10

−10

−8

−6

−4

−2

−2

−4

−4

−6

−6

−8

−8

−10

−10

x 2

4

6

8

10

6. ___________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 3, Form C

70

Find the relative extrema and points of inflection of the function on [0,2π].

List your answers in terms of ordered pairs. Then sketch a graph of the function.

7. f ( x)=

1.0

sin x −6 + 2 sin x

8.

y

f(x)= sin2x+cosx

2.0

0

0

0.7

1.5

5 π/2

−00.2

π

0 −0.

50.2−

0.5 50

5 0 0.5

0 −1.

− 0.7

00

5

−

−1.

1.0 0

50

Sketch a graph of the function.

9.

00

10

π/2

π

2π

3π/2

10. f ( x) =

−6 x + 2x − 3 2

y

10

8

8

6

6

4

−8

−6

−4

11. f ( x) =

−2

2

x 2

−2

4

6

8

10

−10

−8

−6

−4

−2

−4

−6

−6

−8

−8

−10

−10

2x2− 1

12. f ( x) =

x

y

10

8

8

6

6

−6

−4

−2

−2

4

6

8

10

y

4

2 −8

2

x −1 3 x+

4

−10

x

−2

−4

10

y

4

2 −10

2. ________________________

−2.

3 x−6

f(x)=

x

1.0

2π

3π/2

_______________________ 1. __

0

x

0.5

y

2

x 2

4

6

8

10

−10

−8

−6

−4

−2

−2

−4

−4

−6

−6

−8

−8

−10

−10

x 2

4

6

8

10

CALCULUS FOR THE LIFE SCIENCES, Chapter 3, Form C 71 Find absolute maximum and minimum values of the function, if they exist, over the indicated interval. Where no interval is specified, use the real line. List your answers in terms of ordered pairs.

13.

f(x) = x(5− x)

13. ______________________

14.

2;[− f ( x) = x 3 + 3 x 2 − 6 x −4,2] 2

14. ______________________

15. f ( x) = −x 2+3.6 x +12

15. ______________________

( )

16. f x =cosx−sin2x;[0,2 π]

16. ______________________

( ) 17. f x = sec 2 x + 9 csc 2πx2;)(0,

17. ______________________

2 2x,;π( −2) 18. f x = sin 2 x + tanπ

18. ______________________

19. f ( x)= x 2+250;(0, ∞ ) x

19. ______________________

( )

20. Of all numbers whose difference is 8, find the two that have the minimum product.

20. ______________________

21. Minimize Q = 3 x2 + y2 , where x+ y = 4.

21. ______________________

22. From a thin piece of cardboard 84 in. by 84 in., square corners are cut out so that the sides can be folded up to make a box.

What dimensions will yield a box of maximum volume? What is the maximum volume?

22. ______________________

23. Find the linearization tof( x) =

24. Find the linearization tof( x) =

1

at a = −2.

23. ______________________

cosx at a = 0.

1 − sin x

24. ______________________

x −1

72

CALCULUS FOR THE LIFE SCIENCES, Chapter 3, Form C

17.

25. Use linearization to approximate

25. ___________________________

26. Use Newton’s method and x1 = 2to find an approximate

solution of x3−7x = −3.

26. ___________________________

27. Use Newton’s method and x1 = −1to find an approximate

solution of cos2x=sinx−x .

27. ___________________________

dydx . Then 28. Differentiate the following implicitly to find find the slope of the curve at the given point.

2x2+y3 = −15;(−2,1)

28. ___________________________

29. A board 13 ft long leans against a vertical wall. If the lower end is being moved away from the wall at a rate of 0.25 ft/sec, how fast is the upper end coming down when the lower end 29. ___________________________ is 12 ft from the wall?

________________________________________________________________ 30. Find the absolute maximum and minimum values of the function, if they exist, over the indicated interval.

2 x2 ;[0,∞ ) 4 + x3

f(x) = −

30. ___________________________

31. Use a grapher to estimate the relative extrema of the function.

f ( x )=3 x 3 − 25 x 2+30 xx +

31. ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Find the relative extrema and points of inflection of the function. List your answers in terms of ordered pairs. Then sketch a graph of the function.

1. f ( x) = x2 − 7 x + 6

10

2. f ( x) = x4 − 4 x2 + 4

y

10

8

8

6

6

4

−8

−6

−4

−2

3. f ( x) = ( x − 4

2

x 2

−2

4

6

8

10

−10

−8

−6

−4

−2

−4

−6

−6

−8

−8

−10

−10

) 1 2 − 3

4. f ( x) =

−6

−4

−2

y

10 8

6

6

2

4

6

8

10

−10

−8

−6

−4

−2

−4 −6

−8

−8

−10

−10

y

10

8

8

6

6

−2

−2

__________________ 3. __ x 2

4

6

8

10

4. ___________________

y

__________________ 5. __

4

2 −4

2. ___________________

6. f ( x) = x 25 − 4 x2

4

−6

10

y

−2

−6

3

−8

8

2

−4

5. f ( x) = ( x − 4 )

−10

6

4

x

−2

10

4

36 x +36

8

2 −8

2

2

4

−10

x

−2

−4

10

__________________ 1. __

y

4

2 −10

Chapter 3, Form D

2

x 2

4

6

8

10

−10

−8

−6

−4

−2

−2

−4

−4

−6

−6

−8

−8

−10

−10

x 2

4

6

8

10

6. ___________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 3, Form D

74

Find the relative extrema of the function on [0,2π]. List your answers in terms of ordered pairs. Then sketch a graph of the function.

7. f ( x) =

1.0

8. f ( x)= cos2 x+2sinx

1 2sin x − 6

y

2.0

0

0

0.7

1.5

5 π/2

−00.2

π

0 −0.

50.2−

0.5 50

5 0 0.5

0 −1.

− 0.7

00

5

−

−1.

1.0 0

50

Sketch a graph of the function.

9.

00

10

π/2

π

2π

3π/2

10. f ( x) =

−2 x + 2x + 1 2

y

10

8

8

6

6

4

−8

−6

−4

11. f ( x) =

−2

2

x 2

−2

4

6

8

10

−10

−8

−6

−4

−2

−4

−6

−6

−8

−8

−10

−10

x2− 9

12. f ( x) =

x

y

10

8

8

6

6

−6

−4

−2

−2

4

6

8

10

y

4

2 −8

2

x− 2 3 x+

4

−10

x

−2

−4

10

y

4

2 −10

2. ________________________

−2.

2 x−3

f(x)=

x

1.0

2π

3π/2

_______________________ 1. __

0

x

0.5

y

2

x 2

4

6

8

10

−10

−8

−6

−4

−2

−2

−4

−4

−6

−6

−8

−8

−10

−10

x 2

4

6

8

10

CALCULUS FOR THE LIFE SCIENCES, Chapter 3, Form D 75 Find absolute maximum and minimum values of the function, if they exist, over the indicated interval. Where no interval is specified, use the real line. List your answers in terms of ordered pairs.

13.

f(x) = x(3− x)

13. ______________________

14.

f ( x) = −8x3 x2 + +6x9− 2; [−1, 1]

14. ______________________

15. f (

16. f

( (

17. f

x) = −x2 +

4.2x+ 6

15. ______________________

x )= sinx− cos2x;[0, π] 2

16. ______________________

x) = sec2 tan x +2x; (−π2,π2)

17. ______________________

x) = cos2 x +

cotx; (0,π)

18. ______________________

250 ,0) 19. f ( x ) = x 2 x− ; (−∞

19. ______________________

20. Of all numbers whose difference is 18, find the two that have the minimum product.

20. ______________________

21. Minimize Q = x2 + 2 y2 , where x− y = 4.

21. ______________________

18. f (

22. From a thin piece of cardboard 100 in. by 100 in., square corners are cut out so that the sides can be folded up to make a box.

What dimensions will yield a box of maximum volume? What 22. ______________________ is the maximum volume?

23. Find the linearization tof( x) =

1 at a = 2. x+2

23. ______________________

24. Find the linearization tof( x) =

cos x at a = π .

2+sin x

24. ______________________

76

CALCULUS FOR THE LIFE SCIENCES, Chapter 3, Form D

25. Use linearization to approximate3123.

25. ___________________________

26. Use Newton’s method and x1 =1to find an approximate

solution of 2x+x3=4

27. Use Newton’s method and x1 =1to find an approximate solution of cosx+sinx=2 x

26. ___________________________

27. ___________________________

dydx . Then 28. Differentiate the following implicitly to find find the slope of the curve at the given point.

2x3−y3 = −11;(2,3)

29. A board 26 ft long leans against a vertical wall. If the lower end is being moved away from the wall at a rate of 0.4 ft/sec, how fast is the upper end coming down when the lower end is 10 ft from the wall?

28. ___________________________

29. ___________________________

________________________________________________________________ 30. Find the absolute maximum and minimum values of the function, if they exist, over the indicated interval.

5 x2 ;[0,∞ ) 1 + x3

f(x) = −

30. ___________________________

31. Use a grapher to estimate the relative extrema of the function.

f ( x )=2 x 3 − 15 x 2+10 x +10 x

31. ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Find the relative extrema and points of inflection of the function. List your answers in terms of ordered pairs. Then sketch a graph of the function.

1. f ( x) = x2 + 7 x +12

10

2. f ( x) = x4 − 6 x2 + 9

y

10

8

8

6

6

4

−8

−6

−4

−2

2

x 2

−2

4

6

8

10

−10

−8

−6

−4

−2

−4

−6

−6

−8

−8

−10

−10

3. f ( x) = ( x − 1 )23+ 2

6 4. f ( x) = x− 2+

y

10

8

8

6

6

−6

−4

−2

2

4

6

8

10

−10

−8

−6

−4

−2

−4

−6

−6

−8

−8

−10

−10

3

y

10

8

8

6

6

−4

−2

−2

2. ___________________

__________________ 3. __ x 2

4

6

8

10

4. ___________________

y

__________________ 5. __

4

2 −6

10

6. f ( x) = x 49 − 16 x 2

4

−8

8

y

−2

−4

5. f ( x) = ( x + 3 )

−10

6

2

x

−2

10

4

4

2 −8

2

2

4

−10

x

−2

−4

10

__________________ 1. __

y

4

2 −10

Chapter 3, Form E

2

x 2

4

6

8

10

−10

−8

−6

−4

−2

−2

−4

−4

−6

−6

−8

−8

−10

−10

x 2

4

6

8

10

6. ___________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 3, Form E

78

Find the relative extrema of the function on [0,2π]. List your answers in terms of ordered pairs. Then sketch a graph of the function.

7. f ( x)=

1.0

8. f (x)=2sinx+cosx

sin x 2 cosx − 4

y

3

0

2

0.7

1

5 π/2

−00.2

π

x

2π

3π/2

π/2

π

2π

3π/2

1

5 0 0.5

−

− 0.7

2

−

−

1.0 0

Sketch a graph of the function.

3

8 x+5

f(x)=

10. f ( x) =

10

−8 x + 6x + 9 2

y

10

8

8

6

6

4

−8

−6

−4

11. f ( x) =

−2

2

x 2

−2

4

6

8

10

−10

−8

−6

−4

−2

−4

−6

−6

−8

−8

−10

−10

x 2 − 16 x

12. f ( x) =

y

10

8

8

6

6

−6

−4

−2

−2

4

6

8

10

y

4

2 −8

2

x+ 3 2 x−

4

−10

x

−2

−4

10

y

4

2 −10

2. ________________________

−

50.2−

9.

_______________________ 1. __

x

0.5

5

y

2

x 2

4

6

8

10

−10

−8

−6

−4

−2

−2

−4

−4

−6

−6

−8

−8

−10

−10

x 2

4

6

8

10

CALCULUS FOR THE LIFE SCIENCES, Chapter 3, Form E 79 Find absolute maximum and minimum values of the function, if they exist, over the indicated interval. Where no interval is specified, use the real line. List your answers in terms of ordered pairs.

13.

f(x)=x(10−x)

13. ______________________

x3 3

4] + 14. f ( x ) = + x 2 − 3 x 1;[0,

14. ______________________

15. f ( x )= −x 2+2.6 x− 5

15. ______________________

( x) = cos x − cos2 x; [0, 2 16. f π] (

16. ______________________

x )= tanx +2 secx π 2; ,(π−2 )

17. ______________________

17. f

x ) = cos 2 x + cot 2 x ; (0, 18. f ( π)

18. ______________________

1024

19. f ( x)=x2+

x

;(0,∞)

19. ______________________

20. Of all numbers whose difference is 8, find the two that have the minimum product.

20. ______________________

21. Minimize Q = 2 x2 + 2 y2 , where x+ y = 10.

21. ______________________

22. From a thin piece of cardboard 108 in. by 108 in., square corners are cut out so that the sides can be folded up to make a box.

What dimensions will yield a box of maximum volume? What 22. ______________________ is the maximum volume?

23. Find the linearization tof( x) =5 2 − x at a=−7.

24. Find the linearization tof( x

)=

cosx + sin x at a= π 2.

23. ______________________

24. ______________________

80

CALCULUS FOR THE LIFE SCIENCES, Chapter 3, Form E

80.

25 Use linearization to approximate .

1=4xto find an approximate Use Newton’s method and

26

solution of x3−3x2=8.

.

27. Use Newton’s method and x1 = 2to find an approximate solution of 2sinx=x−cosx .

25. ___________________________

26. ___________________________

27. ___________________________

dydx . Then 28. Differentiate the following implicitly to find find the slope of the curve at the given point.

3x3+y3=3;(−2,3) 29. A board 17 ft long leans against a vertical wall. If the lower end is being moved away from the wall at a rate of 0.1 ft/sec, how fast is the upper end coming down when the lower end is 8 ft from the wall?

28. ___________________________

29. ___________________________

________________________________________________________________ 30. Find the absolute maximum and minimum values of the function, if they exist, over the indicated interval.

f () =

4 x2 ;(−∞,0] 1 − x3

30. ___________________________

31. Use a grapher to estimate the relative extrema of the function.

f ( x )=3 x 3 − 20 x 2+25 xx+

31. ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Find the relative extrema and points of inflection of the function. List your answers in terms of ordered pairs. Then sketch a graph of the function.

1. f ( x) = x2 + 2 x − 8

10

2. f ( x) = x4 − 2 x2 + 3

y

10

8

8

6

6

4

−8

−6

−4

−2

2

x 2

−2

4

6

8

10

−10

−8

−6

−4

−2

−4

−6

−6

−8

−8

−10

−10

3. f ( x) = ( x − 3 )23− 1

4. f ( x) =

y

10

8

8

6

6

−6

−4

−2

2

4

6

8

10

−10

−8

−6

−4

−2

−4

−6

−6

−8

−8

−10

−10

3

y

10

8

8

6

6

−4

−2

−2

2. ___________________

__________________ 3. __ x 2

4

6

8

10

4. ___________________

y

__________________ 5. __

4

2 −6

10

6. f ( x) = x 25 − 16 x 2

4

−8

8

y

−2

−4

5. f ( x) = ( x − 2 )

−10

6

2

x

−2

10

4

4

2 −8

2

10 x 2+4

4

−10

x

−2

−4

10

__________________ 1. __

y

4

2 −10

Chapter 3, Form F

2

x 2

4

6

8

10

−10

−8

−6

−4

−2

−2

−4

−4

−6

−6

−8

−8

−10

−10

x 2

4

6

8

10

6. ___________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 3, Form F

82

Find the relative extrema of the function on [0,2π]. List your answers in terms of ordered pairs. Then sketch a graph of the function.

7. f ( x)=

1.0

cosx 6 + 3cosx

8.

y

3

0

2

0.7

1

5 π/2

−00.2

π

_______________________ 1. __

x

2π

3π/2

π/2

π

2π

3π/2

1

5 0 0.5

−

− 0.7

2

−

−

1.0 0

Sketch a graph of the function.

3

3 x−5

f(x)=

10. f ( x) =

10

−6 x − 25 2

y

10

8

8

6

6

4

−8

−6

−4

11. f ( x) =

−2

2

x 2

−2

4

6

8

10

−10

−8

−6

−4

−2

−4

−6

−6

−8

−8

−10

−10

x 2 − 16 x

12. f ( x) =

y

10

8

8

6

6

−6

−4

−2

−2

4

6

8

10

y

4

2 −8

2

x +1 2 x−

4

−10

x

−2

−4

10

y

4

2 −10

2. ________________________

−

50.2−

5

y

x

0.5

9.

f(x)= 3cosx−sinx

2

x 2

4

6

8

10

−10

−8

−6

−4

−2

−2

−4

−4

−6

−6

−8

−8

−10

−10

x 2

4

6

8

10

CALCULUS FOR THE LIFE SCIENCES, Chapter 3, Form F 83 Find absolute maximum and minimum values of the function, if they exist, over the indicated interval. Where no interval is specified, use the real line. List your answers in terms of ordered pairs.

13. f ( x) = x(2− x)

14.

13. ______________________

( )

f x = −8x3 + x92 + 6x − 5; [− 2, 0]

14. ______________________

( ) 5.4x+ 9 15. f x = −x2 +

15. ______________________

16. f ( x) = sin2 x + 2 cosx; [0,2 π ]

16. ______________________

( )

( )

17. f x = tan2 x −cos2x; [0,π ]

17. ______________________

18. f x = sin2 x −cot2x; (0,π)

18. ______________________

19. f ( x)= x2 +

1 6 (0, ; ∞) x

19. ______________________

20. Of all numbers whose difference is 16, find the two that have the minimum product.

20. ______________________

21. Minimize Q = x2 + 3 y2 where x− y = 6.

21. ______________________

22. From a thin piece of cardboard 81 in. by 81 in., square corners are cut out so that the sides can be folded up to make a box.

What dimensions will yield a box of maximum volume? What is the maximum volume?

22. ______________________

23. Find the linearization tof( x) =

x at a = −1. x 4 +

24. Find the linearization tof( x) = 1 ( + cosx) at a = 0.

2

23. ______________________

24. ______________________

84

CALCULUS FOR THE LIFE SCIENCES, Chapter 3, Form F

7.3 25. Use linearization to approximate

25. ___________________________

26. Use Newton’s method and x1 =3to find an approximate

solution of x3−3x2=2.

27. Use Newton’s method and x1 = −1to find an approximate sin solution of −2cosx=x+ x.

26. ___________________________

27. ___________________________

dydx . Then 28. Differentiate the following implicitly to find find the slope of the curve at the given point. −2x3+y3 = −17;(2,−1)

29. A board 13 ft long leans against a vertical wall. If the lower end is being moved away from the wall at a rate of 0.1 ft/sec, how fast is the upper end coming down when the lower end is 12 ft from the wall?

28. ___________________________

29. ___________________________

________________________________________________________________ 30. Find the absolute maximum and minimum values of the function, if they exist, over the indicated interval.

x2

f(x)= 4 −x 3 ;(−∞,0 ]

30. ___________________________

31. Use a grapher to estimate the relative extrema of the function.

f ( x )=4 x 3 − 22 x 2+30 xx+4

31. ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 4, Form A Differentiate.

1. y=3ex

1.______________________

x) lnx 2. y = (sin

2.______________________

3. f ( x) = e− x

3. _____________________

x 4. f ( x) =ln 5

4. ______________________

5. f ( x) = ex − 6 x3

5. ______________________

4

( )

6. f x = 2 ex ln x

6. ______________________

7. y =

ln ( cos x+ e )

7. ______________________

8. y =

ln 5 ex

8. ______________________

x

x

Given logb 6= 1.2925 and logb 12= 1.7925 , find each of the following.

9. logb36

9.______________________

logb 2

10. ______________________

11. logb 72

11. ______________________

10.

86

CALCULUS FOR LIFE SCIENCES, Chapter 4, Form A

12. Find the function that satisfies dEdt=kE. List the answer in terms of E0.

12. ___________________________

13. The doubling time of a certain bacteria culture is 8 hr. What is the growth rate? Round to the nearest tenth of a percent.

13. ___________________________

14. Interest Compounded Continuously .

An investment is made at 6.5% per year, compounded continuously. What is the doubling time? Round to the nearest year.

14. ___________________________

15.Cost of Pizza . The cost Cof a slice of pizza at a certain shop was $1.25 in1990. In 2005 the cost was $3 per slice.

Assuming the exponential-growth model applies:

(a) Find the exponential-growth rate, and write the equation. 15. (a) _________________________

(b) Find the cost of a slice of this pizza in 2020.

(b) _________________________

16. Bacterial Population. After beginning antiseptic treatment, the number of bacteria in a population decreases at a rate of 12% per hour, that is, dN

=−0.12N, dt where Nis the number of bacteria and tis the time, in hours.

(a) When t=0, there are 100,000 bacteria present. Find a function that satisfies this equation.

16. (a) _________________________

(b) How many bacteria will remain after 8 hr?

(b) _________________________

(c) After how long do half of the original bacteria remain?

(c) _________________________

17. Decay Rate. The decay rate of iodine-131 is 8.6% per day.

What is its half-life?

17. ___________________________

18. Half-Life. The half-life of radium-226 is 1600 years. What is the decay rate? As a percent, round to three decimal places.

18. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 4, Form A 87 19. Effect of Advertising. A company introduces a new product on a trial run in a city. They advertised the product on television and found the percentage Pof people who bought the product after tads had run satisfied the function

100%

P ( t ) =1+20e− 0.15t (a) What percentage buys the product after the ad has been been run 1 time, 5 times, 20 times? (b) Find the rate of change P′ (t).

(c) Sketch a graph of the function.

19. (a) _____________________

(b) ____________________

(c)

Differentiate.

f(x)=5x

20. ______________________

21. y=log 15x

21. ______________________

20.

___________________________________________________________ 22. Differentiate:y = x (lnx) − x ln x + 3 x .

3

22. ______________________

23. Find the maximum and minimum values of

f ( x )= x 2 e−x over [0,10].

23. ______________________

88

CALCULUS FOR LIFE SCIENCES, Chapter 4, Form A

24. Use your grapher to graph:f ( x) =

the graph.

e3x e− x 25. Find lim e3x − − x . x→0 +e

e x − e− x . Then sketch 3 x + e− x e 3 24.

25. ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 4, Form B Differentiate.

1. y=ex

2. y =

1.______________________

( cos 2x x) ln

2.______________________

3. f ( x) = e− x

3. _____________________

x 4. f ( x) = ln 5

4. ______________________

5. f ( x) = ex − 3 x4

5. ______________________

3

( )

6. f x = 10e x ln x

6. ______________________

7. y =

lne( + tanx)

7. ______________________

8. y =

6 ln x ex

8. ______________________

x

Given logb 3= 0.5283 and logb 18= 1.3900 , find each of the following.

1 9. logb 3

9.______________________

logb 6

10. ______________________

11. logb54

11. ______________________

10.

90

CALCULUS FOR LIFE SCIENCES, Chapter 4, Form B

12. Find the function that satisfies dJdt=kJ. List the answer in terms of J0.

12. ___________________________

13. The doubling time of a certain bacteria culture is 8 hr. What is the growth rate? Round to the nearest tenth of a percent.

13. ___________________________

14. Interest Compounded Continuously .

An investment is made at 7.625% per year, compounded continuously. What is the doubling time? Round to the nearest year.

14. ___________________________

15.Cost of Ice Cream . The cost Cof an ice cream bar was $0.05 in1950. In 1995 the cost was $1.25. Assuming the exponential-growth model applies:

(a) Find the exponential-growth rate, and write the equation. 15. (a) _________________________

(b) Find the cost of an ice cream bar in 2020.

(b) _________________________

16. Bacterial Population. After beginning antiseptic treatment, the number of bacteria in a population decreases at a rate of 25% per day, that is, dN

=−0.25N, dt where Nis the number of bacteria and tis the time, in days.

(a) When t=0, there are 80,000 bacteria present. Find a function that satisfies this equation.

16. (a) _________________________

(b) How many bacteria will remain after 7 days?

(b) _________________________

(c) After how long do half of the original bacteria remain?

(c) _________________________

17. Decay Rate. The decay rate of krypton-85 is 6.3% per year.

What is its half-life?

17. ___________________________

18. Half-Life. The half-life of phosphorus-32 is 14.3 days. What is the decay rate? As a percent, round to two decimal places.

18. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 4, Form B 91 19. Effect of Advertising. A company introduces a new product on a trial run in a city. They advertised the product on television and found the percentage Pof people who bought the product after tads had run satisfied the function

100%

P ( t ) =1+40e− 0.32t (a) What percentage buys the product after the ad has been been run 1 time, 5 times, 20 times? (b) Find the rate of change P′ (t).

(c) Sketch a graph of the function.

19. (a) _____________________

(b) ____________________

(c)

Differentiate.

f(x)=13x

20. ______________________

21. y=log 25x

21. ______________________

20.

___________________________________________________________ 22. Differentiate:y = x (lnx) − 2 x2 ln x + x2 .

3

22. ______________________

23. Find the maximum and minimum values of

f ( x )= x 5 e−x over [0,10].

23. ______________________

92

CALCULUS FOR LIFE SCIENCES, Chapter 4, Form B

24. Use your grapher to graph:f ( x) =

the graph.

e−5x −e5x . Then sketch e5 x + e−x5 24.

e 2x e−2 x 25. Find lim e2x − −2 x . x→0 +e

25. ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 4, Form C Differentiate.

1. y=ex

1.______________________

2. y =cot ( x) lnx

2.______________________

3. f ( x) = e 2x

3. _____________________

4x 4. f ( x) =ln 3

4. ______________________

5. f ( x) = ex − 4 x2

5. ______________________

( )

6. f x = 5 ex ln x

6. ______________________

7. y =

lne( + secx)

7. ______________________

8. y =

ln x 2 ex

8. ______________________

2x

Given logb 5= 0.7740 and logb 3= 1.5283 , find each of the following.

9. logb 1 5

9.______________________

logb 25

10. ______________________

1 11. logb 3

11. ______________________

10.

94

CALCULUS FOR LIFE SCIENCES, Chapter 4, Form C

12. Find the function that satisfies dQdt=kQ. List the answer 0. in terms of Q

12. ___________________________

13. The doubling time of a certain bacteria culture is 6 hr. What is the growth rate? Round to the nearest tenth of a percent.

13. ___________________________

14. Interest Compounded Continuously .

An investment is made at 5.768% per year, compounded continuously. What is the doubling time? Round to the nearest year.

14. ___________________________

15.Cost of Candy . The cost Cof a certain candy bar in 1970 was $0.05. In 1995 the cost was $0.35. Assuming the exponential-growth model applies:

(a) Find the exponential-growth rate, and write the equation. 15. (a) _________________________

(b) Find the cost of this candy in 2020.

(b) _________________________

16. Bacterial Population. After beginning antiseptic treatment, the number of bacteria in a population decreases at a rate of 10% per hour, that is, dN

=−0.10N, dt where Nis the number of bacteria and tis the time, in hours.

(a) When t=0, there are 80,000 bacteria present. Find a function that satisfies this equation.

16. (a) _________________________

(b) How many bacteria will remain after 24 hr?

(b) _________________________

(c) After how long do half of the original bacteria remain?

(c) _________________________

17. Decay Rate. The decay rate of phosphorus-32 id 4.85% per day. What is its half-life?

17. ___________________________

18. Half-Life. The half-life of uranium-238 is 4.5 billion years.

What is the decay rate? As a percent, round to nine decimal places.

18. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 4, Form C 95 19. Effect of Advertising. A company introduces a new product on a trial run in a city. They advertised the product on television and found the percentage Pof people who bought the product after tads had run satisfied the function

100%

P ( t ) =1+20e− 0.25t (a) What percentage buys the product after the ad has been been run 1 time, 5 times, 20 times? (b) Find the rate of change P′ (t).

(c) Sketch a graph of the function.

19. (a) _____________________

(b) ____________________

(c)

Differentiate.

f(x)=20x

20. ______________________

21. y=log 15x

21. ______________________

20.

___________________________________________________________

22. Differentiate:y = x ln x + 3 xln x − 3 x .

22. ______________________

23. Find the maximum and minimum values of

f ( x )= x 3 e− 3 x over [0,10].

23. ______________________

96

CALCULUS FOR LIFE SCIENCES, Chapter 4, Form C

24. Use your grapher to graphf: ( x) =

the graph.

e3x e−3 x 25. Find lim e3x − −3 x . x→0 +e

e x − e−3 x . Then sketch 3 x + e−3 x e 3 24.

25. ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 4, Form D Differentiate.

1. y=5ex

1.______________________

x) lnx 2. y = (sec

2.______________________

3. f ( x) = ex3

3. _____________________

4. f ( x) = ln 4 x

4. ______________________

2

( )

5. f x = e x− 4 x5

5. ______________________

( ) 6. f x = 5ex ln x

7. y =

8. y =

6. ______________________

ln(cos x + e2 ) x

ex lnx

7. ______________________

8. ______________________

Given logb 6= 1.2925 and logb 10= 1.6610 , find each of the following.

1 9. logb 6

9.______________________

logb 60

10. ______________________

11. logb 100

11. ______________________

10.

98

CALCULUS FOR LIFE SCIENCES, Chapter 4, Form D

12. Find the function that satisfies dFdt=kF. List the answer in terms of F0.

12. ___________________________

13. The doubling time of a certain bacteria culture is 12 hr. What is the growth rate? Round to the nearest tenth of a percent.

13. ___________________________

14. Interest Compounded Continuously .

An investment is made at 8.085% per year, compounded continuously. What is the doubling time? Round to the nearest year.

14. ___________________________

15.Cost of Bread . The costCof a loaf of bread at a particular bakery was $0.70 in 1990. In 2002 the cost was $2.75 per loaf. Assuming the exponential-growth model applies:

(a) Find the exponential-growth rate, and write the equation. 15. (a) _________________________

(b) Find the cost of a loaf of bread at this bakery in 2020.

(b) _________________________

16. Bacterial Population. After beginning antiseptic treatment, the number of bacteria in a population decreases at a rate of 40% per day, that is,

dN

=−0.40N, dt where Nis the number of bacteria and tis the time, in days.

(a) When t=0, there are 5,000,000 bacteria present. Find a function that satisfies this equation.

16. (a) _________________________

(b) How many bacteria will remain after 3 days?

(b) _________________________

(c) After how long do half of the original bacteria remain?

(c) _________________________

17. Decay Rate. The decay rate of hydrogen-3 is 5.6% per year.

What is its half-life?

17. ___________________________

18. Half-Life. The half-life of cobalt-60 is 5.3 years. What is the decay rate? As a percent, round to two decimal places.

18. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 4, Form D 99 19. Effect of Advertising. A company introduces a new product on a trial run in a city. They advertised the product on television and found the percentage Pof people who bought the product after tads had run satisfied the function

100%

P ( t ) =1+20e− 0.15t (a) What percentage buys the product after the ad has been been run 1 time, 5 times, 20 times? (b) Find the rate of change P′ (t).

(c) Sketch a graph of the function.

19. (a) _____________________

(b) ____________________

(c)

Differentiate.

20.

f(x)=20x

21. y=log 8x

20. ______________________

21. ______________________

___________________________________________________________

22. Differentiate:y = x ln x − 3 xln x + 3 x .

22. ______________________

23. Find the maximum and minimum values of

f(x)=x4e−4x over [0,5].

23. ______________________

100 CALCULUS FOR LIFE SCIENCES, Chapter 4, Form D 24. Use your grapher to graph:f ( x) =

the graph.

e−3 x e x 25. Find lim −3 x − x . x→ 0 e +3 e 3

e−3 x − e x . Then sketch e−3 x + 3 x e 3 24.

25. ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 4, Form E Differentiate.

1. y=ex

2. y =

1.______________________

(sinx)2lnx

2.______________________

3. f ( x) = ex

3. _____________________

x 4. f ( x) = ln 2

4. ______________________

5

( ) 5. f x = −3x4 + xe

5. ______________________

6. f x =5exlnx

6. ______________________

( )

7. y =

lne(3x+3tanx)

7. ______________________

8. y =

4 ln x ex

8. ______________________

Given logb 6= 1.2925 and logb 8= 1.5 , find each of the following.

9. logb 48

9.______________________

logb 2

10. ______________________

1 11. logb 8

11. ______________________

10.

102 CALCULUS FOR LIFE SCIENCES, Chapter 4, Form E 12. Find the function that satisfies dNdt=kN. List the answer in terms of N0.

12. ___________________________

13. The doubling time of a certain bacteria culture is 18 hr. What is the growth rate? Round to the nearest tenth of a percent.

13. ___________________________

14. Interest Compounded Continuously .

An investment is made at 3.55% per year, compounded continuously. What is the doubling time? Round to the nearest year.

14. ___________________________

15.Cost of a Movie Ticket . The costCof a movie ticket at a certain theatre was $3.75 in 1982. In 1999 the cost was $7.50 per ticket. Assuming the exponential-growth model applies:

(a) Find the exponential-growth rate, and write the equation. 15. (a) _________________________

(b) Find the cost of a movie ticket at this theatre in 2020.

(b) _________________________

16. Bacterial Population. After beginning antiseptic treatment, the number of bacteria in a population decreases at a rate of 20% per hour, that is, dN

=−0.20N, dt where Nis the number of bacteria and tis the time, in hours.

(a) When t=0, there are 150,000 bacteria present. Find a function that satisfies this equation.

16. (a) _________________________

(b) How many bacteria will remain after 6 hr?

(b) _________________________

(c) After how long do half of the original bacteria remain?

(c) _________________________

17. Decay Rate. The decay rate of cobalt-60 is 13.08% per year. What is its half-life?

17. ___________________________

18. Half-Life. The half-life of radium-226 is 1600 years. What is the decay rate? As a percent, round to three decimal places.

18. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 4, Form E 103 19. Effect of Advertising. A company introduces a new product on a trial run in a city. They advertised the product on television and found the percentage Pof people who bought the product after tads had run satisfied the function

100%

P ( t ) =1+80e− 0.32t (a) What percentage buys the product after the ad has been been run 1 time, 5 times, 20 times? (b) Find the rate of change P′ (t).

(c) Sketch a graph of the function.

19. (a) _____________________

(b) ____________________

(c)

Differentiate.

20.

f(x)=15x

20. ______________________

21. y=log 5x

21. ______________________

___________________________________________________________ 22. Differentiate:y = 3 x( lnx) − 4 x ln x + 4 x .

4

22. ______________________

23. Find the maximum and minimum values of

f ( x )=4 x 2 e− 2 x over [0,10].

23. ______________________

104 CALCULUS FOR LIFE SCIENCES, Chapter 4, Form E 24. Use your grapher to graph:f ( x) =

the graph.

e−5 x e x 25. Find lim e−5 x − x . x→0 +5 e 5

e−5 x − e x . Then sketch e−5 x + 5 x e 5 24.

25. ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 4, Form F Differentiate.

1. y=2ex

1.______________________

x) lnx 2. y = (csc

2.______________________

3. f ( x) = ex−

3. _____________________

4. f ( x) = ln 3 x

4. ______________________

2

( )

5. f x = e x− 4 x3

5. ______________________

( ) 6. f x = 8ex ln x

6. ______________________

7. y =

lne( − cotx)

7. ______________________

8. y =

5 ex ln x

8. ______________________

2x

Given logb 4= 0.7124 and logb 12= 1.2770 , find each of the following.

9. logb 3

9.______________________

1 logb 12

10. ______________________

11. logb 8

11. ______________________

10.

106 CALCULUS FOR LIFE SCIENCES, Chapter 4, Form F 12. Find the function that satisfies dSdt=kS. List the answer in terms of S0.

12. ___________________________

13. The doubling time of a certain bacteria culture is 4 hr. What is the growth rate? Round to the nearest tenth of a percent.

13. ___________________________

14. Interest Compounded Continuously .

An investment is made at 5.676% per year, compounded continuously. What is the doubling time? Round to the nearest year.

14. ___________________________

15.Cost of Gasoline. The cost Cof a gallon of gasoline at a certain station in 1974 was $0.49. In 2005 the cost was $2.85.

Assuming the exponential-growth model applies:

(a) Find the exponential-growth rate, and write the equation. 15. (a) _________________________

(b) Find the cost of a gallon of gasoline at this station in 2020.

(b) _________________________

16. Bacterial Population. After beginning antiseptic treatment, the number of bacteria in a population decreases at a rate of 60% per day, that is,

dN = −0.60N, dt t time, in days. whereN is the number of bacteria and is the

(a) When t=0, there are 250,000 bacteria present. Find a function that satisfies this equation.

16. (a) _________________________

(b) How many bacteria will remain after 5 days?

(b) _________________________

(c) After how long do half of the original bacteria remain?

(c) _________________________

17. Decay Rate. The decay rate of radium-226 is 0.043% per year. What is its half-life?

17. ___________________________

18. Half-Life. The half-life of iodine-131 is 8.1 days. What is the decay rate? As a percent, round to two decimal places.

18. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 4, Form F 107 19. Effect of Advertising. A company introduces a new product on a trial run in a city. They advertised the product on television and found the percentage Pof people who bought the product after tads had run satisfied the function

100%

P ( t ) =1+50e− 0.17t (a) What percentage buys the product after the ad has been been run 1 time, 5 times, 20 times? (b) Find the rate of change P′ (t).

(c) Sketch a graph of the function.

19. (a) _____________________

(b) ____________________

(c)

Differentiate.

f(x)=5x

20. ______________________

21. y=log 4x

21. ______________________

20.

___________________________________________________________ 22. Differentiate:y = x2 (lnx) − x ln x + 2 x .

2

22. ______________________

23. Find the maximum and minimum values of

f(x)=x2e−2x over [0,5].

23. ______________________

108 CALCULUS FOR LIFE SCIENCES, Chapter 4, Form F 24. Use your grapher to graph:f ( x) =

the graph.

e−0.5x − e0.5x . Then sketch e−0.5x + e0.5x 24.

e−0.5x e0.5x 25. Find lim −0.5x − 0.5x . x→ 0 e +e

25. ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 5, Form A

Evaluate.

1.

dx ∫

1.______________________

2.

∫ 200 x 4 dx

2.______________________

3.

⎛ x +ex+x3 ∫ ⎜ 3

⎝

5

⎞ dx ⎟ ⎠

3. _____________________

Find the area under the curve over the indicated interval.

4.

y=3x−x2;[0,3]

2 5. y = x ; [1, 8]

4. ______________________

5. ______________________

6. Give two interpretations of the shaded region.

6.______________________

Evaluate.

7.

8.

9.

5 x 4 x ) dx ∫ ∫( + 2

3

−1

5 0

e− dx 4x

dx

∫x 2a a

7. ______________________

8. ______________________

9. ______________________

110 CALCULUS FOR LIFE SCIENCES, Chapter 5, Form A is positive, negative, or zero.

10. Decide whether∫f ( x ) dx b

a

10. ___________________________

Evaluate using substitution. Do not use Table 1 or CAS.

dx

11.

∫ x 5

11. ___________________________

12.

∫ xedx

12. ___________________________

13.

∫x2sin(x3)dx

13. ___________________________

+

0.1

Evaluate using integration by parts. Do not use Table 1 or CAS.

14.

xcosx(3) dx ∫

14. ___________________________

15.

∫x4lnx5dx

15. ___________________________

Evaluate using Table 1 or CAS.

16.

3xdx ∫

16. ___________________________

17.

∫ dxx x

17. ___________________________

(5− )

[ −2,3]. y = 3t5 + 2 tover 18. Find the average value of

18. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 5, Form A 111 19. Find the area of the region bounded yby= 2 x , y = x3 ,

x = 0, and x = 2.

5 16 20. Approximate ∫ 2 dxusing a Reimann sum withn = 4.

1 x

19. ______________________

20. ______________________

21.Area of a Healing Wound. The area A of a healing would is decreasing at a rate of

A′(t ) = −37.76t−2,1 ≤ t ≤ 7 wheretis time in days, and Ais in square centimeters. On 34.5 day one, the wound had an area of cm2. What will the area of the wound be in 5 days?

21. ______________________

22. Distance. A particle is initially at the origin. Its velocity, in meters per second, at any time t, t≥0, is given by

v(t)=3t2+2t. Find the distance that the particle travels

in the first 4 seconds (from t=0 to t=4).

22. ______________________

Integrate using any method.

dx

23.

∫ x (6 − 5 x )

23. ______________________

24.

5 x 4 exdx ∫

24. ______________________

25.

xe dx ∫

25. ______________________

26.

∫x ln x d x

26. ______________________

27.

x x + 6 dx ∫

27. ______________________

28.

∫ 16 − x 2

3 x4

1

3

2

dx

28. ______________________

112 CALCULUS FOR LIFE SCIENCES, Chapter 5, Form A 29.

∫xe0.5xdx

29. ___________________________

30.

∫x2sinxdx

30. ___________________________

4 −

Find the volume generated by revolving about the x-axis the regions bounded by the following graphs.

31. y = 3 x,x = 1,x = 4

32. y =

2 x2

31. ___________________________

,x = 1,x = 4

32. ___________________________

33. Find the volume of a pyramid 10 ft high with a 5-ft-by-5ft square base.

33. ___________________________

Determine whether the improper integral is convergent or divergent, and calculate its value if it is convergent.

34.

x ∫e−2dx

34. ___________________________

35.

∫∞x4dx

35. ___________________________

∞

0

1

________________________________________________________________ Evaluate using any method.

⎡⎣3 + 2( ln)2x+ (lnx) ⎤ ⎦ dx 36. ∫ x 5

37.

x 9

ln⎛⎜ + ⎞⎟ dx ∫ x 2

36. ___________________________

37. ___________________________

⎝ + ⎠

38. Use a grapher to approximate the area between the following curves:

y = 4 xx −2, y = 3 x3x −2 − 8 x , 0, x = x2. =

38. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 5, Form A 113

∫ 2sin xdxusing 39. Approximate the integral 2

0

(a) the Trapezoid rule with = n 6.

39. (a) _____________________

n = 6. (b) Simpson’s rule with

39. (b) ____________________

CALCULUS FOR THE LIFE SCIENCES

Name: ______________________

Chapter 5, Form B

Evaluate.

1.

dx ∫

1.___________________________

2.

9 x 4 dx ∫

2.___________________________

3.

⎛x ∫ ⎜

32

⎝

+ex+4 ⎞ x ⎟ dx ⎠

3. __________________________

Find the area under the curve over the indicated interval.

4.

y=9x−x2;[0,9]

3 5. y = x ; [1, 3]

4. ___________________________

5. ___________________________

6. Give two interpretations of the shaded region.

6.___________________________

Evaluate.

7.

8.

9.

4 x 6 x ) dx ∫ ∫( + 2

5

−3

5

7. ___________________________

e− dx

8. ___________________________

∫ xdx

9. ___________________________

0

a

0

3x

CALCULUS FOR THE LIFE SCIENCES, Chapter 5, Form B 115 is positive, negative, or zero.

10. Decide whether∫f ( x ) dx b

a

10. ______________________

Evaluate using substitution. Do not use Table 1 or CAS.

dx

11.

∫ x 5

11. ______________________

12.

e−0.8 x dx ∫

12. ______________________

13.

∫xcos(x2)dx

13. ______________________

−

Evaluate using integration by parts. Do not use Table 1 or CAS.

14.

15.

x

x sin⎛⎜ ⎞⎟ dx ∫ 2

14. ______________________

∫x2lnx3dx

15. ______________________

⎝ ⎠

Evaluate using Table 1 or CAS.

16.

5xdx ∫

16. ______________________

17.

dx ∫ x x)

17. ______________________

(3+

5,4]. y = 2 t − 6 t2 over−[ 18. Find the average value of

18. ______________________

116 CALCULUS FOR LIFE SCIENCES, Chapter 5, Form B 19. Find the area of the region bounded yby= 4 x , y = x3 , x = 0, and x = 2.

20. Approximate ∫9 − x2 3

0

(

) dx using a Reimann sum withn = 3.

19. ___________________________

20. ___________________________

21.Memory. In a certain memory experiment, the rate of memorizing is given by:

M ′(t=)−0.007t2+0.2t

where M(t)is the number of Spanish words memorized in tminutes. The subject has no Spanish words memorized at the beginning of the experiment. How many words are memorized in 14 minutes? (Round answer to the nearest whole number.)

21. ___________________________

22. Distance. A particle is initially at the origin. Its velocity, in meters per second, at any time t, t≥0, is given by

v(t)=4t2+t. Find the distance that the particle travels

in the first 6 seconds (from t=0 to t=6).

22. ___________________________

Integrate using any method.

dx

23.

∫ x (5+2 x )

23. ___________________________

24.

3 x 3 exdx ∫

24. ___________________________

25.

x 9 exdx ∫

25. ___________________________

26.

∫x ln x d x

26. ___________________________

27.

x x + 2 dx ∫

27. ___________________________

28.

dx ∫ 25 − x 2

28. ___________________________

10

3

5

3

CALCULUS FOR THE LIFE SCIENCES, Chapter 5, Form B 117 29.

∫xexdx

29. ______________________

30.

∫ cos x (2x)dx

30. ______________________

2 −0.8

2

Find the volume generated by revolving about the x-axis the regions bounded by the following graphs.

31. y=14x3,x=0,x=1

31. ______________________

4 ,x=1,x=5 x

32. ______________________

32. y=

33. Find the volume of a pyramid 20 ft high with an 8-ft-by-8ft square base.

33. ______________________

Determine whether the improper integral is convergent or divergent, and calculate its value if it is convergent.

34.

∞ 3 dx ∫ x 3

34. ______________________

35.

∫e3xdx

35. ______________________

1

∞

1

___________________________________________________________ Evaluate using any method.

⎡3( lnx) + 5 lnx( − 2 )⎤⎦ ⎣ dx 36. ∫ x 2

37.

36. ______________________

x 4

ln⎛⎜ + ⎞⎟ dx ∫ x 3

37. ______________________

⎝ − ⎠

38. Use a grapher to approximate the area between the following curves:

y = 4 xx−

2

,

x+3 x2 − 4 x , x = 0,x = 2. y=

38. ______________________

118 CALCULUS FOR LIFE SCIENCES, Chapter 5, Form B 39. Approximate the integral ∫ 4− x 2 dxusing 2

0

n=6. (a) the Trapezoid rule with

39. (a) _________________________

n = 6. (b) Simpson’s rule with

39. (b) _________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 5, Form C

Evaluate.

1.

dx ∫

1.______________________

2.

∫ 250 x 4 dx

2.______________________

3.

⎛x ∫ ⎜

12

⎝

+2ex+1 ⎞ x ⎟ dx ⎠

3. _____________________

Find the area under the curve over the indicated interval.

4.

y=4x−x2;[0,4]

6 5. y = x ; [1, 4]

4. ______________________

5. ______________________

6. Give two interpretations of the shaded region.

6.______________________

Evaluate.

7.

8.

9.

2 x 4 x ) dx ∫ ∫( + 2

3

−3

4

7. ______________________

e−6 x dx

8. ______________________

∫ xdx

9. ______________________

0

0

−a

3

120 CALCULUS FOR LIFE SCIENCES, Chapter 5, Form C is positive, negative, or zero.

10. Decide whether∫f ( x ) dx b

a

10. ___________________________

Evaluate using substitution. Do not use Table 1 or CAS.

dx

11.

∫ x 6

11. ___________________________

12.

∫ edx

12. ___________________________

13.

∫excos(ex)dx

13. ___________________________

+

−0.2x

Evaluate using integration by parts. Do not use Table 1 or CAS.

14.

7 xcosx() dx ∫

14. ___________________________

15.

x8lnx4dx ∫

15. ___________________________

Evaluate using Table 1 or CAS.

16.

∫8dx

17.

∫

x

dx

x + 25 2

−[ 3,4]. y = 2 t3 − 3tover 18. Find the average value of

16. ___________________________

17. ___________________________

18. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 5, Form C 121 19. Find the area of the region bounded yby= x , y = x5 , x = 0, and x = 1.

20. Approximate ∫x2 + 2 4

0

(

) dx using a Reimann sum withn = 4.

19. ______________________

20. ______________________

21.Weight-Height. The rate of change of weight for an average adult male is given by:

W′ ( h )=0.0018 h 2 where W(h)is the weight with respect to height, h, in inches.

An average 60 in male weighs 112 pounds. What is the weight of an average adult male who is 6 ft tall?

21. ______________________

22. Distance. A particle is initially at the origin. Its velocity, in meters per second, at any time≥t,0, t is given by

v(t)=t2+2t. Find the distance that the particle travels

in the first 4 seconds (from t=0 to t= 4).

22. ______________________

Integrate using any method.

dx

23.

∫ x

24.

∫x10 4 exdx

24. ______________________

25.

xe dx ∫

25. ______________________

26.

x ln xdx ∫

26. ______________________

27.

x x +1 d x ∫

27. ______________________

28.

dx ∫ 49 − x 2

28. ______________________

29.

x4e ∫

29. ______________________

(3−x)

6

x7

1

5

3

dx

−1.1 x

23. ______________________

122 CALCULUS FOR LIFE SCIENCES, Chapter 5, Form C 30.

∫ sin x (4x)dx 2

30. ___________________________

Find the volume generated by revolving about the x-axis the regions bounded by the following graphs.

31. y=

7−x,x=2,x=6

31. ___________________________

32. y=

2 ,x=1,x=5 x

32. ___________________________

33. Find the volume of a pyramid 15 ft high with a 3-ft-by-3ft square base.

33. ___________________________

Determine whether the improper integral is convergent or divergent, and calculate its value if it is convergent.

34.

∫4e−xdx

34. ___________________________

35.

∫

35. ___________________________

∞

0

∞

1

1

x

dx

________________________________________________________________ Evaluate using any method.

⎡10 + 5( ln)5x− 2( lnx)10 ⎤ ⎦ dx x

36.

⎣ ∫

37.

ln⎛⎜ + ⎞⎟ dx ∫ x 2

x 8

36. ___________________________

37. ___________________________

⎝ − ⎠

38. Use a grapher to approximate the area between the following curves:

y = x2 − 2 x , y − 2 x3 − x 2+6 x , x = 0, x = 2. −

38. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 5, Form C 123

3 x + 5dx using 39. Approximate the integral ∫ 1 3

(a) the Trapezoid rule with n = 6.

39. (a) ____________________

n = 6. (b) Simpson’s rule with

39. (b) ____________________

CALCULUS FOR THE LIFE SCIENCES

Name: ______________________

Chapter 5, Form D

Evaluate.

1.

dx ∫

1.___________________________

2.

∫ 60 x 4 dx

2.___________________________

3.

⎛ ∫ ⎜

3ex+5+x x ⎝

56

⎞ dx ⎟ ⎠

3. __________________________

Find the area under the curve over the indicated interval.

4.

y=3x−x2;[0,3]

2 5. y = x ; [1, 4]

4. ___________________________

5. ___________________________

6. Give two interpretations of the shaded region.

6.___________________________

Evaluate.

7.

8.

9.

4 x +10 x ) dx ∫ ∫( 3

3

−2

5

7. ___________________________

e− dx

8. ___________________________

∫x dx

9. ___________________________

1

3a

a

2x

3

CALCULUS FOR THE LIFE SCIENCES, Chapter 5, Form D 125 is positive, negative, or zero.

10. Decide whether∫f ( x ) dx b

a

10. ______________________

Evaluate using substitution. Do not use Table 1 or CAS.

dx

11.

∫ x 2

11. ______________________

12.

e−0.25 dxx ∫

12. ______________________

13.

∫3x2sin(x3)dx

13. ______________________

+

Evaluate using integration by parts. Do not use Table 1 or CAS.

14.

x () dx − 2 xsin ∫

14. ______________________

15.

∫x5lnx6dx

15. ______________________

Evaluate using Table 1 or CAS.

16.

∫1xd1x

17.

∫

dx

x2 − 9

3,4]. y = 5t 4 + 2tover−[ 18. Find the average value of

16. ______________________

17. ______________________

18. ______________________

126 CALCULUS FOR LIFE SCIENCES, Chapter 5, Form D 19. Find the area of the region bounded yby= 9 x , y = x3 , x = −3, and x = 0.

(16− x ) dx using a Reimann sum withn = 4.

20. Approximate ∫ 4

0

2

19. ___________________________

20. ___________________________

21.Area of a Healing Wound. The area Aof a healing would is decreasing at a rate of

A′ ( t )= − 3.5 t− 3,1 ≤t≤ 10 where tis time in days, and Ais in square centimeters. On day one, the wound had an area of 2cm2. What will the area of the wound be in 8 days?

21. ___________________________

22. Distance. A particle is initially at the origin. Its velocity, in meters per second, at any time≥t,0, t is given by

v(t)=6t2+2t. Find the distance that the particle travels

in the first 4 seconds (from t=0 to t= 4).

22. ___________________________

Integrate using any method.

dx

23.

∫ x (12 −x )

23. ___________________________

24.

x 4 exdx ∫

24. ___________________________

25.

xe dx ∫

25. ___________________________

26.

x 2 ln xdx ∫

26. ___________________________

27.

x x + 3 dx ∫

27. ___________________________

28.

∫ 81 − x 2

28. ___________________________

29.

x 2 e− dx ∫

29. ___________________________

8 x9

1

5

3

dx

0.8x

CALCULUS FOR THE LIFE SCIENCES, Chapter 5, Form D 127 30.

∫3x2cos(x)dx

30. ______________________

Find the volume generated by revolving about the x-axis the regions bounded by the following graphs.

31. y = 3 x − 4,x = 4,x=9

32. y =

5

x

,x = 1,x = 5

31. ______________________

32. ______________________

33. Find the volume of a pyramid 18 ft high with a 10-ft-by-10-ft square base.

33. ______________________

Determine whether the improper integral is convergent or divergent, and calculate its value if it is convergent.

34.

dx ∫ x1.01

34. ______________________

35.

∫4e2xdx

35. ______________________

∞

7

1

∞

1

___________________________________________________________ Evaluate using any method.

⎡(lnx) − 3( lnx) + 4⎤ ⎦ ⎣ dx 36. ∫ x 4

37.

2

ln (x − 5 )(x + 3 ) dx ∫

36. ______________________

37. ______________________

38. Use a grapher to approximate the area between the following curves:

y = 4 x − 2 x 2 + 4, y = 4 x 3 − 6 x 2+ x +4, x = 0, x = 1.5.

38. ______________________

128 CALCULUS FOR LIFE SCIENCES, Chapter 5, Form D 4 x3 16 dx using 39. Approximate the integral ∫ 2 6. (a) the Trapezoid rule with n=

n = 6. (b) Simpson’s rule with

39. (a) _________________________

39. (b) _________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 5, Form E

Evaluate.

1.

dx ∫

1.______________________

2.

∫180 x8dx

2.______________________

3.

⎛ x +ex+x3 ∫ ⎜ 3

4

⎝

⎞ dx ⎟ ⎠

3. _____________________

Find the area under the curve over the indicated interval.

4.

y=6x−x2;[0,6]

5 5. y = x ; [1, 10]

4. ______________________

5. ______________________

6. Give two interpretations of the shaded region.

6.______________________

Evaluate.

7.

8.

9.

2 x +12 x ) dx ∫ ∫( 2

5

−1

6 0

e− dx 12 x

dx

∫x 2 a2

1

7. ______________________

8. ______________________

9. ______________________

130 CALCULUS FOR LIFE SCIENCES, Chapter 5, Form E is positive, negative, or zero.

10. Decide whether∫f ( x ) dx b

a

10. ___________________________

Evaluate using substitution. Do not use Table 1 or CAS.

dx

11.

∫ x 21

11. ___________________________

12.

e−0.2 x dx ∫

12. ___________________________

13.

ecost sintdt ∫

13. ___________________________

−

Evaluate using integration by parts. Do not use Table 1 or CAS.

14.

∫ 4 xcos (2 x ) dx

14. ___________________________

15.

x8lnx7 dx ∫

15. ___________________________

Evaluate using Table 1 or CAS.

16.

∫11dx

17.

dx ∫ 5+2 x

x

x

y = 4 t3 + 6 t2 over [−3, − 1].

18. Find the average value of

16. ___________________________

17. ___________________________

18. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 5, Form E 131 4 19. Find the area of the region boundedyby = x, y = x , x = −1, and x = 0.

5 4 20. Approximate ∫x dx using a Reimann sum withn = 4.

1

19. ______________________

20. ______________________

21.Memory. In a certain memory experiment, the rate of memorizing is given by:

M ′(t=)−0.005t2+0.2t

where M(t)is the number of Spanish words memorized in tminutes. The subject has no Spanish words memorized at the beginning of the experiment. How many words are memorized in 15 minutes? (Round answer to the nearest whole number.)

21. ______________________

22. Distance. A particle is initially at the origin. Its velocity, in meters per second, at any time t, t≥0, is given by

v(t)=2t2+6t. Find the distance that the particle travels

in the first 3 seconds (from t=0 to t=3).

22. ______________________

Integrate using any method.

dx

23.

∫ x(2−x)

23. ______________________

24.

x 3 exdx ∫

24. ______________________

25.

xe dx ∫

25. ______________________

26.

∫ ln xdx

26. ______________________

27.

x x + 6dx ∫

27. ______________________

28.

∫ 16 − x 2

x5

4

1

x

5

3

dx

28. ______________________

132 CALCULUS FOR LIFE SCIENCES, Chapter 5, Form E 29.

∫x3e0.25xdx

29. ___________________________

30.

∫5x2sin(5x)dx

30. ___________________________

Find the volume generated by revolving about the x-axis the regions bounded by the following graphs.

31. y = 3 x + 1,x = −1,x=2

1

32. y =

x2

,x = 1,x = 2

31. ___________________________

32. ___________________________

33. Find the volume of a pyramid 25 ft high with a 50-ft-by-50-ft square base.

33. ___________________________

Determine whether the improper integral is convergent or divergent, and calculate its value if it is convergent.

34.

x ∫5e−9dx

34. ___________________________

35.

∫

35. ___________________________

∞

0

∞

5

1

x

dx

________________________________________________________________ Evaluate using any method.

⎡8( lnx) − 2( lnx) − 6⎤ ⎦ ⎣ dx 36. ∫ x 3

37.

2

x 5

ln⎛⎜ − ⎞⎟ dx ∫ x 2

36. ___________________________

37. ___________________________

⎝ + ⎠

38. Use a grapher to approximate the area between the following curves:

y = 3 x − 2 x 2 +2, y = 4 x3 − 2 x 2 − x + 2, x = 0, x = 1.

38. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 5, Form E 133 39. Approximate the integral ∫2sin xdxusing 3

1

(a) the Trapezoid rule with =n 6.

39. (a) ____________________

n = 6. (b) Simpson’s rule with

39. (b) ____________________

CALCULUS FOR THE LIFE SCIENCES

Name: ______________________

Chapter 5, Form F

Evaluate.

1.

dx ∫

1.___________________________

2.

∫100 x 4 dx

2.___________________________

3.

⎛ ∫ ⎜

3ex+1+x x ⎝

32

⎞ dx ⎟ ⎠

3. __________________________

Find the area under the curve over the indicated interval.

4.

y=8x−x2;[0,8]

4 5. y = x ; [1, 10]

4. ___________________________

5. ___________________________

6. Give two interpretations of the shaded region.

6.___________________________

Evaluate.

7.

8.

9.

6 x 2 x ) dx ∫ ∫( + 3

2

−2

4

7. ___________________________

e− d x

8. ___________________________

∫ xdx

9. ___________________________

0

a

1

8x

CALCULUS FOR THE LIFE SCIENCES, Chapter 5, Form F 135 is positive, negative, or zero.

10. Decide whether∫f ( x ) dx b

a

10. ______________________

Evaluate using substitution. Do not use Table 1 or CAS.

dx

11.

∫ x

11. ______________________

12.

∫ edx

12. ______________________

13.

∫x2cos(x3+5)dx

13. ______________________

−1

−0.4x

Evaluate using integration by parts. Do not use Table 1 or CAS.

14.

∫ 3 xsin (5 x ) dx

14. ______________________

∫x9lnx10dx

15. ______________________

15.

Evaluate using Table 1 or CAS.

16.

2xdx ∫

17.

∫

dx

x 1+ x2

y = −3t2 + 5tover [− 2,1].

18. Find the average value of

16. ______________________

17. ______________________

18. ______________________

136 CALCULUS FOR LIFE SCIENCES, Chapter 5, Form F 5 19. Find the area of the region boundedyby = x, y = x , x = −1, and x = 0.

19. ___________________________

20. Approximate ∫x2

20. ___________________________

4

0

dxusing a Reimann sum withn = 4. ( +10)

21.Population Growth. The population of an urban city grows at a rate given by:

N′t( ) =500+550 t ,0

≤t≤9

where N(t)is the population of the city at time t, given in years. The population at the time of incorporation was 7500.

What is the population 7 years later?

21. ___________________________

22. Distance. A particle is initially at the origin. Its velocity, in meters per second, at any time≥t,0, t is given by

v(t)=t2+2t. Find the distance that the particle travels

in the first 3 seconds (from t=0 to t= 3).

22. ___________________________

Integrate using any method.

dx

23.

∫ x(4−x)

23. ___________________________

24.

3 x 5 exdx ∫

24. ___________________________

25.

xe dx ∫

25. ___________________________

26.

∫x ln x d x

26. ___________________________

27.

x x + 2dx ∫

27. ___________________________

28.

dx ∫ 25 − x 2

28. ___________________________

29.

x2e ∫

29. ___________________________

2

x3

4

7

4

dx

−0.5 x

CALCULUS FOR THE LIFE SCIENCES, Chapter 5, Form F 137 30.

x2cos(3x)dx ∫ 2 1

30. ______________________

Find the volume generated by revolving about the x-axis the regions bounded by the following graphs.

31. y = 2 5 − x,x = 0,x=5

32. y =

31. ______________________

1 , x = 1,x = 3 4x

32. ______________________

33. Find the volume of a pyramid 12 ft high with a 20-ft-by-20-ft square base.

33. ______________________

Determine whether the improper integral is convergent or divergent, and calculate its value if it is convergent.

34.

∞ 6 dx ∫ x

34. ______________________

35.

∫ 4 x dx

35. ______________________

1

5

∞

0

2+

___________________________________________________________ Evaluate using any method.

⎡⎣3 − 2( lnx )2+ 4( lnx) ⎤ ⎦ dx 36. ∫ x 3

37.

ln (x − 1)(x + 6 ) dx ∫

36. ______________________

37. ______________________

38. Use a grapher to approximate the area between the following curves:

y = 10xx+ 2 , y = x3x + 2 + x, x = 0,x =3.

38. ______________________

138 CALCULUS FOR LIFE SCIENCES, Chapter 5, Form F 2 6 dxusing 39. Approximate the integral ∫ 0 x +2 (a) the Trapezoid rule with n = 6.

n = 6. (b) Simpson’s rule with

39. (a) _________________________

39. (b) _________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 6, Form A

Compute.

8 5 4 1. ⎡⎢ −2 ⎤⎡ ⎣ 1 3 ⎦⎥ ⎣1 2 ⎢

⎡0 6 2. ⎢1 4 ⎢ ⎢3 4 ⎣

−3⎤ 7 ⎥⎦

1.______________________

−2⎤ ⎡11 −6 ⎤ 2 ⎥ ⎢ 5 −1 ⎥ ⎥ ⎥⎢ 1 ⎦⎢⎣ ⎥ 3 7⎥ ⎦

2.______________________

⎡ 4⎤ 3. ⎢ −7⎥ [6 −3 ⎣ ⎦

⎡−2 1 ⎣9 4

2 +⎢ −]

4⎤ 0 ⎥⎦

3. ______________________

Solve the following systems of linear equations using Gaussian elimination.

4.

x− 2y= 1 5 2x− y =

+ x + 2 y +2 z = 7 5. 6 x + 11 y +9 z = 35 x y − 2 z = −3

4. ______________________

5.______________________

Find the determinant of the matrix and determine if the matrix is invertible.

3 6 6. ⎡⎢ −2 4 ⎤⎥ ⎣− ⎦

6.______________________

⎡9 6⎤ ⎥ ⎢ 2⎦ ⎣3

7.______________________

⎡6 −3 −12 ⎤ ⎥ 8. ⎢ ⎥ −1 −2 ⎢2 ⎢⎣6 −2 −1 ⎥⎦

8.______________________

⎡ 5 0 3⎤ 9. ⎢ −2 1 4 ⎥ ⎢ ⎥ ⎢ 4 6 9⎥ ⎣ ⎦

9.______________________

7.

140 CALCULUS FOR THE LIFE SCIENCES, Chapter 6, Form A Find the inverse matrix.

1 1 10. ⎡⎢2 ⎤⎥ ⎣ 3⎦

10. ___________________________

⎡2 1 1 ⎤ 11. ⎢ 2 1 0 ⎥ ⎢ ⎥ ⎢3 0 3 ⎥ ⎣ ⎦

11. ___________________________

Find the eigenvalues and the corresponding eigenvectors.

2 12. ⎡⎢1 ⎣

−12 ⎤ ⎥ −5 ⎦

12. ___________________________

⎡4 −5⎤ ⎥ ⎢ −3⎦ ⎣2

13. ___________________________

3 2⎤ 14. ⎡⎢ ⎥ ⎣ −1 0 ⎦

14. ___________________________

⎡ 5 0 1⎤ 15. ⎢ 1 1 0 ⎥ ⎢ ⎥ ⎢ −7 1 0 ⎥ ⎣ ⎦

15. ___________________________

⎡2 0 1 ⎤ 16. ⎢ 0 3 4 ⎥ ⎢ ⎥ ⎢0 0 1 ⎥ ⎣ ⎦

16. ___________________________

13.

Find the particular solution to the difference equation.

xn+1 = 2xn+ 3xn−1;x0= 1,x1 = −9

17. ___________________________

18. xn+1 = −5xn+ 14xn+1+ 16;x0 = 3,x1 = −18

18. ___________________________

17.

CALCULUS FOR THE LIFE SCIENCES, Chapter 6, Form A 1 41

19.Home Air Quality.Chemicals enter a house’s basement air. LetF (t and Btbe ( ) the amount of chemical

)

(in mgm3)in the first floor air and the basement air

after tminutes, respectively. Let’s suppose that the rate of change of Fand Bare given by the equations: B′=−0.012B+0.12

and

F′=0.015B−0.02F. Find the equilibrium values for BandF.

19. ______________________

. Leslie matrix for a population of 20. Bird PopulationThe birds is ⎡ 0.7 2 ⎤ G = ⎢ 0.5 0.4⎥ ⎣ ⎦ The population vector for year 1 is

⎡hatchlings⎤ ⎡180⎤ . ⎥= p=⎢ ⎣ adults ⎦ ⎢⎣ 150⎥ ⎦ (a) Estimate the population vector for year 2.

20. (a) ____________________

(b) Estimate the population vector for year 3.

20. (b) ____________________

21. Bird Population. The 2×2Leslie matrix for a population of birds has eigenvalues r=1.2 and r=0.8. Find the long-term growth rate for the population.

21. ______________________

___________________________________________________________ ⎡1 ⎡1 3⎤ 22. Let A=⎢ 2⎤ and B=⎢ ⎥. ⎣3 1⎥ ⎣5 1⎦ ⎦ 1. (a) Compute A−

22.

(b) Compute B−1.

__________________

(c)

Compute

A−1B−1.

1 (d) Compute (AB )− . (e) Compute ( BA)−1 .

(a)

_______ 22.

(b)

22. (d) ____________________ __________________

_______ 22. (c) 22. (e) ____________________ __________________ _______

142 CALCULUS FOR THE LIFE SCIENCES, Chapter 6, Form A 23. Bird Population. The Leslie matrix for a population of birds is ⎡0.6 3.2⎤ G=⎢ ⎣0.8 0.4⎥ The population vector⎦ for year 1 is ⎡hatchlings⎤ ⎡210⎤ . ⎥= p=⎢ ⎣ adults ⎦ ⎢⎣ 180⎥ ⎦ (a) Estimate the population vector for year 2.

(b) Estimate the population vector for year 3. (c) Estimate the population vector for year 15. (d) Estimate the long-term growth rate.

23. (a) _________________________

23. (b) _________________________

23. (c) _________________________

23. (d) _________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 6, Form B

Compute.

⎡ 4 0⎤ 2 −1 7 ⎤⎢ ⎡ 1. ⎢ 3 1⎥ ⎥⎢ ⎥ 3 0 5 ⎣ ⎦ ⎣⎢ −1 −2 ⎦⎥

1.______________________

9 7⎤ 1 2. ⎡⎢ ⎡ ⎣2 5 ⎥ 0 ⎢ ⎦ ⎣

2 −9⎤ 3 4 ⎥⎦

2.______________________

⎡2 0 ⎤ 3 3. ⎢6 1 ⎥⎡ ⎢ ⎥⎢ 6 ⎢7 5 ⎥⎣ ⎣ ⎦

⎡ 0 −3⎤ −7⎤ ⎢ 2 5 ⎥ ⎥ 0 ⎥⎦ + ⎢ − ⎢1 4⎥ ⎣ ⎦

3. _____________________

Solve the following systems of linear equations using Gaussian elimination.

4.

7 x + 3 y = 14 4 3x + y =

4.______________________

− 2 x − 2 y − z = 13 2 x + y + z − 14 5.

3x + y + 2 z = 25

5.______________________

= Find the determinant of the matrix and determine if the matrix is invertible.

10 6. ⎡⎢ ⎣6

5⎤ 3⎥ ⎦

6.______________________

⎡ −1 −2⎤ ⎢2 4 ⎥ ⎣ ⎦

7.______________________

⎡0.3 2 2.1⎤ 04 5.1⎥⎥ 8. ⎢ 3.1 ⎢ ⎢ 8.2⎥⎦ ⎣7.2

8.______________________

⎡1 9. ⎢0 ⎢ ⎢2 ⎣

9.______________________

7.

−3 0 ⎤ 1 1⎥ ⎥ −1 4 ⎥⎦

144 CALCULUS FOR THE LIFE SCIENCES, Chapter 6, Form B

Find the inverse matrix.

5 1 10. ⎡⎢ 2 0 ⎤⎥ ⎣ ⎦

⎡ 3 1 11. ⎢ −1 0 ⎢ ⎢ 1 1 ⎣

10. ___________________________

0⎤ 2⎥ ⎥ −2⎥⎦

11. ___________________________

Find the eigenvalues and the corresponding eigenvectors.

2 12. ⎡⎢1 ⎣

6⎤ −3⎥ ⎦

12. ___________________________

13.

⎡1 −1⎤ ⎥ ⎢ 4⎦ ⎣2

13. ___________________________

14.

⎡7 2⎤ ⎥ ⎢⎣19 4⎦

14. ___________________________

⎡1 −2 1⎤ 1 15. ⎢ ⎢ 0 0 ⎥4⎥ 3⎥⎦ ⎢ ⎣0

15. ___________________________

⎡5 16. ⎢0 ⎢ ⎢1 ⎣

16. ___________________________

6

2⎤ ⎥ −1 −8 ⎥ 0 −2 ⎥ ⎦

Find the particular solution to the difference equation.

xn+1 = 3xn+ 2xn−1;x0= 3,x1 = 2

17. ___________________________

18. xn+1 = 11xn+ 18xn+1− 84;x0 = 5,x1 = −14

18. ___________________________

17.

CALCULUS FOR THE LIFE SCIENCES, Chapter 6, Form B 1 45

19.Home Air Quality.Chemicals enter a house’s basement air. LetF (t and Btbe ( ) the amount of chemical

)

(in mgm3)in the first floor air and the basement air

after tminutes, respectively. Let’s suppose that the rate of change of Fand Bare given by the equations: B′=−0.017B+0.85

and F′=0.015B−0.0125F. Find the equilibrium values for BandF.

19. ______________________

. Leslie matrix for a population of 20. Bird PopulationThe birds is ⎡1.2 2. ⎤ G=⎢ ⎥ ⎣ 0.7 5 ⎦ 0.

The population vector 3 for year 1 is hatchlings ⎤ ⎡ 72 ⎤ p = ⎡⎢ ⎥ = ⎢ 40 ⎥ . ⎣ adults ⎦ ⎣ ⎦ (a) Estimate the population vector for year 2.

20. (a) ____________________

(b) Estimate the population vector for year 3.

20. (b) ____________________

21.Bird Population.The 2 × 2Leslie matrix for a r=2.1 and population of birds has eigenvalues r=0.8. Find the long-term growth rate for the population.

21. ______________________

___________________________________________________________ ⎡1

⎡1 5⎤ −3⎤ ⎥ and B= . ⎣−2 1 ⎦ ⎢4 1⎥ ⎣ ⎦ (a) Compute A−1.

22.

(b) Compute B−1. (c)

__________________

22. Let A=⎢

Compute A−1B−1.

(a)

_______ 22.

(b)

(d) Compute (AB )− .

22. (d) _____________________ __________________

(e) Compute ( BA)−1 .

_______ 22. (c) 22. (e) ____________________

1

__________________ _______

146 CALCULUS FOR THE LIFE SCIENCES, Chapter 6, Form B 23. Bird Population. The Leslie matrix for a population of birds is ⎡0.4 1.5⎤ G=⎢ ⎣0.5 0.3⎥ The population vector⎦ for year 1 is ⎡hatchlings⎤ ⎡160⎤ . ⎥= p=⎢ ⎣ adults ⎦ ⎢⎣ 150⎥ ⎦ (a) Estimate the population vector for year 2.

(b) Estimate the population vector for year 3. (c) Estimate the population vector for year 10. (d) Estimate the long-term growth rate.

23. (a) _________________________

23. (b) _________________________

23. (c) _________________________

23. (d) _________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 6, Form C

Compute.

8 1 ⎤ −1 4 5 ⎤ 1. ⎡⎢ ⎡ ⎥ ⎣2 0 ⎥ 3 2 −4⎦ ⎢ ⎦ ⎣

⎡ −34 2. ⎢ 510 ⎢ ⎢ 01 ⎣

⎤ ⎥⎡ 7 ⎥⎢ 0 ⎥⎣ ⎦

8⎤ −2⎥ −1 ⎦ 1

⎡ 4 −1 3 ⎤ 2 9⎥ ⎢ ⎥ ⎢2 8 0 ⎥ ⎣ ⎦

⎡ −3⎤ 3. ⎢ 1 ⎥ 7 8 ⎢ ⎥ ⎢2[ ⎣ ⎥ ⎦

⎢ 11] + 4

1.______________________

2.______________________

3. _____________________

Solve the following systems of linear equations using Gaussian elimination.

4.

11x + 5 y

33

3x + =

5

4.______________________

y − 2 x −= y 5.

= 1 x + y − z =− 6 2 x + y − 4 z = −13

5.______________________

Find the determinant of the matrix and determine if the matrix is invertible.

3 −12 ⎤ 6. ⎡⎢ ⎥ ⎣ −1 4 ⎦

6.______________________

−8 4 7. ⎡⎢ 4 2 ⎤⎥ ⎣− ⎦

7.______________________

⎡0.2 3.4 0.3⎤ 8. ⎢ 0.5 ⎢ 1.2 ⎥ ⎥⎥⎦ 0.2 1.7 8.2 ⎢ ⎣1.3

8.______________________

⎡4 9. ⎢1 ⎢ ⎢3 ⎣

9.______________________

4 0⎤ ⎥ −2 7 ⎥ −5 6 ⎥⎦

148 CALCULUS FOR THE LIFE SCIENCES, Chapter 6, Form C

Find the inverse matrix.

2 10. ⎡⎢7 ⎣

−1⎤ 8 ⎥⎦

10. ___________________________

⎡1 11. ⎢2 ⎢ ⎢0 ⎣

−2 −1⎤ 1 1⎥ ⎥ 1 0⎥ ⎦

11. ___________________________

Find the eigenvalues and the corresponding eigenvectors.

5 3 12. ⎡⎢ −6 − ⎤⎥ −2 ⎦ ⎣

12. ___________________________

− 9⎤ ⎥ −2⎦

13. ___________________________

8 2⎤ 14. ⎡⎢ ⎥ − 17 −5⎦ ⎣

14. ___________________________

10 13. ⎡⎢ ⎣4

4⎤ ⎥ −2⎥ 3⎥ ⎦

15. ___________________________

−2 −2⎤ 1 0⎥ ⎥ 0 1⎥ ⎦

16. ___________________________

⎡ −5 3 7 15. ⎢ 0 ⎢ 0 0 ⎢ ⎣

⎡4 16. ⎢0 ⎢ ⎢1 ⎣

Find the particular solution to the difference equation.

17.

xn+1 = −xn + 20xn−1;x0= 5,x1 = −7

18. xn+1 =3xn− 4xn+1+5;x0 = 4,x1 = −34

17. ___________________________

18. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 6, Form C 1 49

19.Home Air Quality.Chemicals enter a house’s basement air. LetF (t and Btbe ( ) the amount of chemical

)

(in mgm3)in the first floor air and the basement air

after tminutes, respectively. Let’s suppose that the rate of change of Fand Bare given by the equations: B′=−0.018B+0.72

and

F′ =0.015 B− 0.036 F .

Find the equilibrium values for BandF.

19. ______________________

20. Bird Population. The Leslie matrix for a population of birds is

⎡ 1.9 2 ⎤ G=⎢ ⎥ ⎣0.6 1.8⎦ The population vector for year 1 is ⎡hatchlings⎤ ⎡120⎤ ⎥. ⎥=⎢ p=⎢ ⎣ adults ⎦ ⎣68⎦

(a) Estimate the population vector for year 2.

20. (a) ____________________

(b) Estimate the population vector for year 3.

20. (b) ____________________

21. Bird Population. The 2×2Leslie matrix for a population of birds has eigenvalues r=1.7 and r=0.6. Find the long-term growth rate for the population.

21. ______________________

___________________________________________________________ ⎡1 6⎤ ⎡1 2⎤ and B=⎢ ⎥. ⎣3 7⎦ ⎣1 7⎥ (a) Compute ⎦A−1.

22.

(b) Compute B−1.

__________________

22. Let A=⎢

(c)

Compute

A−1B−1.

1 (d) Compute (AB )− . (e) Compute ( BA)−1 .

(a)

_______ 22.

(b)

22. (d) ____________________ __________________

_______ 22. (c) 22. (e) ____________________ __________________ _______

150 CALCULUS FOR THE LIFE SCIENCES, Chapter 6, Form C 23. Bird Population. The Leslie matrix for a population of birds is ⎡0.8 1.2⎤ G=⎢ ⎣0.7 0.4⎥ The population vector⎦ for year 1 is ⎡hatchlings⎤ ⎡225⎤ . ⎥= p=⎢ ⎣ adults ⎦ ⎢⎣ 185⎥ ⎦ (a) Estimate the population vector for year 2.

(b) Estimate the population vector for year 3. (c) Estimate the population vector for year 25. (d) Estimate the long-term growth rate.

23. (a) _________________________

23. (b) _________________________

23. (c) _________________________

23. (d) _________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 6, Form D

Compute.

6 ⎤ −10 ⎥⎦

1.______________________

⎡ 0 1 3 2 ⎤⎢ ⎡ 2. ⎢ 9 ⎥⎢ ⎣ −1 5 −3⎦ 4 ⎢⎣

−5⎤ 1⎥ ⎥ 6⎥ ⎦

2.______________________

⎡ −2 6 ⎤ 4 5 3. ⎢ ⎡ ⎣ 1 4 ⎥ 3 −7 ⎢ ⎦ ⎣

−1⎤ ⎡−5 −2 4 ⎤ +⎢ 0 ⎥⎦ ⎣ 8 1 7 ⎥⎦

3. ______________________

7 1. ⎡⎢ ⎣4

11 ⎤ −1 1 ⎡ −2⎥ 3 13 ⎢ ⎦ ⎣

Solve the following systems of linear equations using Gaussian elimination.

x + 3 y = 12 2 x + 5 y =19

4. ______________________

14 x − 3y 4z = 88 5. 7 x + 4z = 47 + 7x − 3 4 z = 53

5.______________________

4.

y Find the determinant of the matrix and determine if the matrix is invertible. + 16 −8 ⎤ ⎡ 6. ⎢ 6.______________________ ⎥ ⎣ 8 −4 ⎦ ⎡ 4 10 ⎤ ⎢2 5 ⎥ ⎣ ⎦

7.______________________

⎡ 2 −1 −2⎤ 8⎥ 8. ⎢ −4 2 ⎢ ⎥ ⎢ 6 −2 −1⎥ ⎦ ⎣

8.______________________

⎡1.2 0.7 2.1⎤ 9. ⎢ 6.2 3.1 0.5⎥ ⎢ ⎥ ⎢⎣ 2.4 1.3 8.1⎥⎦

9.______________________

7.

152 CALCULUS FOR THE LIFE SCIENCES, Chapter 6, Form D Find the inverse matrix.

3 10. ⎡⎢ − ⎤⎥ 8 ⎣ 2 ⎦ −

10. ___________________________

5 ⎡ −2 −3 −1⎤ 11. ⎢ −3 −3 −1⎥ ⎢ ⎥ ⎢ −2 −4 −1⎥ ⎦ ⎣

11. ___________________________

Find the eigenvalues and the corresponding eigenvectors.

2 7⎤ 12. ⎡⎢ − ⎥ ⎣ −7 12 ⎦

12. ___________________________

4 13. ⎡⎢ ⎣ −7

3⎤ ⎥ −6⎦

13. ___________________________

−1 14. ⎡⎢ ⎣3

5⎤ −3⎥ ⎦

14. ___________________________

⎡1 0 −4⎤ 15. ⎢ 30 1⎥⎥ ⎢ 0 1⎥⎦ ⎢ ⎣2

15. ___________________________

⎡2 16. ⎢ 1 ⎢ ⎢ −1 ⎣

16. ___________________________

0

0⎤ ⎥ −1 0 ⎥ 2 −1⎥⎦

Find the particular solution to the difference equation.

17.

xn+1 = 7xn− 10xn−1;x0= 18,x1 = −41

18. xn+1 = 12xn + 28xn+1− 1;x0=14,x1 = 33

17. ___________________________

18. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 6, Form D 1 53

19. Equilibrium. Let’s suppose 2.5 mg of glactosyl human serum (Tc-GSA) is injected into the bloodstream. LetB and Lbe the amount of Tc-GSA in the blood and liver, respectively. The quantities Band Lare related by:

2.5=B+L dB dt =−0.06B+0.03L =0.06B−0.03L. dL dt

and

At equilibrium, the quantities Band Lare constant. Find the equilibrium amount of Tc-GSA in the blood and liver.

19. ______________________

. Leslie matrix for a population of 20. Bird PopulationThe birds is ⎡ 2.1 G=⎢ ⎣1.8

3. ⎤ 2 ⎥ 1. ⎦ 6

The population vector for year 1 is

⎡hatchlings⎤ ⎡220⎤ . ⎥= p=⎢ ⎣ adults ⎦ ⎢⎣ 145⎥ ⎦ (a) Estimate the population vector for year 2.

20. (a) ____________________

(b) Estimate the population vector for year 3.

20. (b) ____________________

21. Bird Population. The 2×2Leslie matrix for a population of birds has eigenvalues r=1.2 and r=1.4. Find the long-term growth rate for the population.

21. ______________________

___________________________________________________________ ⎡2 ⎡1 3⎤ 22. Let A=⎢ 4⎤ and B=⎢ ⎥. ⎣5 2⎥ ⎣2 5⎦ ⎦ 1. (a) Compute A−

22.

(b) Compute B−1.

__________________

(c)

Compute

A−1B−1.

1 (d) Compute (AB )− . (e) Compute ( BA)−1 .

(a)

_______ 22.

(b)

22. (d) ____________________ __________________

_______ 22. (c) 22. (e) ____________________ __________________ _______

154 CALCULUS FOR THE LIFE SCIENCES, Chapter 6, Form D 23. Bird Population. The Leslie matrix for a population of birds is ⎡0.9 1.3⎤ G=⎢ ⎣0.6 0.4⎥ The population vector⎦ for year 1 is ⎡hatchlings⎤ ⎡140⎤ . ⎥= p=⎢ ⎣ adults ⎦ ⎢⎣ 110⎥ ⎦ (a) Estimate the population vector for year 2.

(b) Estimate the population vector for year 3. (c) Estimate the population vector for year 15. (d) Estimate the long-term growth rate.

23. (a) _________________________

23. (b) _________________________

23. (c) _________________________

23. (d) _________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 6, Form E

Compute.

⎡ −3 6 ⎤ 1 1. ⎢ 0 4 ⎥⎡ −5 ⎤⎥ ⎢ ⎥⎢ 2 4 ⎦ ⎢ 2 −1⎥⎣ ⎦ ⎣

2 1 ⎤ −3 1 2. ⎡⎢ − ⎡ ⎣ −1 4 ⎥ 4 6 ⎢ ⎦ ⎣

1.______________________

5⎤ ⎥ −2⎦

⎡11 4 ⎤ 1 7 6⎤ ⎥ ⎡5 3. ⎡⎢ ⎢ 6 −3⎥ + ⎢ 2 5 4 ⎣ −2 ⎥ ⎢ 0 1 ⎥ ⎣ ⎦ ⎢⎣ ⎦

2.______________________

1⎤ ⎥ −3⎦

3. _____________________

Solve the following systems of linear equations using Gaussian elimination.

4.

6x+ 7 y = 7 70 5x− 7 y =

+ + 2 x − 3 y − z = 10 5. 2 x − 3 y + 3 z = 22 4 x 2 y 3 z = −2

4.______________________

5.______________________

Find the determinant of the matrix and determine if the matrix is invertible.

1 −2 ⎤ 6. ⎡⎢ 1⎥0 ⎣5 − ⎦

7.

⎡8 ⎢⎣16

2⎤ 4⎥ ⎦

6.______________________

7.______________________

⎡ 0.2 −1 6.3⎤ 8. ⎢ 2.4 0.7 1.4⎥ ⎢ ⎥ ⎢ −2 0.1 1.7⎥ ⎣ ⎦

8.______________________

⎡ 3 2 2⎤ 9. ⎢ 2 2 2 ⎥ ⎢ ⎥ ⎢ −4 4 3 ⎥ ⎣ ⎦

9.______________________

156 CALCULUS FOR THE LIFE SCIENCES, Chapter 6, Form E Find the inverse matrix.

7 5⎤ 10. ⎡⎢4 ⎥ ⎣ 3⎦

⎡1 11. ⎢2 ⎢ ⎢3 ⎣

10. ___________________________

2 1⎤ 5 0⎥ ⎥ 3 8⎥ ⎦

11. ___________________________

Find the eigenvalues and the corresponding eigenvectors.

3 2 12. ⎡⎢ 4 1 ⎤⎥ ⎣ ⎦

13.

12. ___________________________

⎡7 −1⎤ ⎢1 ⎥ ⎣ 5⎦

13. ___________________________

1 8⎤ 14. ⎡⎢ − ⎥ ⎣ −4 11 ⎦

14. ___________________________

⎡ 1 −2 2 ⎤ 15. ⎢ 0 −2 0 ⎥ ⎢ ⎥ ⎢ −1 −6 4 ⎥ ⎣ ⎦

15. ___________________________

⎡1 0 16. ⎢1 2 ⎢ ⎢⎣1 1

16. ___________________________

0⎤ 0⎥ ⎥ 2⎥ ⎦

Find the particular solution to the difference equation.

xn+1 = 16xn − 63xn−1;x0=18,x1 = 146

17. ___________________________

18. xn+1 = −9xn+ 26xn+1−80;x0=9,x1 = 58

18. ___________________________

17.