Type:
Test Bank
Resource:
Calculus for the Life Sciences
Edition:
1st Edition
Author(s):
Marvin L. Bittinger Neal Brand John Quintanilla
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: ______________________
Chapter 1, Form A 1. Heart Attacks and Cholesterol. The 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)
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 by f ( x ) = x 3 − 4 . Find
(a) f ( −2 ) .
2. (a) _________________________
(b) f ( x + h ) .
2. (b) _________________________
3. What are the slope and the y-intercept of y = −
4. Find an equation of the line with slope
( 3, −6 ) .
1 x −5 ? 2
3. ____________________________
2 , containing the point 3
5. Find the slope of the line containing the points ( 3, −2 ) and ( −6 , −8 ) .
4. ____________________________
5. ____________________________
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: x 2 + 4 x − 8 = 0.
11. Convert to rational exponents:
10. ____________________________
1 5
x4
12. Convert to radical notation: y − 2 3 .
.
11. ____________________________
12. ____________________________
CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form A
13. Graph: f ( x ) =
2 x
3
13. 5
y
4 3 2 1 −5
−4
−3
−2
−1
x 1
−1
2
3
4
5
−2 −3 −4 −5
14. Graph: f ( x ) =
x 2 − 16 x−4
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 ( − 5π 6 )
15. __________________________
16. sin ( 5π 6 )
16. __________________________
17. tan (π 2 )
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. 4 sin 2 t = 3
19. ____________________________
20. 2 sin 2 t − sin t − 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 sin bt + k or y = a cos bt + k , Then find a, b, and k. y
1 −π
−3π/4
−π/2
−π/4
−1
t
π/4
π/2
3π/4
π
−2 −3 −4 −5 −6 −7 −8
23. ______________________________
2 ⎪⎧ x + 1, for x ≥ 1 24. Graph: f ( x ) = ⎨ ⎪⎩ x − 2, for x < −1
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 25. (b) __________________________ in 1988. ______________________________________________________________________________________________ 26. Solve for x: 2 +
1 = −2 x . x
26. ____________________________
27. Use your grapher to graph: f ( x ) = x 3 − 4 x 2 + 2 x − 3. Then sketch the graph below. 27.
28. Find the zeros of the function: f ( x ) = 3 x 2 − 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 by f ( x ) = 2 x 3 + 4 . Find
(a) f ( −2 ) .
2. (a) _________________________
(b) f ( x + a ) .
2. (b) _________________________
3. What are the slope and the y-intercept of y = −4 x + 5 ?
4. Find an equation of the line with slope
( 5, −2 ) .
3. ____________________________
2 , containing the point 5
5. Find the slope of the line containing the points ( 4 , −2 ) and ( 8,5 ) .
4. ____________________________
5. ____________________________
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) ; (b) the domain; (c) all x-values such that f ( x ) = 0 ; and (d) the range.
9.(a) _________________________
9.(b) _________________________
9.(c) _________________________
9.(d) _________________________
10. Solve for x: x 2 − 2 x − 4 = 0.
10. ____________________________
1
11. Convert to rational exponents: 3
y2
12. Convert to radical notation: p − 3 4 .
.
11. ____________________________
12. ____________________________
8
CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form B
11.
Graph: f ( x ) =
3 x
11. 5
y
4 3 2 1 −5
−4
−3
−2
−1
x 1
−1
2
3
4
5
−2 −3 −4 −5
14.Graph: f ( x ) =
x2 − x − 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. sin ( 3π 4 )
15. __________________________
16. cos ( 5π 6 )
16. __________________________
17. tan ( −π 2 )
17. __________________________
18. Solve for the missing side, x.
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. 4 cos 2 t = 3
19. ____________________________
20. 2 cos 2 t + 3cos t + 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 ( 3π t ) + 3 2
21. ____________________________
⎛t ⎞ ⎟−2 ⎝3⎠
22. ____________________________
22. y = 2 sin ⎜
23. Determine if the graph should be modeled by either y = a sin bt + k or y = a cos bt + k , Then find a, b, and k. y
3 2 1 t
−4
−3
−2
−1
1
2
3
4
−1 −2
2 ⎪⎧ x − 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: 9 x +
8 = 12 . x
26. ____________________________
27. Use your grapher to graph: f ( x ) = 5 x 3 − 6 x 2 + x − 2. Then sketch the graph below. 27.
28. Find the zeros of the function: f ( x ) =
x2 − 4 − 5 − 6 .
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 by f ( x ) = 3x 2 − 5 . Find
(a) f ( −7 ) .
2. (a) _________________________
(b) f ( x + 1) .
2. (b) _________________________
3. What are the slope and the y-intercept of y = −
2 x +8 ? 3
3. ____________________________
5 4. Find an equation of the line with slope − , containing the point 8 ( 4 , 0) .
4. ____________________________
5. Find the slope of the line containing the points ( 2 ,9 ) and ( −5, −6 ) .
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) the domain; (c) all x-values such that f ( x ) = 1 ; and (d) the range.
9.(a) _________________________
9.(b) _________________________
9.(c) _________________________
9.(d) _________________________
10. Solve for x: x 2 + 8 x + 13 = 0.
11. Convert to rational exponents: 3
10. ____________________________ 5 x
12. Convert to radical notation: a − 6 7 .
.
11. ____________________________
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
−1
2
3
4
5
−2 −3 −4 −5
14.
Graph: f ( x ) =
x 2 − 3 x − 10 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 ( 7π 6 )
15. __________________________
16. sin (π 2 )
16. __________________________
17. tan ( 3π 2 )
17. __________________________
18. Solve for the missing side, x.
7.93
x
35
18. ____________________________
14
CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form C
Find all values of t that satisfy the equation. 19. 2 sin 2 t = 1
19. ____________________________
20. cos 2 t + cos t = 0
20. ____________________________
For each of the following functions, find the amplitude, period, and midline. Also, find the maximum and minimum. ⎛t ⎞ ⎟−5 ⎝2⎠
21. ____________________________
1 3 sin ( 3t ) + 2 2
22. ____________________________
21. y = 2 cos ⎜ 22. y =
23. Determine if the graph should be modeled by either y = a sin bt + k or y = a cos bt + k , Then find a, b, and k. y
1 t
−3
−2
−1
1
2
3
−1 −2 −3 −4 −5
23. _____________________________
−6
2 ⎪⎧ − x − 3, for x ≥ −2 24. Graph: f ( x ) = ⎨ ⎪⎩ 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: 4 x +
11 = 12 . x
26. ____________________________
27. Use your grapher to graph: f ( x ) = 2 x 3 − 5 x 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 seconds t 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 by f ( x ) = 2 x 2 + 3 . Find
(a) f ( −1) .
2. (a) _________________________
(b) f ( a − 3) .
2. (b) _________________________
3. What are the slope and the y-intercept of y = 5 x − 4 ?
4. Find an equation of the line with slope
( 6, −2 ) .
3. ____________________________
5 , containing the point 8
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: x 2 + 8 x + 13 = 0.
11. Convert to rational exponents: 3
10. ____________________________ 5 n
.
11. ____________________________
18
CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form D
12. Convert to radical notation: y − 4 7 .
13.
Graph: f ( x ) =
−3 x−4
12. ____________________________
13. 5
y
4 3 2 1 −5
−4
−3
−2
−1
x 1
−1
2
3
4
5
−2 −3 −4 −5
14.Graph: f ( x ) =
x2 − 9 x+3
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. sin ( −5π 4 )
15. __________________________
16. cos ( − 2π 3)
16. __________________________
17. tan ( 3π 2 )
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. 3 tan 2 t = 1
19. ____________________________
20. 2 sin 2 t + sin t − 1 = 0
20. ____________________________
For each of the following functions, find the amplitude, period, and midline. Also, find the maximum and minimum 21. y = 4 sin (π t ) + 1
21. ____________________________
22. y = 3cos ( 3t ) − 1
22. ____________________________
23. Determine if the graph should be modeled by either y = a sin bt + k or y = a cos bt + k , Then find a, b, and k.
3
y
2
1 t
−4
−3
−2
−1
1
2
3
4
−1
23. _____________________________
2 ⎪⎧ x − 4, for x ≥ 1 24. Graph: f ( x ) = ⎨ ⎪⎩ x + 3, for x < 1
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: f ( x ) = x 3 − 2 x 3 − 2 x + 3. Then sketch the graph below. 27.
28. Find the zeros of the function: f ( x ) = 3 x 2 + 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 by f ( x ) = 4 x 2 + x . Find
(a) f ( −4 ) .
2. (a) _________________________
(b) f ( x − 2 ) .
2. (b) _________________________
3. What are the slope and the y-intercept of y = −2 x − 6 ?
4. Find an equation of the line with slope −
( 2 , −8) .
3. ____________________________
1 , containing the point 4
5. Find the slope of the line containing the points ( −1,1 ) and ( 2, −8 ) .
4. ____________________________
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 lb ft 2 , 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 the surface of the liquid, the pressure is 187.5 lb ft 2 . Find an equation of variation expressing P as a function of d.
8. ____________________________
9. For the following graph of function, f, determine (a) f ( 3) ; (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: x 2 + 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 notation: y −1 2 .
13. Graph: f ( x ) =
4 x−4
12. ____________________________
13.
5
y
4 3 2 1 −5
−4
−3
−2
−1
x 1
−1
2
3
4
5
−2 −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 ( − 4π 3)
15. __________________________
16. sin (π 6 )
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. 4 sin 2 t = 1
19. ____________________________
20. 2 cos 2 t + 3cos t + 1 = 0
20. ____________________________
For each of the following functions, find the amplitude, period, and midline. Also, find the maximum and minimum 21. y = 3cos ⎜
⎛t ⎞ ⎟−2 ⎝3⎠
21. ____________________________
22. y = 2 cos (π t ) + 3
22. ____________________________
23. Determine if the graph should be modeled by either y = a sin bt + k or y = a cos bt + k , Then find a, b, and k.
6
y
5 4 3 2 1 −4π
−3π
−2π
−π
t
π
2π
3π
4π
−1 −2
⎧⎪ x 2 − 3, for x ≥ 0 24. Graph: f ( x ) = ⎨ ⎪⎩ −2 x, 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
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: 6 x +
2 = −3 . x
26. ____________________________
27. Use your grapher to graph: f ( x ) = 2 x 3 − 3x 2 + x + 2. Then sketch the graph below. 27.
28. Find the zeros of the function: f ( x ) = 2 2 − x 2 − 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 by f ( x ) = 3x 2 − x . Find
(a) f ( −4 ) .
2. (a) _________________________
(b) f ( x + a ) .
2. (b) _________________________
3. What are the slope and the y-intercept of y = −2 x + 3 ?
4. Find an equation of the line with slope −
( 5, −3) .
3. ____________________________
2 , containing the point 3
5. Find the slope of the line containing the points ( −4, −2 ) and ( 3,10 ) .
4. ____________________________
5. ____________________________
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 ) ; (b) the domain; (c) all x-values such that f ( x ) = 4 ; and (d) the range.
9.(a) _________________________
9.(b) _________________________
9.(c) _________________________
9.(d) _________________________
10. Solve for x: x 2 − 6 x + 4 = 0.
11. Convert to rational exponents:
10. ____________________________
5 x
.
11. ____________________________
28
CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form F
12. Convert to radical notation: m − 5 4 .
13. Graph: f ( x ) = −
2 x
12. ____________________________
13. 5
y
4 3 2 1 −5
−4
−3
−2
−1
x
−1
1
2
3
4
5
2
4
6
8
10
−2 −3 −4 −5
14. Graph: f ( x ) =
x2 + 2x − 8 x+4
14. 10
y
8 6 4 2 −10
−8
−6
−4
−2
−2
x
−4 −6 −8 −10
Determine the following exactly. 15. sin ( 2π 3)
15. __________________________
16. cos (11π 6 )
16. __________________________
17. tan ( 0 )
17. __________________________
18. Solve for the missing side, x.
7.93
x
35
18. ____________________________
CALCULUS FOR THE LIFE SCIENCES, Chapter 1, Form F 29
Find all values of t that satisfy the equation. 19. 4 cos 2 t = 1
19. ____________________________
20. 2 sin 2 x + 3sin x + 1 = 0
20. ____________________________
For each of the following functions, find the amplitude, period, and midline. Also, find the maximum and minimum 21. y =
1 sin ( 3t ) − 3 2
21. ____________________________
⎛t ⎞ ⎟ −1 ⎝2⎠
22. ____________________________
22. y = sin ⎜
23. Determine if the graph should be modeled by either y = a sin bt + k or y = a cos bt + k , Then find a, b, and k.
6
y
5 4 3 2 1 −2π
−3π/2
−π
−π/2
−1
t
π/2
π
3π/2
2π
−2 −3 −4
2 ⎪⎧ x + 1, for x > 0 24. Graph: f ( x ) = ⎨ ⎪⎩ x − 2, 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
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: 25 x +
12 = 30 . x
26. ____________________________
27. Use your grapher to graph: f ( x ) = − x 3 + 6 x − 3. Then sketch the graph below. 27.
28. Find the zeros of the function: f ( x ) = 3 10 − x 2 − 4 .
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 Limits Graphically. Consider the following graph of function f for Questions 1 – 6.
Find the limit, if it exists.
lim f ( x )
1. ____________________________
lim f ( x )
2. ____________________________
3. lim f ( x )
3. ____________________________
4. lim f ( x )
4. ____________________________
5. lim f ( x )
5. ____________________________
6. lim f ( x )
6. ____________________________
1. 2.
x →−3
x →−2
x →1
x→ 2
x→ 4
x →5
Determine whether the function is continuous. 7.
8.
7. ____________________________
8. ___________________________
32
CALCULUS FOR LIFE SCIENCES, Chapter 2, Form A
For the function in Question 7, answer the following. 9. Find lim f ( x ) .
9. ____________________________
10. Find f (1) .
10. ____________________________
11. Is f continuous at 1 ?
11. ____________________________
12. Find lim f ( x ) .
12. ____________________________
13. Find f ( 4 ) .
13. ____________________________
14. Is f continuous at 4 ?
14. ____________________________
x →1
x→ 4
Find the limit if it exists.
(
15. lim 2 x3 − x + 4 x→ 2
16.
lim
)
x+2
x →−2 + x 2 − 4
8 x→0 x
15. ____________________________
16. ____________________________
17. lim
17. ____________________________
18. Find a simplified difference quotient for f ( x ) = 4 x2 − 3 .
18. __________________________
19. Find an equation of the tangent line to the graph of
( )
y = 2 x + 3x at the point (1, 5 ) .
19. ____________________________
CALCULUS FOR LIFE SCIENCES, Chapter 2, Form A 33 20. Find the points on the graph of y = x3 − 3x 2 at which the tangent line is horizontal.
20. ____________________________
Find dy dx . 21.
y=3 x
22. y = −
21. ____________________________
8
22. ____________________________
x3
23. y = x 4 5
23. ____________________________
24. y = −5 x 2 + 0.4 x + 16
24. ____________________________
25. y = cot ( 3x )
25. ____________________________
Differentiate. 26.
f ( x) =
x 4− x
(
27. y = 3x 4 − 4 x 2 + 6
26. ____________________________
)
−5
27. ____________________________
28.
f ( x ) = x 2 x3 + 1
29.
f ( x ) = cos ⎜
29. ____________________________
30.
f ( x ) = cot ( x ) csc ( 4 x )
30. ____________________________
31.
f ( x) =
32.
f ( x ) = cos tan x3
⎛x⎞ ⎟ ⎝2⎠
sin 2 ( x )
tan 2 ( x ) − x
( ( ))
28. ___________________________
31. ____________________________
32. ____________________________
34 33.
CALCULUS FOR LIFE SCIENCES, Chapter 2, Form A
(
f ( x ) = x 2 + 4 + cos x 2 − x
34. For y = 4 x 6 − 9 x3 , find
d3 y dx3
)
33. ____________________________
.
34. ____________________________
35. Given that an object is at position s ( t ) = 4t + 2sin(t ) at any time t , find each of the following. a) v ( t ) .
35. (a) _________________________
b) a ( t ) .
35. (b) _________________________
c) The position, velocity and acceleration when t =
5π . 6
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 = 12 n 2 − n . a) 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) _________________________
c) Find the rate of change at n = 15 teams.
36. (c) _________________________
37. Find ( f g )( x ) and ( g f )( x ) , given that f ( x ) = x − 3x 2 and g ( x ) = x3 .
38. Differentiate y = ( 5 − 4 x )
x3 − 8 . x→ 2 x − 2
39. Find lim
40. Find lim
t
t →0 cos
(t )
.
43
( 5 + 4 x )2 3 .
37. ____________________________
38. ____________________________
39. ____________________________
40. ____________________________
CALCULUS FOR LIFE SCIENCES, Chapter 2, Form A 35 41. Using a grapher, graph f and f ′ over the given interval.
Then estimate points at which the tangent line to f is horizontal. Sketch the graphs. f ( x ) = 4 x3 − 25 x 2 + 32 x + 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, and 0.0001. When you think you know the limit, graph and use the TRACE feature to further verify your assertion.
lim
x→0
16 − 4 x − 4 x
42. _____________________________
CALCULUS FOR THE LIFE SCIENCES Name: ______________________
Chapter 2, Form B Limits Graphically. Consider the following graph of function f for Questions 1 – 6.
Find the limit, if it exists.
lim f ( x )
1. ____________________________
lim f ( x )
2. ____________________________
3. lim f ( x )
3. ____________________________
4. lim f ( x )
4. ____________________________
5. lim f ( x )
5. ____________________________
6. lim f ( x )
6. ____________________________
1. 2.
x →−5
x →−3
x→0
x →3
x →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. Find lim f ( x ) .
9. ____________________________
10. Find f ( −2 ) .
10. ____________________________
11. Is f continuous at −2 ?
11. ____________________________
12. Find lim f ( x ) .
12. ____________________________
13. Find f ( 2 ) .
13. ____________________________
14. Is f continuous at 2 ?
14. ____________________________
x →−2
x→ 2
Find the limit if it exists.
(
15. lim x 4 − 2 x 2 + 3 x →3
16.
lim −
x →−7
17. lim
)
7+ x 49 − x 2
15
15. ____________________________
16. ____________________________
17. ____________________________
x→0 x
18. Find a simplified difference quotient for f ( x ) = 3x 2 − 6 .
18. __________________________
19. Find an equation of the tangent line to the graph of
( )
y = 3 x + 8x at the point ( 2, 10 ) .
19. ____________________________
38
CALCULUS FOR LIFE SCIENCES, Chapter 2, Form B
20. Find the points on the graph of y = x3 − 2 x 2 at which the tangent line is horizontal.
20. ____________________________
Find dy dx . 21.
y = 45 x
22. y = −
21. ____________________________
11
22. ____________________________
x2
23. y = x3 5
23. ____________________________
24. y = −0.6 x 2 + 4.1x − 100
24. ____________________________
25. y = x 2 sin ( x )
25. ____________________________
Differentiate. 26.
f ( x) =
2x x+2
(
27. y = 6 x 2 + 2 x5 + x 6
26. ____________________________
)
−4
28.
f ( x ) = x 2 x3 − 2
29.
f ( x ) = sin
30.
f ( x ) = tan ( 2 x ) csc ( x )
31.
f ( x) =
32.
f ( x ) = sin sec
( x)
cos 2 ( x )
sec 2 ( x ) − x
( ( x ))
27. ____________________________
28. ___________________________
29. ____________________________
30. ____________________________
31. ____________________________
32. ____________________________
CALCULUS FOR LIFE SCIENCES, Chapter 2, Form B 39 33.
(
f ( x ) = 2 x + 5 − cos x 2 x + 1
34. For y = 5 x 4 − x 2 + 6 , find
)
d3 y dx3
33. ____________________________
.
34. ____________________________
35. Given that an object is at position s ( t ) = 2t − 5cos(t ) at any time t , find each of the following. a) v ( t ) .
35. (a) _________________________
b) a ( t ) .
35. (b) _________________________
c) The position, velocity and acceleration when t =
π 3
.
35. (c) _________________________
36. Surface Area. The surface area S of an 8-in. high can
is given by S = 2π r 2 + 16π r , where r is 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 2 in.?
36. (b) _________________________
c) Find the rate of change at r = 2 in.
36. (c) _________________________
37. Find ( f g )( x ) and ( g f )( x ) , given that f ( x ) = x 2 + 4 x and g ( x ) = x .
38. Differentiate y = ( 2 − 3x )
27 − x3 . x →3 3 − x
39. Find lim
40. Find lim t →0
cot ( t ) + cos ( t ) t
23
( 5 + x )1 2 .
37. ____________________________
38. ____________________________
39. ____________________________
.
40. ____________________________
40
CALCULUS FOR LIFE SCIENCES, Chapter 2, Form B
41. Using a grapher, graph f and f ′ over the given interval.
Then estimate points at which the tangent line to f is horizontal. Sketch the graphs. f ( x ) = 6 x3 − 20 x 2 + 10 x + 3 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, and 0.0001. 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 Limits Graphically. Consider the following graph of function f for Questions 1 – 6.
Find the limit, if it exists.
lim f ( x )
1. ____________________________
lim f ( x )
2. ____________________________
3. lim f ( x )
3. ____________________________
lim f ( x )
4. ____________________________
5. lim f ( x )
5. ____________________________
6. lim f ( x )
6. ____________________________
1. 2.
x →−5
x →−3
x→0
4.
x →−2
x →5
x→6
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. Find lim f ( x ) .
9. ____________________________
10. Find f ( 2 ) .
10. ____________________________
11. Is f continuous at 2 ?
11. ____________________________
12. Find lim f ( x ) .
12. ____________________________
13. Find f ( 3) .
13. ____________________________
14. Is f continuous at 3 ?
14. ____________________________
x→ 2
x →3
Find the limit if it exists. 15.
16.
(
lim 3x 2 − x + 10
x →−1+
)
x 2 + 3 x − 18 x →−6 x+6 lim
17. lim − x→0
3 x
15. ____________________________
16. ____________________________
17. ____________________________
18. Find a simplified difference quotient for f ( x ) = 2 x2 + 5x .
18. __________________________
19. Find an equation of the tangent line to the graph of
( )
y = 4 x + −10 at the point ( 5, 18 ) . x
19. ____________________________
CALCULUS FOR LIFE SCIENCES, Chapter 2, Form C 43 20. Find the points on the graph of y = 2 x3 − 3x 2 at which the tangent line is horizontal.
20. ____________________________
Find dy dx . 21.
y=6 x
22. y = −
21. ____________________________
7
22. ____________________________
x7
23. y = x 2 5
23. ____________________________
24. y = −3x 2 + 10.1x − 6
24. ____________________________
( )
25. ____________________________
3x 1− x
26. ____________________________
25. y = x cos x 2
Differentiate. 26.
f ( x) =
(
)
27. y = 6 x 2 − 10 x + 1
−4
27. ____________________________
28.
f ( x ) = x3 x 4 − 3
28. ___________________________
29.
f ( x ) = cos ( 3 x )
29. ____________________________
30.
f ( x ) = sin ( 4 x ) cos ( 3x )
30. ____________________________
31.
f ( x) =
cos ( x )
31. ____________________________
32.
f ( x ) = cos ( tan ( 2 x ) )
sin 2 ( x ) − x
32. ____________________________
44 33.
CALCULUS FOR LIFE SCIENCES, Chapter 2, Form C
(
f ( x ) = x 2 − 3 + sin x x − 3
34. For y = 3 x 6 − 4 x3 , find
d3 y dx3
)
33. ____________________________
.
34. ____________________________
35. Given that an object is at position s ( t ) = sin ( 3t ) − 2t at any time t , find each of the following. a) v ( t ) .
35. (a) _________________________
b) a ( t ) .
35. (b) _________________________
c) The position, velocity and acceleration when t =
5π . 3
35. (c) _________________________
36. Electing Officers. For a club consisting of n people, the number of ways in which a president, vice-president, and treasurer can be elected is given by P = n3 − 3n 2 + 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) _________________________
c) Find the rate of change at n = 20 .
36. (c) _________________________
37. Find ( f g )( x ) and ( g f )( x ) , given that f ( x ) = 3 − x 2 and g ( x ) = x 2 + 5 .
38. Differentiate y = ( 6 − 3x )
13
x3 + 8 . x →−2 x + 2
39. Find lim
40. Find lim t →0
1 − cos ( t ) t
(10 + x )1 6 .
37. ____________________________
38. ____________________________
39. ____________________________
.
40. ____________________________
CALCULUS FOR LIFE SCIENCES, Chapter 2, Form C 45
41. Using a grapher, graph f and f ′ over the given interval.
Then estimate points at which the tangent line to f is horizontal. Sketch the graphs. f ( x ) = 3x5 − 15 x 2 + 5 x ; [ −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, and 0.0001. 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 Limits Graphically. Consider the following graph of function f for Questions 1 – 6.
Find the limit, if it exists.
lim f ( x )
1. ____________________________
lim f ( x )
2. ____________________________
3. lim f ( x )
3. ____________________________
4. lim f ( x )
4. ____________________________
5. lim f ( x )
5. ____________________________
lim f ( x )
6. ____________________________
1. 2.
x →−5
x →−2
x→0
x →3
x→ 4
6.
x →−4
Determine whether the function is continuous. 7.
8.
7. ____________________________
8. ___________________________