Calculus for the Life Sciences Marvin L. Bittinger Test Bank.pdf (69)

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

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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,

38. ______________________

x = 0,x =3.

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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.

CALCULUS FOR THE LIFE SCIENCES, Chapter 6, Form E 1 57

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:

4=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 ⎡ 0.8 2. ⎤ G=⎢ ⎥ ⎣ 0.6 5 ⎦ 0.

The population vector 6 for year 1 is hatchlings ⎤ ⎡150 ⎤ p = ⎡⎢ ⎥ = ⎢120 ⎥ . 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=2.2 and population of birds has eigenvalues r=1.8. Find the long-term growth rate for the population.

21. ______________________

___________________________________________________________ ⎡5 3⎤ ⎡6 3⎤ ⎥ and B=⎢ ⎥. ⎣3 2⎦ ⎣5 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

__________________ _______

158 CALCULUS FOR THE LIFE SCIENCES, Chapter 6, Form E 23. Bird Population. The Leslie matrix for a population of birds is ⎡0.8 2.8⎤ G=⎢ ⎣0.5 0.4⎥ The population vector⎦ for year 1 is ⎡hatchlings⎤ ⎡160⎤ . ⎥= p=⎢ ⎣ adults ⎦ ⎢⎣ 140⎥ ⎦ (a) Estimate the population vector for year 2.

(b) Estimate the population vector for year 3. (c) Estimate the population vector for year 20. (d) Estimate the long-term growth rate.

23. (a) _________________________

23. (b) _________________________

23. (c) _________________________

23. (d) _________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 6, Form F

Compute.

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

1.______________________

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

2.______________________

⎡3 1. ⎢0 ⎢ ⎢1 ⎣

⎡8 1 ⎤ 5 3 −3 ⎤ ⎡ 4 −1 6 ⎤ 3. ⎢ ⎡ ⎥−⎢ ⎥ ⎣5 −2 ⎥ 0 1 7 ⎦ ⎣ 2 −3 5 ⎦ ⎢ ⎦ ⎣

3. ______________________

Solve the following systems of linear equations using Gaussian elimination.

5x +2 y = 8 6 x+ y =−3

4. ______________________

x + 2 y + 2 z = −1 5. x + 3 y + 2 z = 3 x + 3 y + z =4

5.______________________

4.

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

6 2 6. ⎡⎢3 − ⎤⎥ ⎣ −1 ⎦

6.______________________

⎡ −1 − 3⎤ ⎥ ⎢ 9⎦ ⎣3

7.______________________

⎡1.2 0.5 3 ⎤ ⎥ 8. ⎢ −1 ⎥ ⎢2.3 0.8 3.1 4.5⎥ ⎢⎣6.7 ⎦

8.______________________

⎡3 6 9. ⎢1 1 ⎢ ⎢2 4 ⎣

9.______________________

7.

−5 ⎤ 2⎥ ⎥ −3 ⎥⎦

160 CALCULUS FOR THE LIFE SCIENCES, Chapter 6, Form F Find the inverse matrix.

1 10. ⎡⎢ −1⎤⎥ ⎣5 4 ⎦

⎡1 3 11. ⎢2 7 ⎢ ⎢2 9 ⎣

10. ___________________________

−1 ⎤ ⎥ −4 ⎥ −7 ⎥⎦

11. ___________________________

Find the eigenvalues and the corresponding eigenvectors.

10 ⎤ 5 12. ⎡⎢ −3 2 ⎥⎦ ⎣

12. ___________________________

⎡12 −6⎤ ⎥ ⎢ −1⎦ ⎣7

13. ___________________________

5 1⎤ 14. ⎡⎢ ⎥ ⎣ −2 2 ⎦

14. ___________________________

13.

⎡ 0 −4 15. ⎢ −3 4 ⎢ ⎢ 5 −10 ⎣

⎡6 16. ⎢0 ⎢ ⎢0 ⎣

0⎤ 0⎥ ⎥ 4⎥ ⎦

15. ___________________________

−1 ⎤ ⎥ −5 −1 ⎥ 0 −5 ⎥ ⎦

16. ___________________________

2

Find the particular solution to the difference equation.

17.

xn+1 = 10xn − 21xn−1;x0=618,x1 = 38

18. xn+1 = −3xn+ 40xn+1+72;x0 = −5,x1=74

17. ___________________________

18. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 6, Form F 1 61

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:

1.8=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 ⎡1.5 3 ⎤ G=⎢ ⎥ ⎣ 0.9 0.6⎦ The population vector for year 1 is

hatchlings ⎤ ⎡90 ⎤ p = ⎡⎢ ⎥ = ⎢60 ⎥ . 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=1.6 and population of birds has eigenvalues r=0.7. 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

__________________ _______

162 CALCULUS FOR THE LIFE SCIENCES, Chapter 6, Form F 23. Bird Population. The Leslie matrix for a population of birds is ⎡0.7 1.1⎤ G=⎢ ⎣0.5 0.2⎥ The population vector⎦ for year 1 is ⎡hatchlings⎤ ⎡110⎤ . ⎥= p=⎢ ⎣ adults ⎦ ⎢ 90⎥ ⎦ (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: _________________

Given f ( x,y ) =

cose ( ), find each of the following.

Chapter 7, Form A

2 x+ y

1. f (2, − 3 )

1.______________________

f 2. ∂ ∂x

2.______________________

f 3. ∂ ∂y

3.______________________

∂ f ∂x 2

4.______________________

f 5. ∂ ∂x∂y

5.______________________

∂2f ∂y 2

6.______________________

4.

2

2

6.

For the following functions, use linearization and the valuesz,ofzx , and zy,at ( a , b ) to estimatez ( x , y ).

7. z = f ( x , y=) y2 x ; a= 9, b = 3, x = 2.71, y = 4.05

)=

8. z = f ( x , yy x; a = 25, b = 5, x = 7.28, y = 8.51

7. ______________________

8.______________________

) 3 and 9. Find the Jacobian off ( x , y=xy

g ( x , y )= cos ( xey ).

9.______________________

Find the relative maximum and minimum values. f ( x,y) = x2 + xy +y3+ 2x

10. ______________________

11. f ( x,y ) = 3x 2+7 y 2 = 12xy

11. ______________________

10.

164 CALCULUS FOR THE LIFE SCIENCES, Chapter 7, Form A

12. Age of Men at First Marriage. Consider the data in the following table regarding the median age of men at their first marriage in the U.S. (U.S. Bureau of the Census.)

Year, x

Age at First Marriage

0, 1980

24.7

10, 1990

26.1

20, 2000

26.8

(a) Find the regression liney=mx+b.

12. (a) _________________________

(b) Use the regression line to predict the median age of men at their first marriage in 2020.

12. (b) _________________________

13. Evaluate

∫ ∫ 3 ydydx ) .

0 2 (x3+ 1

x

13. ___________________________

________________________________________________________________

14. Evaluate

∫ ∫∫ (x−2y+z)dzdydx.

0 0 0 2

x 2 −x

14. ___________________________

15. Use a 3D grapher to graph

(1 fx,y)=x+y2 Sketch the graph.

3

−

1 3 x .

2

15.

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Given f ( x , y=) e

sin (2−xy)

Chapter 7, Form B , find each of the following.

(1,2)

1. f

1.______________________

f 2. ∂ ∂x

2.______________________

f 3. ∂ ∂y

3.______________________

∂ f ∂x 2

4.______________________

f 5. ∂ x ∂ ∂y

5.______________________

∂2f ∂y 2

6.______________________

4.

2

2

6.

For the following functions, use linearization and the valuesz,ofzx , and zy,at ( a , b ) to estimatez ( x , y ).

7. z = f ( x , y=) x2 y3 ; a = 5, b =1, x = 6.01, y = 3.12

)=

8. z = f ( x , y x 2 y ; a = 3, b = 16, x = 5.27, y = 7.93

7. ______________________

8.______________________

9. Find the Jacobian off ( x , y=2) x 2 y 3 and

g ( x , y= )sine2(x y ).

9.______________________

Find the relative maximum and minimum values.

10.

f(x,y)= x2 + 2y2−4xy+y

11. f ( x , y=)3x 2+ y2

10. ______________________

11. ______________________

166 CALCULUS FOR THE LIFE SCIENCES, Chapter 7, Form B 12. MLB Salaries. Consider the data in the following table regarding the average salary of an MLB player over a recent 10-year period. (asp.usatoday.com) Year, x

Average Salary of an MLB Player, y (in millions)

0, 1995

0.32

5, 2000

0.96

10, 2000

1.19

(a) Find the regression liney=mx+b.

12. (a) _________________________

(b) Use the regression line to predict the average MLB 12. (b) _________________________

player salary in 2015.

13. Evaluate

0∫x41(+4ydydx ∫ ) .

2

x

13. ____________________________ ________________________________________________________________ 14. Evaluate

(3x+y+2z)dzdydx. ∫ ∫∫ 0 0 0 2 3x

x−y

14. ___________________________

15. Use a 3D grapher to graph 1 1 f ( x,y)=2x+ y2 − x2 .

4 5 Sketch the graph.

15.

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Given f ( x , y=)

π 8

1. f ⎛⎜ ⎝

ln ( sin− y(4))x, find each of the following.

Chapter 7, Form C

2

, 0⎞⎟ ⎠

1.______________________

f 2. ∂ ∂x

2.______________________

f 3. ∂ ∂y

3.______________________

∂ f ∂x 2

4.______________________

f 5. ∂2 ∂x∂y

5.______________________

∂2f ∂y 2

6.______________________

4.

6.

2

For the following functions, use linearization and the valuesz,ofzx , and zy,at ( a , b ) to estimatez ( x , y ).

7. z = f ( x , y=) y3 x ; a= 4, b = 2, x = 1.87, y = 2.05

)=

8. z = f ( x , y x 2 3y ; a = 1, b = 8, x = 8.31, y = 1.44

7. ______________________

8.______________________

)=( x+1 2 ) y and

9. Find the Jacobian off ( x , y g ( x,y ) =

cosxe(2 ). y

9.______________________

Find the relative maximum and minimum values.

10.

f ( x,y) = y3 − 6xy+x2−48y

11. f ( x,y ) = 2x 2+3 y2

10. ______________________

11. ______________________

168 CALCULUS FOR THE LIFE SCIENCES, Chapter 7, Form C 12. Age of Women at First Marriage. Consider the data in the following table regarding the median age of women at their first marriage in the U.S. (U.S. Bureau of the Census.)

Year, x

Age at First Marriage

0, 1980

22.0

10, 1990

23.9

20, 2000

25.1

(a) Find the regression liney=mx+b.

12. (a) _________________________

(b) Use the regression line to predict the median age of 12. (b) _________________________

women at their first marriage in 2020.

13. Evaluate 2 −y dydx .

0∫3x1( ∫ ) 2

x

13. ___________________________

________________________________________________________________ 14. Evaluate

(x−y+2z)dzdydx. ∫ ∫∫ 0 0 0 2

4x

y−x

14. ___________________________

15. Use a 3D grapher to graph

f ( x , y ) = 2 x 2 +−x3.y 2 Sketch the graph.

15.

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Given f ( x,y ) =

sine ( y ), find each of the following.

4x

Chapter 7, Form D

3

1. f (2,1 )

1.______________________

f 2. ∂ ∂x

2.______________________

f 3. ∂ ∂y

3.______________________

∂ f ∂x 2

4.______________________

f 5. ∂ ∂x∂y

5.______________________

∂2f ∂y 2

6.______________________

4.

2

2

6.

For the following functions, use linearization and the valuesz,ofzx , and zy,at ( a , b ) to estimatez ( x , y ).

7. z = f ( x , y ) =x3 y; a= 2, b =−3,x= 2.99, y = 0.08

7. ______________________

6.31, y = 0.09

8.______________________

(x, y ) = y 3 x; a = 27, b=4,x= 8. z = f

)=(1− x2 ) y and g( x ,) y = sin (.x 2 e 2 y )

9. Find the Jacobian off ( x , y

2

9.______________________

Find the relative maximum and minimum values.

10.

f (x,y) = 3x2 + y2 + x − xy

11. f ( x,y ) = 8y2 − 2x2

10. ______________________

11. ______________________

170 CALCULUS FOR THE LIFE SCIENCES, Chapter 7, Form D

12. Cost of Movie Tickets. Consider the data in the following table regarding the average U.S. ticket prices for movie admissions.

Year, x

Age at First Marriage

0, 1980

$2.69

10, 1990

$4.22

20, 2000

$5.39

(a) Find the regression liney=mx+b.

12. (a) _________________________

(b) Use the regression line to predict the average price of a U.S. movie ticket in 2010?.

12. (b) _________________________

13. Evaluate x. ∫ 0∫1x3 −(4 y ) dyd 3

x

13. ___________________________

________________________________________________________________

14. Evaluate

∫ ∫∫ (2x+y− z)dx dy dz.

0 0 0 1 2z

y+z

14 .____________________________

15. Use a 3D grapher to graph

f(x,y)= 1 x3−8x2 + x .

2

Sketch the graph.

15.

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

x 2 cosy) (2

Given f ( x,y ) = e

Chapter 7, Form E , find each of the following.

1. f (3,0 )

1.______________________

f 2. ∂ ∂x

2.______________________

f 3. ∂ ∂y

3.______________________

∂ f ∂x 2

4.______________________

f 5. ∂ ∂x∂y

5.______________________

∂2f ∂y 2

6.______________________

4.

2

2

6.

z ( x , y.) to estimate For the following functions, use linearization and the valuesz,ofzx , and zy , at ( a ,)b 1 7. z = f ( x , y) = x 2 y 3 ;=a4, b = 2 , x = 2.67, y = 1.01

(

8. z = f x , y) = y 2 x ; a= 4, b = 3, x = 1.01, y = 8.17

7.______________________

8.______________________

9. Find the Jacobian of f( x , y=) y3 − 2 x 2 and

sin(y2e3x).

(

)

g ( x,y ) =

9.______________________

Find the relative maximum and minimum values.

10.

f (x,y ) = 2x2 + 4xy−y3+x

11. f ( x,y ) = 2x2 − 5y2

10. ______________________

11. ______________________

172 CALCULUS FOR THE LIFE SCIENCES, Chapter 7, Form E

12. 10th Graders Who Have Smoked. Consider the data in the following table regarding the number of 10th-graders per 100 (in a particular U.S. region) who have smoked cigarettes in a 30-day period. (www.monitoringthefuture.org) Year, x

10th Graders who have smoked in the last 30 days, per 100

0, 1996

30.4

4, 2000

23.9

8, 2004

16.0

(a) Find the regression liney=mx+b.

12. (a) _________________________

(b) Use the regression line to predict the number of 10th- grade cigarette smokers in 2012?.

12. (b) _________________________

13. Evaluate

∫ ∫4 x− 3(y 2)dydx .

0 1 2

x

13. ___________________________

________________________________________________________________

14. Evaluate

∫ ∫∫ (4x−y+z)dxdydz.

0 0 0 1

z

z− 3

14. ___________________________

15. Use a 3D grapher to graph 1 1 f ( x,y)=5x− y2 − x2 .

5 2 Sketch the graph.

15.

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

ln ( cos ( find each of the following. x 2 y 4)),

Given f ( x,y ) =

Chapter 7, Form F

1. f (−1,0 )

1.______________________

f 2. ∂ ∂x

2.______________________

f 3. ∂ ∂y

3.______________________

∂ f ∂x 2

4.______________________

f 5. ∂ x ∂ ∂y

5.______________________

∂2f ∂y 2

6.______________________

4.

2

2

6.

z ( x , y.) to estimate For the following functions, use linearization and the valuesz,ofzx , and zy , at ( a ,)b 7. z = f ( x , y=) xy3; a =9, b

=1, x = 6.11, y = 7.09 3

7.______________________

yy 3 x; a = 9, b = 2, x = 3.01, y = 7.84 8. z = f ( x ,) =

8.______________________

9. Find the Jacobian off ( x , y=3) x − 2 y 3and

cosy3(e ) .

g ( x,y ) =

xy

9.______________________

Find the relative maximum and minimum values. f ( x,y ) = 2y2 − 3x2

10. ______________________

11. f ( x,y) = x 3 +2xy+ y2 + y

11. ______________________

10.

174 CALCULUS FOR THE LIFE SCIENCES, Chapter 7, Form F 12. Marijuana Use. Consider the data in the following table regarding the number of 10th-graders per 100 (in a particular U.S. region) who have used marijuana in a 30-day period. (www.monitoringthefuture.org) 10th Graders who have smoked in the last 30 days, per 100

Year, x

0, 1996

20.4

4, 2000

19.7

8, 2004

15.9

(a) Find the regression liney=mx+b.

12. (a) _________________________

(b) Use the regression line to predict the number of 10th- graders using marijuana in 2016?.

12. (b) _________________________

13. Evaluate

0∫y31(−2xdydx ∫ ) .

2

x

13. ___________________________

________________________________________________________________ 14. Evaluate

∫ ∫∫ ( z+3y−2 x)dxdydz.

0 0 0 2

z +1

y−z

14. ___________________________

15. Use a 3D grapher to graph

f(x,y)=y− 2 x2 − 3 y2 .

3

Sketch the graph.

4

15.

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Solve the differential equation.

1. y′ = 3 − 6x2

(

Chapter 8, Form A

1.______________________

)

2.______________________

3. y′ = xcos (2 )x

3. _____________________

2. y′ = e− x 7 + e− x

3

′−2y=8 4. y

4.______________________

5. x 2 y′+3xy= 2 x−2

5.______________________

6. y′−xy=5x

6.______________________

7. y′ = e x + 5 y

7.______________________

8. y′ = 3 x 2 y 2

8.______________________

9. y′ =

x 3y2 + 4y

9.______________________

Solve the initial-value problem.

10. y′ = 3x2 − x + 4;y1= ()

11 2

10. ______________________

5 11. y′ = x2 cos⎛⎜ x3 − π ⎞⎟; y (0 ) = 3 2 ⎝ ⎠

11. ______________________

12. 3 xy′ +2 y = 0; y (27)= 2

12. ______________________

176 CALCULUS FOR THE LIFE SCIENCES, Chapter 8, Form A 13. y′ + ycosx = cosx;y ⎛⎜ π ⎞⎟ = 2 ⎝2⎠

13. ___________________________

3−x 14. y′ = y 2 ; y (1)= 3

14. ___________________________

15. y′ =

5 ; y (1)= 0 2 − ey

15. ___________________________

For #16 and #17, (a) find the equilibrium value(s) of the differential equation; (b) access the stability of each equilibrium value; (c) determine the point(s) of inflection; and (d) sketch sample solutions of the differential equation.

16.y′=3y2−6y 16. (a) _________________________

16. (b) _________________________

16. (c) _________________________

16. (d) _________________________ 17. y′=y2−y−2 17. (a) _________________________

17. (b) _________________________

17. (c) _________________________

17. (d) _________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 8, Form A 1 77 18. Crop Yield. A farmer is growing peanuts in a field. The more fertilizer he spreads, the greater his yieldY, up to a limiting value of 80 bushels per acre. If he spreads x pounds of fertilizer per acre, the yield follows the differential equation d Y = k (80 −Y ),

dx where k is a constant.

(a) Write this first-order linear differential equation in standard form.

18. (a) _____________________

(b) Solve for Y(x)given that Y(0)=20.

(b)

18.

__________________ (c) Suppose that for Y(10)=24. Use this to determinek . _______

(d) Find Y(20).

18.

(c)

__________________

(e) Find xso that Y(x)=50. In other words, how many _______ pounds of fertilizer per acre must be spread to yield 50 bushels per acre?

19. Mixing Chemicals. A tank initially contains 150 lbs of salt dissolved in 600 gal of water. Saltwater containing 2 lbs of salt per gal enters the tank at the rate of 5 gal per min. The mixture (kept uniform by stirring) is removed at the same rate. How many pounds of salt are in the tank after an hour?

18.

(d)

__________________

(e) ____________________ 18. _______

19. ______________________

20. Dye Degradation. A dye is placed into a mixture of acetone and triethylamine solution. Let ( )Dtbe the concentration of

the dye after tseconds. Let’s suppose the acetone causes the dye to degrade with a decay rate of 0.244 per sec, while the triethylamine causes a downtake rate of 3×10−8molars per second.

(a) Draw a one-compartment model forD.

0.244D−(3×10−8).

(b) Show that D′ = −

20. (a) ____________________

20. (b) ____________________

(

D (0 ) = D0 . (c) Solve for D t) given the initial condition

20. (c) ____________________

178 CALCULUS FOR THE LIFE SCIENCES, Chapter 8, Form A 21. Bacteria Growth. During a fermentation experiment, the pop-

ulation P(t) (in millions per mL) of bacteria Lactobacilli reuteri after thours in a barley medium satisfied the differential equation dP

=0.14 P⎜⎛1 − P ⎞⎟ ⎝ 20⎠ dt Initially, 1.6 million bacteria per mL were present. (a) Determine the number of bacteria per mL after 5 hr. (b) Determine when 10 million bacterial per mL will be present.

21. (a) _________________________ 21. (b) _________________________

________________________________________________________________

22. Yeast Growth. Let P(t)be the number of yeast cells in a laboratory experiment. Suppose that P(t)follows a logistic model of inhibited growth, given by dP

= kP⎜⎛1 − P ⎞ ⎝ 600⎠⎟ . dt

Find kifP(0)=20 and P(7)=500.

22. ___________________________

For the initial-value problem, use (a) Euler’s method and (b) the Runge-Kutta

method with the given step size ∆xto approximatey(2).

23.

y′=2y−1,y(0)=3,∆x=0.2

(a)

23.

__________________ _______ 24. y′ = sinx( y), y (0 ) = 1, ∆x= 0.1

23.

(b)

__________________ 24. (a) _________________________ _______ 24. (b) _________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Solve the differential equation.

1. y′ = 8x3 + 4

(

Chapter 8, Form B

1.______________________

)

2.______________________

3. y′ = xsin (2 x )

3. _____________________

2. y′ = ex 3 ex 3 + 1

5

′+2y=5 4. y

4.______________________

5. x 4 y′ − 7 x 3 y = 3 x10

5.______________________

6. y′− 6 xy =2 x

6.______________________

7. y′ = e y + 4 x

7.______________________

x2 8. y′ = 2 y

8.______________________

9. y′ =

x3−x y + 3y2 + y 3

9.______________________

Solve the initial-value problem.

10. y′ = 3 x 2 − 8 x+1; y (1)= 6

10. ______________________

11. y′ = x2 sin⎛⎜ x 3 − π ⎞⎟; y (0 ) = 3 2⎠ ⎝

11. ______________________

12. 3 xy′− 5y = 0; y (8 ) = 64

12. ______________________

180 CALCULUS FOR THE LIFE SCIENCES, Chapter 8, Form B 13. y′ + ycotx = 1; y ⎛⎜ π ⎞⎟ = 4 ⎝2⎠

13. ___________________________

14. y′ =

3 x − x2 ; y (1)= 2 3 y2

14. ___________________________

15. y′ =

2x ; y (2 ) = 0 7 + ey

15. ___________________________

For #16 and #17, (a) find the equilibrium value(s) of the differential equation; (b) access the stability of each equilibrium value; (c) determine the point(s) of inflection; and (d) sketch sample solutions of the differential equation.

16. y′=y2−5y 16. (a) _________________________

16. (b) _________________________

16. (c) _________________________

16. (d) _________________________ 17. y′=y2−16 17. (a) _________________________

17. (b) _________________________

17. (c) _________________________

17. (d) _________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 8, Form B 1 81 18. Acceptance of New TreatmentThe . proportion Pof doctors aware of a new treatment after tmonths may be modeled by the differential equation

dP = k (0.9 −P )

dt where k is a constant.

(a) Write the first-order linear differential equation in standard form.

(b) Solve for P(t)given that P(0)=0

18. (a) _____________________ (b)

18.

__________________ (c) Suppose that forP(6)=0.5. Use this to determinek .

(d) Find P(18).

_______

(c)

__________________

(e) Find tso that P(t)=0.85. In other words, how many _______ months will it take for 85% of doctors to become aware of this treatment?

18.

18.

(d)

__________________ 18. (e) _____________________ _______

19. Mixing Chemicals. A tank initially contains 100 lbs of salt dissolved in 500 gal of water. Saltwater containing 1.5 lbs of salt per gal enters the tank at the rate of 4 gal per min. The mixture (kept uniform by stirring) is removed at the same rate. How many pounds of salt are in the tank after 1/2 hour?

19. ______________________

20. Biohazards. A mussel is placed into water polluted with polychlorinated biphenyls (PCBs). Let ( Qtbe ) the concen-

tration of PCB in the mussel in micrograms (per gram of tissue) after tdays. For certain concentrations of pollution let’s suppose that the mussel absorbs PCBs at the rate of 14 micrograms of PCB per gram of tissue per day. Also, let’s suppose the elimination rate of PCBs from the mussel is 0.20 micrograms per gram of tissue per day.

(a) Draw a one-compartment model forQ.

(b) Show that D′ = −0.20D+14.

(

D (0 ) = D0 (c) Solve for D t) given the initial condition

20. (a) ____________________

20. (b) ____________________

20. (c) ____________________

182 CALCULUS FOR THE LIFE SCIENCES, Chapter 8, Form B 21. Bacteria Growth. During a fermentation experiment, the pop-

ulation P(t) (in millions per mL) of bacteria Lactobacilli reuteri after thours in a wheat medium satisfied the differential equation dP

=0.13 P⎜⎛1 − P ⎟⎞ ⎝ 16⎠ dt Initially, 1.5 million bacteria per mL were present. (a) Determine the number of bacteria per mL after 3 hr. (b) Determine when 5 million bacterial per mL will be present.

21. (a) _________________________ 21. (b) _________________________

________________________________________________________________

22. Yeast Growth. Let P(t)be the number of yeast cells in a laboratory experiment. Suppose that P(t)follows a logistic model of inhibited growth, given by dP

= kP⎜⎛1 − P ⎞ ⎝ 800 ⎠⎟ . dt

Find kifP(0)=10 and P(8)=400.

22. ___________________________

For the initial-value problem, use (a) Euler’s method and (b) the Runge-Kutta

method with the given step size ∆xto approximatey(2).

23.

y′=y+4,y(0)=2,∆x=0.2

(a)

23.

__________________ _______

( ,) y (0 ) = 1, ∆x= 0.1 24. y′ = cosxy

23.

(b)

__________________ 24. (a) _________________________ _______ 24. (b) _________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Solve the differential equation.

Chapter 8, Form C

1. y′ = 7 x3 + 2

1.______________________

ln(2x )

2.______________________

2. y′ =

x

3. y′ = xe 2 x

3. _____________________

4. y′ + 8 y = 1

4.______________________

5. x 3 y′ + 4 x 2 y = 3 x−2

5.______________________

6. y′ +3 x 2 y =6 x 2

6.______________________

7. y′ = e 2 x− 8 y

7.______________________

8. y′ = 6 x 2 y

8.______________________

9. y′ =

34 x2 − 6 x y − 3y2 +1

9.______________________

Solve the initial-value problem.

10. y′ = 4x3 − x + 3;y(2)=25

10. ______________________

⎛π 11. y′ = x4 sin − x5 ⎞⎟; y (0 ) = 2 ⎠ ⎜ ⎝6

11. ______________________

12. x 2 y′ + 2 y = 0; y (4 ) = 1

12. ______________________

184 CALCULUS FOR THE LIFE SCIENCES, Chapter 8, Form C 13. y′ − y sin x = sin x; y (0 ) = 3

13. ___________________________

x2 14. y′ = 1 − ; y (1)= 2 4 y3

14. ___________________________

15. y′ =

2

e +1 2y

; y (1)= 1

15. ___________________________

For #16 and #17, (a) find the equilibrium value(s) of the differential equation; (b) access the stability of each equilibrium value; (c) determine the point(s) of inflection; and (d) sketch sample solutions of the differential equation.

16. y′=3y−6y2 16. (a) _________________________

16. (b) _________________________

16. (c) _________________________

16. (d) _________________________

17. y′=y2−6y−7 17. (a) _________________________

17. (b) _________________________

17. (c) _________________________

17. (d) _________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 8, Form C 1 85 18.Crop Yield. A farmer is growing corn in a field. The more fertilizer he spreads, the greater his yieldY, up to a limiting value of 100 bushels per acre. If he spreads x pounds of fertilizer per acre, the yield follows the differential equation d Y = k (100 −Y ),

dx where k is a constant.

(a) Write this first-order linear differential equation in standard form.

18. (a) _____________________

(b) Solve for Y(x)given that Y(0)=40.

(b)

18.

__________________ (c) Suppose that for Y(10)=70. Use this to determinek . _______

(d) Find Y(20).

18.

(c)

__________________

(e) Find xso that Y(x)=80. In other words, how many _______ pounds of fertilizer per acre must be spread to yield 80 bushels per acre?

19. Mixing Chemicals. A tank initially contains 75 lbs of salt dissolved in 400 gal of water. Saltwater containing 2 lbs of salt per gal enters the tank at the rate of 8 gal per min. The mixture (kept uniform by stirring) is removed at the same rate. How many pounds of salt are in the tank after 1/2 hour?

18.

(d)

__________________

(e) ____________________ 18. _______

19. ______________________

20. Dye Degradation. A dye is placed into a mixture of acetone and triethylamine solution. Let ( )Dtbe the concentration of

the dye after tseconds. Let’s suppose the acetone causes the dye to degrade with a decay rate of 0.288 per sec, while the triethylamine causes a downtake rate of 3×10−8molars per second.

(a) Draw a one-compartment model forD.

0.288D−(3×10−8).

(b) Show that D′ = −

20. (a) ____________________

20. (b) ____________________

(

D (0 ) = D0 . (c) Solve for D t) given the initial condition

20. (c) ____________________

186 CALCULUS FOR THE LIFE SCIENCES, Chapter 8, Form C 21. Bacteria Growth. Many bacteria strains are used by the dairy industry to produce different types of fermented milks and yogurts. During a fermentation experiment, the population y (in millions per mL) of bacteria Lactobacillus fermentum after thours in a wheat medium satisfied the differential equation y ⎞ ⎛ y′ =0.532y⎜1− 1900 ⎠ ⎝ Initially, 10 million bacteria per mL were present. (a) Determine the number of bacteria per mL after 4 hr.

21. (a) _________________________

(b) Determine when 100 million bacterial per mL will be present.

21. (b) _________________________

________________________________________________________________

22. Yeast Growth. Let P(t)be the number of yeast cells in a laboratory experiment. Suppose that P(t)follows a logistic model of inhibited growth, given by dP

= kP⎜⎛1 − P ⎞ ⎝ 500 ⎠⎟ . dt

Find kifP(0)=20 and P(9)=400.

22. ___________________________

For the initial-value problem, use (a) Euler’s method and (b) the Runge-Kutta

method with the given step size ∆xto approximatey(2).

23.

y′=3+y,y(0)=3,∆x=0.2

(a)

23.

__________________ _______ 24. y′ = cosx( y), y (0 ) = 2, ∆x= 0.1

23.

(b)

__________________ 24. (a) _________________________ _______ 24. (b) _________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Solve the differential equation.

1. y′ = 4 x2 − 7

Chapter 8, Form D

1.______________________

(

2. y′ = e− x 2 − e− x

)

5

2.______________________

3. y′ = x ln x

3. _____________________

4. y′ − 3y = 5

4.______________________

2 5. x 4 y′ + 4 x 3 y = x

5.______________________

6. y′ −xy=xex

6.______________________

2

7. y′ = e 7 x− 5 y

7.______________________

8. y′ = − 3 x 2 y 4

8.______________________

3 x 7 9. y′ = 4 x − + y 2 −1

9.______________________

Solve the initial-value problem.

10. y′=x2−x+ 7; y (1)= 8

10. ______________________

⎛ 2 π⎞ y 0 1 11. y′= 2xcos⎜x + ⎟; ( ) = 3⎠ ⎝

11. ______________________

12. x−1 y ′ + 2 y = 0; y (3 ) = 1

12. ______________________

188 CALCULUS FOR THE LIFE SCIENCES, Chapter 8, Form D

13. y′+sec2x=secx;

14.

y′=

15. y′ =

y(0)=5

1 −x 2 ;y2)=1 3y8 − 2 (

4x3 y

; y (3 ) =2

13. ___________________________

14. ___________________________

15. ___________________________

For #16 and #17, (a) find the equilibrium value(s) of the differential equation; (b) access the stability of each equilibrium value; (c) determine the point(s) of inflection; and (d) sketch sample solutions of the differential equation.

16. y′=10y−5y2 16. (a) _________________________

16. (b) _________________________

16. (c) _________________________

16. (d) _________________________

17. y′=y2+6y−16 17. (a) _________________________

17. (b) _________________________

17. (c) _________________________

17. (d) _________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 8, Form D 1 89 18. Hullian Model of Learning.According to the Hullian model of learning, the probability Pof mastering a certain concept after tlearning trials may be modeled by the differential equation d P = k (0.95 −P ), where k is a constant . dt

(a) Write this first-order linear differential equation in standard

form.

18. (a) _____________________

(b) Solve for P(t)given that P(0)=0.

(b)

18.

__________________ (c) Suppose that for P(10)=0.75. Use this to determinek .

(d) Find P(15).

_______

18.

(c)

__________________

(e) Find tso that P(t)=0.9. In other words, how many trials _______

18.

(d)

__________________ trials are necessary in order for the probability of mastering the concept to be 0.90? (e) ____________________ 18. _______

19. Mixing Chemicals. A tank initially contains 125 lbs of salt dissolved in 500 gal of water. Saltwater containing 3 lbs of salt per gal enters the tank at the rate of 5 gal per min. The mixture (kept uniform by stirring) is removed at the same rate. How many pounds of salt are in the tank after 1 1/2 hour? 19. ______________________

20. Biohazards. A mussel is placed into water polluted with polychlorinated biphenyls (PCBs). Let the concen( Qtbe )

tration of PCB in the mussel in micrograms (per gram of tissue) after tdays. For certain concentrations of pollution let’s suppose that the mussel absorbs PCBs at the rate of 10 micrograms of PCB per gram of tissue per day. Also, let’s suppose the elimination rate of PCBs from the mussel is 0.16 micrograms per gram of tissue per day. (a) Draw a one-compartment model forQ.

(b) Show that D′ = −0.16D+10.

(

D (0 ) = D0 (c) Solve for D t) given the initial condition

20. (a) ____________________

20. (b) ____________________

20. (c) ____________________

190 CALCULUS FOR THE LIFE SCIENCES, Chapter 8, Form D 21. Bacteria Growth. During a fermentation experiment, the pop-

ulation P(t) (in millions per mL) of bacteria Lactobacilli reuteri after thours in a barley medium satisfied the differential equation dP

=0.14 P⎜⎛1 − P ⎞⎟ ⎝ 20⎠ dt Initially,1.5 million bacteria per mL were present. (a) Determine the number of bacteria per mL after 7 hr. (b) Determine when 9 million bacterial per mL will be present.

21. (a) _________________________ 21. (b) _________________________

________________________________________________________________

22. Yeast Growth. Let P(t)be the number of yeast cells in a laboratory experiment. Suppose that P(t)follows a logistic model of inhibited growth, given by dP

= kP⎜⎛1 − P ⎞ ⎝ 600⎠⎟ . dt

Find kifP(0)=10 and P(6)=500.

22. ___________________________

For the initial-value problem, use (a) Euler’s method and (b) the Runge-Kutta

method with the given step size ∆xto approximatey(2).

23.

y′=1+2y,y(0)=2,∆x=0.2

(a)

23.

__________________ _______ 24. y′ = y sin x, y (0 ) = 2, ∆x= 0.1

23.

(b)

__________________ 24. (a) _________________________ _______ 24. (b) _________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Solve the differential equation.

1. y′ = x3 − 8

)10 − 5 ′ = x(x2

2. y

Chapter 8, Form E

1.______________________

2.______________________

′=3xsinx 3. y

3. _____________________

4. y′ + y = 2

4.______________________

5. x 2 y′+xy= 8

5.______________________

6. y′ + 7 x y = x

6.______________________

7. y′ = e y − 4 x

7.______________________

8. y′ = x 2 y 3

8.______________________

9. y′ =

x3 + 1 y 4 + 6y2

9.______________________

Solve the initial-value problem.

10. y′ = 3 x2 + x− 5; y (2 ) = − 1

11. y′ = 4 x3

sin ( xπ) 4 ;+ y (0)=3

12. y′ = 3 ( x − y); y (0 ) = 1

10. ______________________

11. ______________________

12. ______________________

192 CALCULUS FOR THE LIFE SCIENCES, Chapter 8, Form E ; y (π 2 ) = 0 13. y′ − y sin (2)x= sin (2)x

13. ___________________________

x3 ; 2 14. y′ = 2 − 1 − 3 y 5 y(1)=

14. ___________________________

′ = x2 ; y (3 ) =4 15. y 2y

15. ___________________________

For #16 and #17, (a) find the equilibrium value(s) of the differential equation; (b) access the stability of each equilibrium value; (c) determine the point(s) of inflection; and (d) sketch sample solutions of the differential equation.

16. y′=4y2−8y 16. (a) _________________________

16. (b) _________________________

16. (c) _________________________

16. (d) _________________________ 17. y′=y2−y−20 17. (a) _________________________

17. (b) _________________________

17. (c) _________________________

17. (d) _________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 8, Form E 1 93 18. Crop Yield. A farmer is growing maize in a field. The more fertilizer he spreads, the greater his yieldY, up to a limiting value of 120 bushels per acre. If he spreads x pounds of fertilizer per acre, the yield follows the differential equation dY = k 120 ( − Y ), dx

where k is a constant.

(a) Write this first-order linear differential equation in standard form.

18. (a) ____________________

(b) Solve for Y(x)given that Y(0)=50.

18. (b) ____________________

(c) Suppose that for Y(10)=72. Use this to determinek.

18. (c) ____________________

(d) Find Y(15).

18. (d) ____________________

(e) Find xso that Y(x)=100. In other words, how many pounds of fertilizer per acre must be spread to yield 100 bushels per acre?

18. (e) ____________________

19. Mixing Chemicals. A tank initially contains 200 lbs of salt dissolved in 600 gal of water. Saltwater containing 2 lbs of salt per gal enters the tank at the rate of 5 gal per min. The mixture (kept uniform by stirring) is removed at the same rate. How many pounds of salt are in the tank after 2 hr?

19. ______________________

20. Pollution. Halibut are exposed to elevated methyl mercury concentrations due to a nearby coal plant. Let ( )Mtbe the

concentration of methyl mercury per gram of fish after tdays.

Suppose that the halibut intake methyl mercury at a rate of m nanograms of methyl mercury per gram of halibut per day.

The elimination rate of methyl mercury from the fish is nper day.

(a) Draw a one-compartment model for M.

20. (a) ____________________

(b) Show that M′=−mM+n.

20. (b) ____________________

(c) Solve for(Mtgiven the initial condition 20. (c) ____________________ 0. M(0)=M )

194 CALCULUS FOR THE LIFE SCIENCES, Chapter 8, Form E 21. Bacteria Growth. During a fermentation experiment, the pop-

ulation P(t) (in millions per mL) of bacteria Lactobacilli reuteri after thours in a barley medium satisfied the differential equation dP

=0.14 P⎜⎛1 − P ⎞⎟ ⎝ 20⎠ dt Initially, 2.1 million bacteria per mL were present. (a) Determine the number of bacteria per mL after 4 hr. (b) Determine when 7 million bacterial per mL will be present.

21. (a) _________________________ 21. (b) _________________________

________________________________________________________________

22. Yeast Growth. Let P(t)be the number of yeast cells in a laboratory experiment. Suppose that P(t)follows a logistic model of inhibited growth, given by dP

= kP⎜⎛1 − P ⎞ ⎝ 800 ⎠⎟ . dt

Find kifP(0)=20 and P(7)=600.

22. ___________________________

For the initial-value problem, use (a) Euler’s method and (b) the Runge-Kutta

method with the given step size ∆xto approximatey(2).

23.

y′=y−2,y(0)=1,∆x=0.2

(a)

23.

__________________ _______

, y (0 ) = 2, ∆x= 0.1 24. y′ = x sin (2)y

23.

(b)

__________________ 24. (a) _________________________ _______ 24. (b) _________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Solve the differential equation.

1. y′ = 3 − x4

1.______________________

′ = ex 3(5+)11 ex

2. y

Chapter 8, Form F

3

2.______________________

′=5xcosx 3. y

3. _____________________

4. y′ − 2 y = 3

4.______________________

5. x 3 y′ − 5 x 2 y = 2 x−3

5.______________________

6. y′− 4 xy =8 x

6.______________________

7. y′ = e 6 x− 3 y

7.______________________

8. y′ = 4 x 2 y 3

8.______________________

x y 4+ y

9.______________________

9. y′ =

Solve the initial-value problem.

10. y′ = 4 x − 3; y (3 ) = 41

10. ______________________

11. y′ = sin x 4 + 3cosx; y (π 2 ) = 119

11. ______________________

12. y′ = 2 ( y + x); y (12)= 1

12. ______________________

196 CALCULUS FOR THE LIFE SCIENCES, Chapter 8, Form F ; y (π 6 ) = 2 13. y′ − y sin (3)x= sin (3)x

13. ___________________________

x3 14. y′ = 5 − ; y (1)= 3 2y

14. ___________________________

15. y′ =

e−y ; y(e)= 1 x

15. ___________________________

For #16 and #17, (a) find the equilibrium value(s) of the differential equation; (b) access the stability of each equilibrium value; (c) determine the point(s) of inflection; and (d) sketch sample solutions of the differential equation.

16. y′=8y−2y2 16. (a) _________________________

16. (b) _________________________

16. (c) _________________________

16. (d) _________________________

17. y′=y2+2y−15 17. (a) _________________________

17. (b) _________________________

17. (c) _________________________

17. (d) _________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 8, Form F 197 18.

Hullian Model of Learning .

According to the Hullian model of learning, the probability Pof mastering a certain assembly line task after tlearning trials may be modeled by the differential equation dP = k (0.7 − P ), dt

where k is a constant.

(a) Write this first-order linear differential equation in standard form.

18. (a) ____________________

(b) Solve for P(t)given that P(0)=0.

18. (b) ____________________

(c) Suppose that for P(10)=0.5. Use this to determinek.

18. (c) ____________________

(d) Find P(20).

18. (d) ____________________

(e) Find tso that P(t)=0.6. In other words, how many trials

trials are necessary in order for the probability of mastering the concept to be 0.60?

18. (e) ____________________

19. Mixing Chemicals. A tank initially contains 50 lbs of salt dissolved in 250 gal of water. Saltwater containing 2.5 lbs of salt per gal enters the tank at the rate of 4 gal per min. The mixture (kept uniform by stirring) is removed at the same rate. How many pounds of salt are in the tank after an hour?

20. Carbon Decomposition. Let C denote the quantity of organic carbon in a sample of soil. Let udenote the decay rate of organic carbon per year, and let v denote the uptake rate of organic carbon into the sample per year. (X. Yang, M. Wang, Y. Huang, and Y. Wang, “A onecompartment model to study soil carbon decomposition rate at equilibrium situation,” Ecological Modeling, Vol. 151, pp. 63-73 (2002).)

(a) Draw a one-compartment model forC. (b) Show that C′=−uC+v.

19. ______________________

20. (a) ____________________

20. (b) ____________________

(c) Solve for( Ctgiven the initial condition 0. C(0)=C 20. (c) ____________________ )

198 CALCULUS FOR THE LIFE SCIENCES, Chapter 8, Form F 21. Bacteria Growth. Many bacteria strains are used by the dairy industry to produce different types of fermented milks and yogurts. During a fermentation experiment, the population y (in millions per mL) of bacteria Lactobacillus fermentum after thours in a wheat medium satisfied the differential equation y ⎞ ⎛ y′ =0.532y⎜1− 1900 ⎠ ⎝

Initially, 9 million bacteria per mL were present.

(a) Determine the number of bacteria per mL after 6 hr.

21. (a) _________________________

(b) Determine when 200 million bacterial per mL will be present.

21. (b) _________________________

________________________________________________________________

22. Yeast Growth. Let P(t)be the number of yeast cells in a laboratory experiment. Suppose that P(t)follows a logistic model of inhibited growth, given by dP

= kP⎜⎛1 − P ⎞ ⎝ 500 ⎠⎟ . dt

Find kifP(0)=10 and P(8)=400.

22. ___________________________

For the initial-value problem, use (a) Euler’s method and (b) the Runge-Kutta

method with the given step size ∆xto approximatey(2).

23.

y′=2y−3,y(0)=2,∆x=0.02

(a)

23.

__________________ _______ 24. y′ = 2 xsin y , y (0 ) = 1, ∆x= 0.1

23.

(b)

__________________ 24. (a) _________________________ _______ 24. (b) _________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 9, Form A Solve.

1. y′′−4 y ′ + 3y = 0

1.______________________

2. y′′−6 y ′ + 9y = 0

2.______________________

3. y′′+ 16 y = 48

3. _____________________

4. y′′−y ′ − 20 y = x + 2

4.______________________

Solve the initial-value problem.

5. y′′− 3y ′ + 2y = 0, y (0 ) = 2, y′(0) = −1

5.______________________

6. y′′−2 y ′ + 5y = 0, y(0)= 1, y′(0) = −1

6.______________________

7. y′′−2 y ′ + y =8, y (0)=4, y′(0)=−1

7.______________________

=4 8. y′′+4 y = 4 x + 2, y(0)= 2, y′(0)

8.______________________

Solve the first-order linear system.

9. x′ = 2x − y, y′ = 3y

9.______________________

10. x′ = x−y , y′ = y − 4x

10. ______________________

11. x′ = x − 4 y , y ′ = x + y

11. ______________________

200 CALCULUS FOR THE LIFE SCIENCES, Chapter 9, Form A Sketch trajectories of solutions of′ z= Azgiven the eigenvalues and associated eigenvectors of A. Classify the origin and assess its stability.

12. r1 = −1 + 4.2i, r2 = −1 − 4.2i

3 ⎡−2 13. r1 =3,r 2 = −2, v1 = ⎢ ⎤⎥ , v2 =⎡⎢ ⎤⎥ ⎣1 ⎦ 2 ⎣ ⎦ 1 ⎡3 ⎤ 14. r1 =−1, r2 = −3, v1 =⎡ ⎤ , v ⎥ ⎢ 1⎦ ⎢ ⎥ 2 =⎣− 2 ⎣ ⎦

12. ___________________________

13. ___________________________

14. ___________________________

15. r1 =6i, r2 = −6 i

_________________________ 15. __

16. Assess the stability of the equilibrium points of the nonlinear system x′ = x−y and y ′ = x2+y2−8

16. ___________________________

17. Solve for x and y in the two-compartment model.

x 2

3

y

17. ___________________________

18. Mixing Chemicals. Tank A contains 200 lbs of salt dissolved in 1000 gal of water. Tank B contains 300 lbs of salt dissolved in 1000 gal of water. The mixture from tank A is pumped to tank B at the rate of 750 gal per hr, while the mixture from tank B is pumped to tank A at the same rate. Assume that the mixture in each tank is uniform by stirring. Let( Atand Btbe ) ( ) the amount of salt in tanks A and B after thours, respectively.

(a) Draw a two-compartment model for ( )Atand Bt ( ).

18. (a) _________________________

(b) Use the initial conditionsA (0 ) = 200 and B (0)=300to solve for A and B.

18. (b) _________________________

A and B? (c) What are the equilibrium values of

18. (c) ________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 9, Form A 2 01 Let x and yrepresent the populations (in thousands) of two species that share a habitat. For each system of equations, find and assess the stability of the equilibrium points. Remember that, for real populations, bothx 0and y0must be

nonnegative0, for x be an equilibrium point. y0)(to x y) , 19. Competing Species.′x= x (0.2 − 0.004 − 0.005 x y) y′ = y (0.1 − 0.0035 − 0.004

19. ______________________

20. Predator and Prey. x′ = 0.6x− 0.2xy , y′ = − y + 0.04 xy

20. ______________________

Competing Species. For the systems of differential equations of problems 21 and 22, use Euler’s method with ∆x=2to plot the trajectory of xand yto plot the trajectory for 0 ≤ t ≤ 500.

21. x′ = x0.05 ( −0.001x−0.0003y ( − 0.002x−0.004y ) , y′=y0.03

x (0 ) = 5,y (0 ) = 3

),

22. x′ = x(0.05 − 0.002 x − 0.004y ) , y′ = y (0.03 − 0.001x − 0.0003 y) , x (0 ) = 5, y 0( ) =5

21. ______________________

22. ______________________

CALCULUS FOR THE LIFE SCIENCES

Name: ______________________

Chapter 9, Form B Solve.

1. y′′−8 y ′ +12 y = 0

1.___________________________

2. y′′− 10 y ′ + 25 y = 0

2.___________________________

3. y′′+y = 4

3. __________________________

4. y′′−2 y′−15 y = 2 x − 5

4.___________________________

Solve the initial-value problem.

5. y′′+ 11y ′ + 18y = 0, y (0 ) = 0, y′(0)= 2

5.___________________________

=1 6. y′′−6 y′ + 25 y = 0, y( = 4, y′(0)

6.___________________________

0)

=1 = 5, y′(0)

7.___________________________

8. y′′+ 81y = 3x−9, y (0)=9, y′(0)= 2

8.___________________________

7. y′′+4 y ′ + 4y = 12,

y( 0)

Solve the first-order linear system.

9. x′ = 3y, y′ = 2x − y

9.___________________________

10. x′=−x−y,

y′=−4x−y

10. ___________________________

11. x′ = x + 5 y , y ′ = − x − 3 y

11. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 9, Form B 2 03 Sketch trajectories of solutions of′ z= Azgiven the eigenvalues and associated eigenvectors of A. Classify the origin and assess its stability.

12. r1 = −2 + 3.9i,r2 = −2 − 3.9i

1 ⎡−2 13. r1 =2, r2 = −1,v1 = ⎡⎢ ⎤⎥ , v2 = ⎢ ⎤⎥ 3 ⎣2⎦ ⎣ ⎦ 1 ⎡2 ⎤ 14. r1 = −1, r2 = −2, v1 =⎡ ⎤ , v ⎥ ⎢ 1⎦ ⎢ ⎥ 2 =⎣− 1 ⎣ ⎦

12. ______________________

13. ______________________

14. ______________________

15. r1 = i , r2=−i

____________________ 15. __

16. Assess the stability of the equilibrium points of the nonlinear system x′ = (x−2)(y−1) and ′ =yx2−2y−8

16. ______________________

17. Solve for xand yin the two-compartment model.

x 4

1

y

17. ______________________

18. Mixing Chemicals. Tank A contains 50 lbs of salt dissolved in 100 gal of water. Tank B contains 100 lbs of salt dissolved in 100 gal of water. The mixture from tank A is pumped to tank B at the rate of 75 gal per hr, while the mixture from tank B is pumped to tank A at the same rate. Assume that the mixture in each tank is uniform by stirring. Let( Atand Btbe ) ( ) the amount of salt in tanks A and B after thours, respectively.

(a) Draw a two-compartment model for ( )Atand Bt ( ).

18. (a) ____________________

(b) Use the initial conditionsA (0 ) = 50 and B (0 ) =100to solve for A and B.

18. (b) ____________________

A and B? (c) What are the equilibrium values of

18. (c) ____________________

204 CALCULUS FOR THE LIFE SCIENCES, Chapter 9, Form B Let x and yrepresent the populations (in thousands) of two species that share a habitat. For each system of equations, find and assess the stability of the equilibrium points. Remember that, for real populations, bothx 0and y0must be

nonnegative0, for x be an equilibrium point. y0)(to 19. Competing Species.x ′ = x(0.02 − 0.0002x − 0.0006y ) ,

x− y) y′ = y (0.01 − 0.0007 0.0001

19. ___________________________

20. Predator and Prey. x′ = 0.8x− 0.2xy , y′ = −0.5 y + 0.1xy

20. ___________________________

Competing Species. For the systems of differential equations of problems 21 and 22, use Euler’s method with ∆x=2to plot the trajectory of xand yto plot the trajectory for 0 ≤ t ≤ 500.

21. x′ = x0.05 ( −0.0025x−0.0003y ( − 0.002x−0.0015y ) , y′=y0.03

x (0 ) = 1,y (0 )= 5

),

21. ___________________________

22. x′ = x(0.05 − 0.002x − 0.0015y ) , y′ = y (0.03 − 0.0025x − 0.0003 y) , x (0 ) = 3, y (0 )= 5 22. ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 9, Form C Solve.

1. y′′−3 y ′ −10 y = 0

1.______________________

2. y′′+6 y ′ + 9y = 0

2.______________________

3. y′′+9 y = 18

3. _____________________

4. y′ + 5y′ + 6 y = 3 x − 2

4.______________________

Solve the initial-value problem.

5. y′′+ y′ − 12y = 0, y(0) = 1, y′(0)=−1

5.______________________

′+34y= 6. y′′− 10 y

7. y′′−6 y′+12y =3,

0, y(0)=2,

y′(0 ) = −1

y (0)=−1, y′(0 )=2

8. y′′+9 y′ = x +3, y (0 ) = 1, y′(0 ) =−1

6.______________________

7.______________________

8.______________________

Solve the first-order linear system.

9. x′ = −2+x 3y, y′ = y

9.______________________

10. x′ = x + 2 y , y′ = 3 x + 2y

10. ______________________

11. x′ = 3 x −3 y , y ′ = 3 x − y

11. ______________________

206 CALCULUS FOR THE LIFE SCIENCES, Chapter 9, Form C Sketch trajectories of solutions of′ z= Azgiven the eigenvalues and associated eigenvectors of A. Classify the origin and assess its stability.

12. r1 = −1 + 3i, r2 = −1 − 3i

12. ___________________________

4 ⎡−2 13. r1 =1, r2 =−2, v1 = ⎢ ⎤⎥ , v2 =⎡⎢ ⎤⎥ ⎣3 1 1= ⎦ ⎣ ⎦

13. ___________________________

14. r1 = −3, r 2 = −4, v

⎡ 0 ⎤ , v ⎡3⎤ 2= ⎢ ⎥ ⎢ ⎥ 1 1 ⎣ ⎦ ⎣ ⎦

14. ___________________________

15. r1 = 5i, r 2 = −5i

_________________________ 15. __

16. Assess the stability of the equilibrium points of the nonlinear system x′ = 2xy and y′ = 3x+y−6

16. ___________________________

17. Solve for xand yin the two-compartment model.

x 1

0.5

y

17. ___________________________

18. Mixing Chemicals. Tank A contains 250 lbs of salt dissolved in 1000 gal of water. Tank B contains 300 lbs of salt dissolved in 1000 gal of water. The mixture from tank A is pumped to tank B at the rate of 400 gal per hr, while the mixture from tank B is pumped to tank A at the same rate. Assume that the mixture in each tank is uniform by stirring. Let( Atand Btbe ) ( ) the amount of salt in tanks A and B after thours, respectively.

(a) Draw a two-compartment model for ( )Atand Bt ( ).

18. (a) _________________________

(b) Use the initial conditionsA (0 ) = 250 and B (0)=300to solve for A and B.

18. (b) _________________________

A and B? (c) What are the equilibrium values of

18. (c) ________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 9, Form C 2 07 Let x and yrepresent the populations (in thousands) of two species that share a habitat. For each system of equations, find and assess the stability of the equilibrium points. Remember that, for real populations, bothx 0and y0must be

nonnegative0, for x be an equilibrium point. y0)(to 19. Competing Species.′x= x ( 0.04 − 0.0008x − 0.0004y ) ,

x y) y′ = y (0.1 − 0.005 − 0.002

19. ______________________

20. Predator and Prey. x′ = x − xy , y′ = −0.6 y + 0.2xy

20. ______________________

Competing Species. For the systems of differential equations of problems 21 and 22, use Euler’s method with ∆x=2to plot the trajectory of xand yto plot the trajectory for 0 ≤ t ≤ 500.

21. x′ = x0.04 ( −0.0016x−0.0005y ( − 0.001x−0.003y ) , y′=y0.002

x (0 ) = 4,y (0 ) = 8

),

21. ______________________

22. x′ = x(0.04 − 0.001x − 0.003y ) , y′ = y (0.002 − 0.0016x − 0.0005y ) , x (0 ) = 4, y 0( ) =2 22. ______________________

CALCULUS FOR THE LIFE SCIENCES

Name: ______________________

Chapter 9, Form D Solve.

1. y′′−7 y′ + 12 y = 0

1.___________________________

2. y′′− 16 y ′ + 64 y = 0

2.___________________________

3. y′′+ 25 y = 25

3. __________________________

4. y′′− 10 y ′ + 21y = 2 x + 4

4.___________________________

Solve the initial-value problem.

0, y (0 ) = 1, y′ (0)= 2

5.___________________________

6. y′′−4 y′+20y =0, y(0)= 5, y(0) ′ =2

6.___________________________

5. y′′+ 3y ′ − 10 =y

7. y′′+2 y′+ y =6,

y (0)=3,y ′ (0)= 1

8. y′′+ 64 y = 4 x −16, y(0)= 2, y(0) ′ = 916

7.___________________________

8.___________________________

Solve the first-order linear system.

9. x′ = − y, y′ = −4x − 3y

9.___________________________

10. x′ = − x − 2 y , y′=−3x−2 y

10. ___________________________

11. x′ = 5 x −6 y , y ′ = 2 x + y

11. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 9, Form D 2 09 Sketch trajectories of solutions of′ z= Azgiven the eigenvalues and associated eigenvectors of A. Classify the origin and assess its stability.

12. r1 = −1 + 2i,r2 = −1 − 2i

1 ⎡−1 13. r1 =4, r2 =−3, v1 = ⎡⎢ ⎤⎥ , v 2 = ⎢ ⎤⎥ ⎣1 ⎦ 2 ⎣ ⎦ 1 3 14. r1 = − 2, r2 = −3, v1 =⎡ ⎤ , v2 =⎡⎢ ⎤⎥ 1 ⎣ ⎦ ⎢ ⎥ 0 ⎣ ⎦ 15. r1 = 7i, r2 = −7i

16. Assess the stability of the equilibrium points of the nonlinear system x′ = x2+y2−32 and y ′ = x−y

12. ______________________

13. ______________________

14. ______________________

____________________ 15. __

16. ______________________

17. Solve for xand yin the two-compartment model.

x 2

1

y

17. ______________________

18. Mixing Chemicals. Tank A contains 50 lbs of salt dissolved in 500 gal of water. Tank B contains 100 lbs of salt dissolved in 500 gal of water. The mixture from tank A is pumped to tank B at the rate of 250 gal per hr, while the mixture from tank B is pumped to tank A at the same rate. Assume that the mixture in each tank is uniform by stirring. Let( Atand Btbe ) ( ) the amount of salt in tanks A and B after thours, respectively.

(a) Draw a two-compartment model for ( )Atand Bt ( ).

18. (a) ____________________

(b) Use the initial conditionsA (0 ) = 50 and B (0 ) =100to solve for A and B.

18. (b) ____________________

A and B? (c) What are the equilibrium values of

18. (c) ____________________

210 CALCULUS FOR THE LIFE SCIENCES, Chapter 9, Form D Let x and yrepresent the populations (in thousands) of two species that share a habitat. For each system of equations, find and assess the stability of the equilibrium points. Remember that, for real populations, bothx 0and y0must be

nonnegative0, for x be an equilibrium point. y0)(to − x − 0.015y) , 19. Competing Species.′x= x (0.60.01 x y) y′ = y (0.009 − 0.001 − 0.003

19. ___________________________

20. Predator and Prey. x′ = 0.5x− 0.2xy , y′ = −0.4 y + 0.1xy

20. ___________________________

Competing Species. For the systems of differential equations of problems 21 and 22, use Euler’s method with ∆x=2to plot the trajectory of xand yto plot the trajectory for 0 ≤ t ≤ 500.

21. x′ = x0.06 ( −0.002x−0.0012y ( − 0.007x−0.0045y ) , y′=y0.03

x (0 ) = 2,y (0 ) = 4

),

22. x′ = x(0.06 − 0.007 x − 0.0045 y) , y′ = y (0.03 − 0.002 x− 0.0012 y ) , x (0 ) = 3, y 0( ) =1

21. ___________________________

22. ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 9, Form E Solve.

1. y′′+ 13 y ′ + 42 y = 0

1.______________________

2. y′′+ 10 y ′ + 25 y = 0

2.______________________

3. y′′+ 49 y = 98

3. _____________________

4. y′′− 10 y ′ + 16 y = 5 x − 2

4.______________________

Solve the initial-value problem.

5. y′′+ 4y ′ − 12y = 0, y (0 )= 3, y′ ( ) = 2

6. y′′−2 y′+10 y = 0, y(0 = 2, y′

7. y′′−8 y′ + 16 y = 4, y

)

0

5.______________________

) =1

6.______________________

)=1

7.______________________

( = 2, y′

(0

0

8. y′′+ 36 y = 54 x + 6, y(0)= 76, (y′(0 ) =1

)

8.______________________

0 Solve the first-order linear system.

9. x′ = − x+ 2y, y′ = 3x

9.______________________

10. x′ = 3 x −4 y , y′ = 4 x − 7y

10. ______________________

11. x′ = − x −5 y , y′ = x +3 y

11. ______________________

212 CALCULUS FOR THE LIFE SCIENCES, Chapter 9, Form E Sketch trajectories of solutions of′ z= Azgiven the eigenvalues and associated eigenvectors of A. Classify the origin and assess its stability.

12. r1 = −2 + 2i,r2 = −2 − 2i

12. ___________________________

3 −1 13. r1 =2, r2 =−2, v1 = ⎡⎢ ⎤⎥ , v2 =⎡⎢ ⎤⎥ 1 ⎣2⎦ ⎣ ⎦

13. ___________________________

⎡1 ⎤ ⎡ −3 ⎤ 14. r1 = −1, r2 = −4, v1 = , v2 = ⎢2 ⎥ ⎢1⎥ ⎣ ⎦ ⎣ ⎦

14. ___________________________

15. r1 =8i, r2 = −8i

_________________________ 15. __

16. Assess the stability of the equilibrium points of the nonlinear system x′ = x(y+3) and′ =yx+y+2

16. ___________________________

17. Solve for xand yin the two-compartment model.

x 1

2

y

17. ___________________________

18.Mixing Chemicals. Tank A contains 50 lbs of salt dissolved in 100 gal of water. Tank B contains 100 lbs of salt dissolved in 100 gal of water. The mixture from tank A is pumped to tank B at the rate of 75 gal per hr, while the mixture from tank B is pumped to tank A at the same rate. Assume that the mixture in each tank is uniform by stirring. Let( Atand Btbe ) ( ) the amount

of salt in tanks A and B after thours, respectively.

(a) Draw a two-compartment model for ( )Atand Bt ( ).

18. (a) _________________________

(b) Use the initial conditionsA (0 ) = 50 and B (0 ) =100to solve for A and B.

18. (b) _________________________

A and B? (c) What are the equilibrium values of

18. (c) ________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 9, Form E 2 13 Let x and yrepresent the populations (in thousands) of two species that share a habitat. For each system of equations, find and assess the stability of the equilibrium points. Remember that, for real populations, bothx 0and y0must be

nonnegative0, for x be an equilibrium point. y0)(to − 0.0004 x − 0.0012y ) , 19. Competing Species.′x= x ( 0.02

x y) y′ = y (0.04 − 0.0024 − 0.0008

19. ______________________

20. Predator and Prey. x′ = 0.8x− 0.4xy , y′ = −0.6 y + 0.2xy

20. ______________________

Competing Species. For the systems of differential equations of problems 21 and 22, use Euler’s method with∆x = 2to plot the trajectory of xand yto plot the trajectory for 0 ≤ t ≤ 500.

21. x′ = x0.1 ( −0.002x−0.0017y ( − 0.07x − 0.005y ) , y′=y0.05

x (0 ) = 1,y (0 )= 2

),

22. x′ = x(0.1 − 0.07 x− 0.005 y) , y′ = y (0.05 − 0.002x − 0.0017y ) , x (0 ) = 3, y (0 )= 7

21. ______________________

22. ______________________

CALCULUS FOR THE LIFE SCIENCES

Name: ______________________

Chapter 9, Form F Solve.

1. y′′−7 y′ − 8y = 0

1.___________________________

2. y′′+ 16 y ′ + 64 y = 0

2.___________________________

3. y′′+4 y = 20

3. __________________________

4. y′′+7 y ′ +12 y = 6 x − 1

4.___________________________

Solve the initial-value problem.

=2 y(0) ′

5.___________________________

6. y′′−8 y′ + 17 y =0, y (0)=1, y′ (0)= 5

6.___________________________

7. y′′− 10 y ′ + 25 y = 5, y(0)=−1, y(0) ′ =1

7.___________________________

8. y′′+ 16 y = 8x − 32, y(0)=4,

8.___________________________

5. y′′− 5y ′ − 24y = 0, y (0 ) =0,

=3 y(0) ′

Solve the first-order linear system.

9. x′ = −4x − 5y,y ′ = − x

9.___________________________

10. x′ = 3 x −4 y , y ′ = 2 x − 3 y

10. ___________________________

11. x′ = x + 5 y , y′ = − x − 3 y

11. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 9, Form F 2 15 Sketch trajectories of solutions of′ z= Azgiven the eigenvalues and associated eigenvectors of A. Classify the origin and assess its stability.

12. r1 = −2 + 4.9i, r2 = −2 − 4.9i 13. r1

=1,

⎡2⎤ ⎡1⎤ r2 =−3, v1 = ⎢ ⎥ , v1 = ⎢ ⎥ ⎣−2⎦ ⎣3⎦

12. ______________________

13. ______________________

⎡1 ⎤ ⎡2⎤ 14. r1 = −2, r2 = −4, v ,v ⎥ ⎥ 1 =⎣− ⎢ 2 ⎦ 1 =⎣− ⎢ 1⎦

14. ______________________

15. r1 =3i, r2 = −3i

____________________ 15. __

16. Assess the stability of the equilibrium points of the nonlinear system x′ = (x−1 )(y−2) and′ =yx2−y−2

16. ______________________

17. Solve for xand yin the two-compartment model.

x 3

4

y

17. ______________________

18. Mixing Chemicals. Tank A contains 25 lbs of salt dissolved in 100 gal of water. Tank B contains 40 lbs of salt dissolved in 100 gal of water. The mixture from tank A is pumped to tank B at the rate of 30 gal per hr, while the mixture from tank B is pumped to tank A at the same rate. Assume that the mixture in each tank is uniform by stirring. Let( Atand Btbe ) ( ) the amount of salt in tanks A and B after thours, respectively.

(a) Draw a two-compartment model for ( )Atand Bt ( ).

18. (a) ____________________

(b) Use the initial conditionsA (0 ) = 25 and B (0 ) =40to solve for A and B.

18. (b) ____________________

A and B? (c) What are the equilibrium values of

18. (c) ____________________

216 CALCULUS FOR THE LIFE SCIENCES, Chapter 9, Form F Let x and yrepresent the populations (in thousands) of two species that share a habitat. For each system of equations, find and assess the stability of the equilibrium points. Remember that, for real populations, bothx 0and y0must be

nonnegative0, for x be an equilibrium point. y0)(to 19. Competing Species.′x= x (0.04 − 0.0008x − 0.004y ) ,

x y) y′ = y (0.1 − 0.005 − 0.002

19. ___________________________

20. Predator and Prey. x′ = x − 0.4xy , y′ = −0.8 y + 0.1xy

20. ___________________________

Competing Species. For the systems of differential equations of problems 21 and 22, use Euler’s method with ∆x=2to plot the trajectory of xand yto plot the trajectory for 0 ≤ t ≤ 500.

21. x′ = x0.1 ( −0.001x−0.001y ( − 0.004x−0.001y ) , y′=y0.05

x (0 ) = 2,y (0 ) = 1

),

22. x′ = x(0.1 − 0.004 x− 0.001 y) , y′ = y (0.05 − 0.001x − 0.001y ) , x (0 ) = 3, y 0( ) =2

21. ___________________________

22. ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 10, Form A Two cards are dealt without replacement from a well-shuffled deck of cards.

Find the probability of each event.

1. The first card is a club.

1.______________________

2.The first card is not a seven.

2.______________________

3. The first card is a face card or an eight.

3. _____________________

4.The first card is not a king and the second card is a three.

4.______________________

Find the probability for the given random variable

5. PX (

= 5;)X is Binomial8,0.2. (

)

6. P ( N = 3 ); N is Poisson(2.3.)

5.______________________

6.______________________

7.

P (1≤ X ≤ 4); Xis Uniform(3,7.)

7.______________________

8.

4).

T Exponential P(0.2 ≤ T ≤ 0.8);is (

8.______________________

9.

P (3≤ X ≤ 5); Xis Normal(4,6.)

9.______________________

X ) the given random variable.

Find E ( X ), Var ( X ),and SD (for 10.

X is Binomial(4,0.3.)

10. ______________________

11. N is Poisson(1.8.)

11. ______________________

12. X is Uniform(10,15.)

12. ______________________

13. T is Exponential ( 7.)

13. ______________________

14.

X is Normal(2.7,0.3.)

14. ______________________

218 CALCULUS FOR THE LIFE SCIENCES, Chapter 10, Form A The probability density function ofX is f ( x) = 5 x6 over [1, ∞ ). Compute the following quantities.

15.

P(X≤3) P(2≤X≤4)

15. ___________________________

16. 17.

18. SD(X) E(X)

16. ___________________________

17. ___________________________

18. ___________________________ 19. The observed level of significance if a value of 3 is observed.

19. ___________________________

20. The critical value corresponding to a level of significance of α= 0.05 .

20. ___________________________

21.

Testing for a Disease. A diagnostic test for a certain disease has a sensitivity of 0.86 and a specificity of 0.92. The probability that a patient has the disease is estimated as 0.3. The diagnostic test is given and returns positive. What is the posterior probability that the patient has the disease?

21. ___________________________

22. Coin Flipping. Ten coins are flipped. Find the probability that exactly six of the coins lands heads.

22. ___________________________

23. Random Sampling. According to the 2000 U.S. Census, Philadelphia, Pennsylvania, has a population of 1,517,550, of whom 812,443 are male. If 500 citizens of Philadelphia are selected at random, what is the probability that at least 250 of them will be male? (U.S. Bureau of the Census.)

23. ___________________________

24. Bird Calls. The loudness, measured in decibels, of a bird

call has a Uniform(50,60)distribution. Find the probability that the loudness is between 50 and 55 decibels.

24. ___________________________

25. Cell Cycle Time. The average cell cycle time for a bacterium is 25 min. Find the probability that the bacterium divides in under 22 min.

25. ___________________________

26. Height of Men. In a certain village, the height of men in

inches follows theNormal(69,2.5)distribution. Find the

probability that a randomly selected man from the village 26. ___________________________ is between 67 in. and 71in. tall. ________________________________________________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 10, Form A 219

The random variable X has the probability density functionf( x) = 0.2 e−0.2 over[0, ∞) .

Compute the following quantities.

27.

P(X>0.6)

E(X) 29. SD(X) 28.

27. ______________________

28. ______________________

29. ______________________

30. SAT Scores. The distribution of the math SAT test

scores at a local college is approximatelyNormal (490,100) . (a) What is the percentage of the scores is below 450? (b) What is the percentage of the scores is above 600?

30. (a) ____________________

30. (b) ____________________

CALCULUS FOR THE LIFE SCIENCES

Name: ______________________

Chapter 10, Form B Two cards are dealt without replacement from a well-shuffled deck of cards.

Find the probability of each event.

1.The first card is a face card.

1.___________________________

2. The first card is not a face card.

2.___________________________

3. The first card is a spade or a heart.

3. __________________________

4.The first card is black and the second card is red.

4.___________________________

Find the probability for the given random variable.

= 10; ) X is Binomial ( 14,0.4).

5. PX (

6. P ( N = 5 ); N is Poisson)(4.7.

≤

5.___________________________

6.___________________________

7.

X Uniform P(3.2 ≤ X 5.7); is ( 5,21).

7.___________________________

8.

0.3).

P (1≤ T ≤ 4); Tis Exponential (

8.___________________________

9.

P (6

9.___________________________

≤ X ≤ 7); Xis Normal(1,4.)

Find E ( X ), Var ( X ) , and SD(X ) for the given random variable

10.

X is Binomial(5,0.2.)

10. ___________________________

11. N is Poisson(3.7.)

11. ___________________________

12. X is Uniform(15,25. )

12. ___________________________

13. T is Exponential ( 8.)

13. ___________________________

14.

X is Normal( − 4,3).

14. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 10, Form B 221 The probability density function ofX is f ( x) = 4

15.

x5

over [1, ∞ ). Compute the following quantities.

P(X≤3)

15. ______________________

16. 17.

P(4≤X≤8) 18. SD(X) E(X)

16. ______________________

17. ______________________

18. ______________________ 19. The observed level of significance if a value of 5 is observed.

19. ______________________

20. The critical value corresponding to a level of significance of α= 0.01 .

20. ______________________

21.

Testing for a Disease. A diagnostic test for a certain disease has a sensitivity of 0.82 and a specificity of 0.76. The probability that a patient has the disease is estimated as 0.2.

The diagnostic test is given and returns positive. What is the posterior probability that the patient has the disease?

21. ______________________

22. True-False Test. A test consists of twelve true-false questions (of which each question has exactly one correct answer). A student guesses at all twelve questions. What is the probability that he gets exactly five of these guessed questions correct? 22. ______________________ 23. Random Sampling. According to the 2000 U.S. Census, Kokomo, Indiana, has a population of 46,113, of whom 24,320 are female. If 200 citizens of Kokomo are selected at random, what is the probability that fewer than 100 of them will be female? (U.S. Bureau of the Census.)

23. ______________________

24. Random Number Generator. A random number generator

is designed to output numbers with theUniform(0,10) distribution. Find the probability that the output is a number between 4.25 and 5.75.

25. Cell Cycle Time. The average cell cycle time for a bacterium is 90 min. Find the probability that the bacterium divides in under 85 min.

24. ______________________

25. ______________________

26. Height of Women. In a certain village, the height of women

in inches follows theNormal(64,2.5). Find the probability

that a randomly selected woman from the village is at least 26. ______________________ 60 in. tall. ___________________________________________________________

222 CALCULUS FOR THE LIFE SCIENCES, Chapter 10, Form B

The random variableX has the probability density functionf( x) = xe−x 2 over[0, ∞) .

2

Compute the following quantities.

27.

P(X>1.5)

E(X) 29. SD(X) 28.

27. ___________________________

28. ___________________________

29. ___________________________

30. SAT Scores.The distribution of the math SAT test

scores at a local college is approximatelyNormal(525,115). (a) What is the percentage of the scores is below 500?

30. (a) _________________________

(b) What is the percentage of the scores is above 600?

30. (b) _________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Two cards are dealt without replacement from a well-shuffled deck of cards.

Find the probability of each event.

Chapter 10, Form C

1. The first card is a nine.

1.______________________

2. The first card is not a two.

2.______________________

3. The first card is a club or a diamond.

3. _____________________

4. The first card is a five and the second card is a ten.

4.______________________

Find the probability for the given random variable.

5. PX (

= 6;)X is Binomial10,0.3. (

)

6. P ( N = 8 ); N is Poisson(5.)

5.______________________

6.______________________

7.

P (3≤ X ≤ 5); Xis Uniform(2,8.)

7.______________________

8.

5).

T Exponential P(0.1 ≤ T ≤ 0.5);is (

8.______________________

9.

P (1≤ X ≤ 3); Xis Normal(5,4.)

9.______________________

Find E ( X ), Var ( X ), and SD(X ) for the given random variable

10.

X is Binomial(7,0.4.)

10. ______________________

11. N is Poisson(8.2.)

11. ______________________

12. X is Uniform(8,22. )

12. ______________________

13. T is Exponential ( 6.)

13. ______________________

X is Normal(−1,2.1.)

14. ______________________

14.

224 CALCULUS FOR THE LIFE SCIENCES, Chapter 10, Form C The probability density function ofX is f ( x) = 81 x4 over [3, ∞ ). Compute the following quantities.

15.

P(X≤9) P(6≤X≤15)

15. ___________________________

16. 17.

18. SD(X) E(X)

16. ___________________________

17. ___________________________

18. ___________________________ 19. The observed level of significance if a value of 9 is observed.

19. ___________________________

20. The critical value corresponding to a level of significance ofα= 0.05 .

20. ___________________________

21.

Testing for a Disease. A diagnostic test for a certain disease has a sensitivity of 0.92 and a specificity of 0.88. The probability that a patient has the disease is estimated as 0.1.

The diagnostic test is given and returns positive. What is the posterior probability that the patient has the disease?

21. ___________________________

22. Rolling a Die. A fair die is rolled seven times. Find the probability that exactly 2 fives appear.

22. ___________________________

23. Random Sampling. According to the 2000 U.S. Census, Santa Fe, New Mexico, has a population of 62,203 of whom 29,744 are Hispanic or Latino (of any race). If 300 citizens of Santa Fe are selected at random, what is the probability that at least 140 of them will be Hispanic or Latino? (U.S. Bureau of the Census.)

23. ___________________________

24. Bird Calls. The loudness, measured in decibels, of a bird

call has a Uniform(45,52)distribution. Find the probability that the loudness is between 47 and 51 decibels.

24. ___________________________

25. Cell Cycle Time. The average cell cycle time for a bacterium is 40 min. Find the probability that the bacterium divides in under 35 min.

25. ___________________________

26. Fish Length. The reported length (measured in inches) of

bass caught at random in a certain lake hasNormal(10,2)

distribution. Find the probability that a randomly caught 26. ___________________________ fish has length at least 6 in. ________________________________________________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 10, Form C 225

The random variableX has the probability density functionf( x) = Compute the following quantities.

27.

P(X>3.7)

E(X) 29. SD(X) 28.

1 xe−x 3 over[0, ∞) .

9

27. ______________________

28. ______________________

29. ______________________

30. SAT Scores.The distribution of the verbal SAT test

scores at a local college is approximatelyNormal (500,110) . (a) What is the percentage of the scores is below 450? (b) What is the percentage of the scores is above 625?

30. (a) ____________________

30. (b) ____________________

CALCULUS FOR THE LIFE SCIENCES

Name: ______________________

Chapter 10, Form D Two cards are dealt without replacement from a well-shuffled deck of cards.

Find the probability of each event.

1.The first card is a queen.

1.___________________________

2. The first card is not a heart.

2.___________________________

3. The first card is a five or a jack.

3. __________________________

4.The first card is a club and the second card is a diamond.

4.___________________________

Find the probability for the given random variable.

5. PX (

)

5.___________________________

6. P ( N = 6 ); N is Poisson) (7.1.

6.___________________________

7.

P 2 ≤ X ≤ 3.7 )X; is Uniform(2,11.)

7.___________________________

8.

(

9.

= 12;)X is Binomial20,0.6. (

2≤T≤

4); Tis Exponential( 2.)

8.___________________________

0); X is Normal3,2. ( )

9.___________________________

−2≤X≤

P (

Find E ( X ), Var ( X ), and SD(X ) for the given random variable

PX is Binomial(2,0.6.)

10. ___________________________

11. N is Poisson(6.1.)

(

11. ___________________________

12. X is Uniform(30,45. )

12. ___________________________

13. T is Exponential ( 3.)

13. __________________________

10.

14.

X is Normal(7.2,1.3.)

14. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 10, Form D 227 The probability density function ofX is f ( x) = 64 x5 over [2, ∞ ). Compute the following quantities.

15.

P(X≤8)

15. ______________________

16. 17.

P(6≤X≤10) 18. SD(X) E(X)

16. ______________________

17. ______________________

18. ______________________ 19. The observed level of significance if a value of 6 is observed.

19. ______________________

20. The critical value corresponding to a level of significance of α= 0.01 .

20. ______________________

21.

Testing for a Disease. A diagnostic test for a certain disease has a sensitivity of 0.86 and a specificity of 0.92. The probability that a patient has the disease is estimated as 0.3.

The diagnostic test is given and returns negative. What is the posterior probability that the patient does not have the disease?

21. ______________________

22. Multiple-Choice Test. A 25-question multiple-choice placement test offers four possible answers for every question, of which only one is the correct answer. After answering all the questions that she knows, the student guesses on the remaining twenty questions. What is the probability that she gets exactly five of these guessed questions correct?

22. ______________________

23. Random Sampling. According to the 2000 U.S. Census, Billings, Montana, has a population of 89,847, of whom 52,192 are high-school graduates or higher. If 400 citizens of Billings are selected at random, what is the probability that between 220 and 240 will be a high-school graduate or higher? (U.S. Bureau of the Census.)

23. ______________________

24. Random Number Generator. A random number generator

is designed to output numbers with theUniform(0,100) distribution. Find the probability that the output is a number between 40 and 60.

25. Cell Cycle Time. The average cell cycle time for a bacterium is 100 min. Find the probability that the bacterium divides in under 95 min.

24. ______________________

25. ______________________

228 CALCULUS FOR THE LIFE SCIENCES, Chapter 10, Form D

26. Fish Length. The reported length (measured in inches) of

trout caught at random in a certain lake hasNormal(12,1.5) distribution. Find the probability that a randomly caught fish has length between 10 in. and 12 in.

26. ___________________________

________________________________________________________________

The random variable X has the probability density functionf(x)=4e−4xover[0,∞).

Compute the following quantities.

27.

P(X>0.3)

E(X) 29. SD(X) 28.

27. ___________________________

28. ___________________________

29. ___________________________

30. SAT Scores. The distribution of the verbal SAT test

scores at a local college is approximatelyNormal (480,110) . (a) What is the percentage of the scores is below 420? (b) What is the percentage of the scores is above 550?

30. (a) _________________________

30. (b) _________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Chapter 10, Form E Two cards are dealt without replacement from a well-shuffled deck of cards.

Find the probability of each event.

1.The first card is a king.

1.______________________

2. The first card is not a club.

2.______________________

3. The first card is three or a ten.

3. _____________________

4.The first card is a diamond and the second is also a diamond.

4.______________________

Find the probability for the given random variable.

5. PX (

= 9;)X is Binomial15,0.8. (

)

6. P ( N = 4 ); N is Poisson (2.5).

5.______________________

6.______________________

7.

X Uniform P 4.2≤X ≤ 5.1); is ( 3,13).

7.______________________

8.

( 3 ≤ T ≤ 4); Tis Exponential( 0.5.)

8.______________________

9.

≤ P − 1 ≤ X 1.5); Xis Normal(2,4. )

9.______________________

(

X ) the given random variable Find E ( X ), Var ( X ),and SD (for PX is Binomial(6,0.7.)

10. ______________________

11. N is Poisson(5.4.)

(

11. ______________________

12. X is Uniform(50,80. )

12. ______________________

13. T is Exponential ( 4.)

13. ______________________

10.

14.

X is Normal( − 4.2,0.7).

14. ______________________

230 CALCULUS FOR THE LIFE SCIENCES, Chapter 10, Form E The probability density function ofX is f ( x) = 3 x4 over [1, ∞ ). Compute the following quantities.

15.

P(X≤3) P(6≤X≤8)

15. ___________________________

16. 17.

18. SD(X) E(X)

16. ___________________________

17. ___________________________

18. ___________________________ 19. The observed level of significance if a value of 2 is observed.

19. ___________________________

20. The critical value corresponding to a level of significance of α= 0.05 .

20. ___________________________

21.

Testing for a Disease. A diagnostic test for a certain disease has a sensitivity of 0.82 and a specificity of 0.76. The probability that a patient has the disease is estimated as 0.2. The diagnostic test is given and returns negative. What is the posterior probability that the patient does not have the disease?

21. ___________________________

22. Coin Flipping. A fair coin is flipped eight times. Find the probability that exactly three of the flips lands tails.

22. ___________________________

23. Random Sampling. According to the 2000 U.S. Census, Cleveland, Ohio, has a population of 478,403, of whom 226,550 are male. If 500 citizens of Cleveland are selected at random, what is the probability that at least 230 will be male? (U.S. Bureau of the Census.)

23. ___________________________

24. Bird Calls. The loudness, measured in decibels, of a bird

call has a Uniform(57,75)distribution. Find the probability that the loudness is between 68 and 72 decibels.

24. ___________________________

25. Cell Cycle Time. The average cell cycle time for a bacterium is 30 min. Find the probability that the bacterium divides in under 28 min.

25. ___________________________

26. Weight of Women. In a certain village, the weight of women in

pounds follows theNormal(150,15)distribution. Find the probability that a randomly selected woman from the village is weighs no more than 135 lb.

26. ___________________________

________________________________________________________________

CALCULUS FOR THE LIFE SCIENCES, Chapter 10, Form E 231

The random variable X has the probability density functionf( x) = 3 e−3 x over[0, ∞) .

Compute the following quantities.

27.

P(X>0.6)

E(X) 29. SD(X) 28.

27. ______________________

28. ______________________

29. ______________________

30. SAT Scores. The distribution of the math SAT test scores

at a local college is approximatelyNormal(630,120). (a) What is the percentage of the scores is below 550? (b) What is the percentage of the scores is above 650?

30. (a) ____________________

30. (b) ____________________

CALCULUS FOR THE LIFE SCIENCES

Name: ______________________

Chapter 10, Form F Two cards are dealt without replacement from a well-shuffled deck of cards.

Find the probability of each event.

1.The first card is a diamond.

1.___________________________

2. The first card is not a seven.

2.___________________________

3. The first card is red.

3. __________________________

4.The first card is a diamond and the second is a spade.

4.___________________________

Find the probability for the given random variable.

5. PX (

= 7;)X is Binomial12,0.7. (

)

6. P ( N = 7 ); N is Poisson(3.4.)

5.___________________________

6.___________________________

7.

P(1.3 ≤ X ≤ 3.5); Xis Uniform ( 30,42).

7.___________________________

8.

P 2 ≤ T ≤ 4); Tis Exponential( 1.)

8.___________________________

9.

( 0 ≤ X ≤ 2.1); Xis Normal(2,1.)

9.___________________________

P

Find E ( X ), Var ( X ), and SD(X ) for the given random variable

10.

(X is Binomial(9,0.5.)

10. ___________________________

11. N is Poisson(2.9.)

11. ___________________________

12. X is Uniform(35,70. )

12. ___________________________

13. T is Exponential ( 2.)

13. ___________________________

X is Normal(5,7.)

14. ___________________________

14.

CALCULUS FOR THE LIFE SCIENCES, Chapter 10, Form F 233 The probability density function ofX is f ( x) = 12 x4 over ⎡ 3 4 , ∞ ⎣

15.

). Compute the following qu

P(X≤5)

15. ______________________

16. 17.

P(3≤X≤7) 18. SD(X) E(X)

16. ______________________

17. ______________________

18. ______________________ 19. The observed level of significance if a value of 7 is observed.

20. The critical value corresponding to a level of significance

ofα=0.01.

Testing for a Disease. A diagnostic test for a certain disease 21. has a sensitivity of 0.92 and a specificity of 0.88. The probability that a patient has the disease is estimated as 0.1.

The diagnostic test is given and returns negative. What is the posterior probability that the patient does not have the disease?

19. ______________________

20. ______________________

21. ______________________

22. Rolling a Die. A fair die is rolled three times. Find the probability that exactly one ace appears.

22. ______________________

23. Random Sampling. According to the 2000 U.S. Census, Boulder, Colorado, has a population of 94,673, of whom 45,778 are female. If 200 citizens of Boulder are selected at random, what is the probability that between 95 and 100 of them will be female? (U.S. Bureau of the Census.)

23. ______________________

24. Random Number Generator. A random number generator

is designed to output numbers with theUniform(0,10) distribution. Find the probability that the output is a number between 6.50 and 8.25.

25. Cell Cycle Time. The average cell cycle time for a bacterium is 50 min. Find the probability that the bacterium divides in under 45 min.

24. ______________________

25. ______________________

26. Weight of Women. In a certain village, the weight of women in

pounds follows theNormal(150,15)distribution. Find the probability that a randomly selected woman from the village is weighs between 140 lb. and 155 lb.

26. ______________________

___________________________________________________________

234 CALCULUS FOR THE LIFE SCIENCES, Chapter 10, Form F

The random variable X has the probability density functionf( x) = 0.02 e−0.02x over[0, ∞).

Compute the following quantities.

27.

P(X>0.5)

E(X) 29. SD(X) 28.

27. ___________________________

28. ___________________________

29. ___________________________

30. SAT Scores. The distribution of the math SAT test

scores at a local college is approximatelyNormal (450,100) . (a) What is the percentage of the scores is below 400? (b) What is the percentage of the scores is above 600?

30. (a) _________________________

30. (b) _________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Final Exam, Form A 3 1.Write an equation of the line with slope − 4 and containing the point(5,−2 ).

2. For f ( x)=

____________________ 1. ___

5x2−2, find the simplified difference quotient.

2. ______________________

3. Solve for the missing side, x.

40

x

3.______________________

4.81 4. Find the amplitude, period and mid-line. Also, find the maximum and minimum.

t y = 2 sin⎛⎜ ⎞⎟ + 2 ⎝3⎠

1, forx ≠ 1 ⎧⎪ 5. (a) Graph:f ( x ) =x + for x = 0 ⎨ 0, ⎪⎩ 2

4. ______________________

5. (a) 5

y

4 3 2 1 −5

−4

−3

−2

−1

−1 −2

x 1

2

3

4

5

−3 −4 −5

(b) Find limf ( x) .

5. (b) ____________________

(c) Find f (0 ) .

5. (c) ____________________

(d) Is f continuous at 0?

5. (d) ____________________

x→ 0

236 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form A Find the limit, if it exists.

6.

lim(2x5+x3+4)

x→−2

6.___________________________

5 7. lim x→0x

7.___________________________

x−3 8. lim 2 + x − x→3x 12

8.___________________________

1 9. lim x 2 x

9.___________________________

→∞

Differentiate.

10. y=x3− 4 x + 5

10. ___________________________

11. y=x56

11. ___________________________

12.

13.

f(x)= sin( x 2)

12. ___________________________

f(x) = ( x + 2 )(x−3)

13. ___________________________

3x − 4 x2 − 2

14. ___________________________

4

14. f ( x) =

lnx(3+ x + 1)

15. ___________________________

16. f (x)= cos (x )tan(x)

16. ___________________________

15. y =

x2

17. y = e +2x 18. y =

17. ___________________________

cos ( cot ( x )) 2

18. ___________________________

d2y . dx2

19. ___________________________

d 20. Differentiate the following function implicitly to find. y x4 − 2xy3=3. d x 21. Find the equation of the tangent line to the graph of y=2ex+x3 − 4at the point (0,−2).

20. ___________________________

19. For y = 10x 3 + 4 x , find

21. __________________________

CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form A 2 37 Find the relative extrema of the function. Find the relative extrema of trigonometric

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

22.

f(x)=x3+6x2+9x+1

22.

10

y

8 6 4 2 −10

−8

−6

−4

−2

x 2

−2

4

6

8

10

−4 −6 −8 −10

23.

f ( x) = 4 − 5 x2 + x4

23.

10

y

8 6 4 2 −10

−8

−6

−4

−2

x 2

−2

4

6

8

10

−4 −6 −8 −10

24.

3x f ( x) = −2 x +2

24.

10

y

8 6 4 2 −10

−8

−6

−4

−2

−2 −4 −6 −8 −10

x 2

4

6

8

10

238 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form A 25.

f ( x) = sin2x + cosx

25.

2.0

y

0 1.5 0

x

1.0 0 −0.

π/2

π

3π/2

2π

0.5 50 0 −1.

00 −1.

50

Find absolute maximum and minimum values of the function, if they exist, over the −2. indicated interval. Where no interval is specified, use the real line. List your answers 00 in terms of ordered pairs.

26.

27.

f(x)=4x2−3x+2 1 f(x)= x3+x2−8x+2;[−1,3]

26. ___________________________ 27. ___________________________ 28. ___________________________

3

28. y=sin2(x)−cos(x);[0,2π] 29. 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 is the maximum volume?

29. ___________________________

35.

30. Use linearization to approximate

30. ___________________________

31. Use Newton’s method and1=5 x to find an approximate

solution of f(x)=3x2−2.

31. ___________________________

32. Population Growth. A town is experiencing a growth pattern of 3% per year in its population P; that is dP

=0.03P, dt where P is the population of the town and t is the time, in years, from 2005. (a) Given that the town’s population in 2005 was 14,000, 0=14,000 find the solution of the equation assuming P and k=0.03.

(b)

What will the population be in 2015?

(c) What is the doubling time of the population?

32. (a) _________________________ 32. (b) _________________________

32. (c) _________________________

CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form A 2 39 Evaluate.

33. ∫ 2 x 7 dx

33. ______________________

x ) dx

34. ______________________

35. ∫ 4 xc x ) dx

35. ______________________

esint costdt 36. ∫

36. ______________________

34. ∫3ex +

(

4

0

os (2 ∫

( x − 2) lnxdx

37.

37. ______________________

18 x dx 38. ∫

38. ______________________

8 2 39. ∫ 1dx 3 x

39. ______________________

y = x3 + 2xover the 40. Find the area under the graph of

interval [1,3].

41. Find the volume generated by revolving about the x-axis (x)=ex2, x= 0, and x = 10. the regions bounded fby

40. ______________________

41. ______________________

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

42. ∫x−2dx

42. ______________________

exdx 43. ∫ 100

43. ______________________

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

44. ______________________

∞

1

∞

1

0

240 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form A 45. Find the determinant of the matrix and determine if the matrix is invertible.

⎡2 ⎢1 ⎢ ⎢4 ⎣

0 3

8⎤ 8 ⎥⎥ −1 2 ⎥⎦

45. ___________________________

46. Find the eigenvalues and corresponding eigenvectors.

⎡1 ⎢2 ⎣

3⎤ 2 ⎥⎦

46. ___________________________

47. 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.

47. (a) _________________________

(b) Estimate the population vector for year 3.

47. (b) _________________________

Given f ( x , y=) 6 y − 2 x 3 y+ 4 e

sin (x) 2

, find each of the following.

48.

∂f ∂y

48. ___________________________

49.

∂ f ∂y ∂x

49. ___________________________

2

( +1)and

50. Find the Jacobian off ( x , y=) y x2

g ( x,y ) =

sinexy ( .)

51. Find the relative maximum and minimum values of f ( x,y ) =2x2+5y2+6.

50. ___________________________

51. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form A 2 41

∫ ( ) dxdy

52. Evaluate:∫ ey + 2 −1 0

52. ______________________

53. Solve the differential equation y ′ := 2 x 3 y.

53. ______________________

54. Solve the initial value problem y :′ = sin x − x;y(0 ) = 3.

54. ______________________

1

55.

1

For the differential equation′ :=y8y − 2 y 2 (a) Find the equilibrium value(s).

55. (a) _____________________

(b) Access the stability of each equilibrium value.

55. (b) _____________________

(c) Determine the point(s) of inflection; and

55. (c) _____________________

(d) Sketch sample solutions of the differential equation.

55. (d) _____________________

56. A tank initially contains 150 lb of salt dissolved in 450 gal of water. Saltwater containing 2 lb of salt per gal enters the tank at the rate of 5 gal per min. The mixture (kept uniform by stirring) is removed at the same rate. How many pounds of salt are in the tank after 2 hours?

56. ______________________

57. Solve: y′′−7 y′ + 6 y = 0.

57. ______________________

58. Solve the first-order linear system:

x′ = −4 x − 5 y , y′ = − x

58. ______________________

59. Two cards are dealt without replacement from a well shuffled deck of cards. Find the probability that the first card is a seven or a king.

59. ______________________

60. Testing for a disease. A diagnostic test for a certain disease has a sensitivity of 0.98 and a specificity of 0.96. The probability that a patient has the disease is estimated as 0.2. The diagnostic test is given and returns positive. What is the posterior probability that the patient has the disease?

60. ______________________

242 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form A 61.

2 The probability density function = xof9 Xis f(x) over [0,3]. Compute each of the following.

(a) E ( X ) (b) SDX (

62.

63.

61. (a) _________________________

)

61. (b) _________________________

Find (P3≤x≤5.2) x is given that the distribution for Normal ( 5,0.8).

62. ___________________________

Multiple-Choice Test. A multiple-choice test offers four possible answers for every question, of which only one is the correct answer. A student guesses on five of the questions. What is the probability that she gets exactly three of the guessed questions correct? 63. ____________________________

________________________________________________________________ 64.

65.

Find lim x→ 3

x3 − 27 3 x−

64. ___________________________

Yeast Growth. Let P(t)be the number of yeast cells in a laboratory experiment. Suppose that P(t)follows a logistic model of inhibited growth, given by P ⎞ dP ⎛ = 500⎟ . dt kP⎜1− ⎝

Find kif P(0)

= 10and P8=275. ( )

65. ___________________________

66. Use your grapher to graphf ( x) = 3 x3 − 2 x4 . Then sketch the graph.

66.

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Final Exam, Form B

3.

2 1.Write an equation of the line with slope − and containing the 3 ). point(3,−1

____________________ 1. ___

2. For f ( x)= 2x2+4, find the simplified difference quotient.

2. ______________________

Solve for the missing side, x.

x

2.81

55

3.______________________

4. Find the amplitude, period and mid-line. Also, find the maximum and minimum.

y =3 cos (4 t ) − 1 ⎧ x2−5, for x ≠ 2 5. (a) Graph: f(x⎪)= for x = 2 ⎨ 0, ⎪ ⎩

4. ______________________

5. (a) 5

y

4 3 2 1 −5

−4

−3

−2

−1

−1 −2

x 1

2

3

4

5

−3 −4 −5

(b) Find limf ( x) .

5. (b) ____________________

(c) Find f (2 ) .

5. (c) ____________________

2? (d) Is f continuous at

5. (d) ____________________

x→ 2

244 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form B Find the limit, if it exists.

6. lim(5x3−3x+4) x→ 2

8

7. lim

6.___________________________

7.___________________________

3 (x+3)

x→−

x +1 8. lim 2 x→−1 x− 4 x − 5

8.___________________________

1 9. lim x5

9.___________________________

x→∞

Differentiate.

10. y=x3 + 2 x −10

10. ___________________________

11. y=x43

11. ___________________________

12.

13.

f(x) = cos2( x )

12. ___________________________

f(x) = ( x + 5 )(x − 2 )

4

13. ___________________________

8 5x 14. f ( x) = − x3+ 3

14. ___________________________

6 x +4) (x2−

15. ___________________________

) x() sec 16. f ( x ) =cos ( x

16. ___________________________

15. y =ln

4x2

17. y = e

−3 x

17. ___________________________

18. y = sin ( cos ))(3 x

18. ___________________________

d2y . dx2

19. ___________________________

d 20. Differentiate the following function implicitly to find. y x10− 5x2y2=4 d x 21. Find the equation of the tangent line to the graph of y=3x2−2x + 4e−xat the point (0,4).

20. ___________________________

19. For y = 8 x3 − 2 x5 , find

21. __________________________

CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form B 245 Find the relative extrema of the function. Find the relative extrema of trigonometric

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

22.

f(x)=2x3+12x2+18x+5

22.

10

y

8 6 4 2 −10

−8

−6

−4

−2

x 2

−2

4

6

8

10

−4 −6 −8 −10

23.

f ( x) = x4 − 6

23.

10

y

8 6 4 2 −10

−8

−6

−4

−2

x 2

−2

4

6

8

10

−4 −6 −8 −10

24.

6x f ( x) = − x2 + 4

24.

10

y

8 6 4 2 −10

−8

−6

−4

−2

−2 −4 −6 −8 −10

x 2

4

6

8

10

246 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form B 25.

f ( x) = cosx + sin x

25.

2.0

y

0 1.5 0

x

1.0 0 −0.

π/2

π

3π/2

2π

0.5 50 0 −1.

00 −1.

50

Find absolute maximum and minimum values of the function, if they exist, over the −2. indicated interval. Where no interval is specified, use the real line. List your answers 00 in terms of ordered pairs.

26.

27.

f(x)=2x2+3x−7 1 f(x)= x3+x2−8x+4;[0,4]

26. ___________________________ 27. ___________________________ 28. ___________________________

3

28. y=sin2(x)+cos(x);[0,2π] 29. 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?

29. ___________________________

30. Use linearization to approximate 122.

30. ___________________________

31. Use Newton’s method and1=1 x to find an approximate

solution of f(x)=3x2−4. (1.154700)

31. ___________________________

32. Population Growth. A town is experiencing a growth pattern of 6% per year in its population P; that is dP

=0.06P, dt where P is the population of the town and t is the time, in years, from 2000. (a) Given that the town’s population in 2000 was 8500, 0=8500 find the solution of the equation assuming P and k=0.06.

(b)

What will the population be in 2015?

(c) What is the doubling time of the population?

32. (a) _________________________ 32. (b) _________________________

32. (c) _________________________

CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form B 247 Evaluate.

33. ∫ 4 x 5 dx 34. ∫4 ex + 2

0

(

3 x ) dx

33. ______________________

34. ______________________

35. ∫ 2 x x ) dx

35. ______________________

etantsec2tdt 36. ∫

36. ______________________

( x + 6) lnxdx

37. ______________________

sin ( ∫

37.

50 x dx 38. ∫

38. ______________________

4 x x 39. ∫ 1 dx 2

39. ______________________

y = − x4 + 8xover the 40. Find the area under the graph of

interval [0,2].

41. Find the volume generated by revolving about the x-axis (x)=e2x, x= 0, and x = 5. the regions bounded fby

40. ______________________

41. ______________________

Determine whether the improper integral is convergent or divergent, and calculate its value if it is convergent. ∞ 42. ∫x−4dx 1 1

42. ______________________

dx 43. ∫ 3x

43. ______________________

44. Compute. ⎡ 3 −1⎤ 3 −1 0 ⎤ ⎢2 5 ⎡ 8 2 6 ⎥ ⎣ ⎦ ⎦⎥ ⎣ ⎢

44. ______________________

∞

0

248 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form B 45. Find the determinant of the matrix and determine if the matrix is invertible. ⎡1 ⎢1 ⎢ ⎢6 ⎣

−1 0 ⎤ 0 −1 ⎥ ⎥ −2 −3 ⎥ ⎦

45. ___________________________

46. Find the eigenvalues and corresponding eigenvectors.

⎡3 2 ⎤ ⎢4 5 ⎥ ⎣ ⎦

46. ___________________________

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

⎡ 1.8 2 ⎤ G=⎢ ⎣0.5 1.7⎥ ⎦ The population vector for year 1 is ⎡hatchlings⎤ ⎡100⎤ ⎥. ⎥=⎢ p=⎢ ⎣ adults ⎦ ⎣59 ⎦

(a) Estimate the population vector for year 2.

47. (a) _________________________

(b) Estimate the population vector for year 3.

47. (b) _________________________

Given f ( x , y=) 6 x2 + xy5− 7 e

cosx) ( , find each of the following.

48.

∂f ∂y

48. ___________________________

49.

∂ f ∂y ∂x

49. ___________________________

2

50. Find the Jacobian of f ( x , y=) x y2 −

cosexy ( .)

(

2)and

g ( x, y) =

51. Find the relative maximum and minimum values of f (x,y ) =3x2−5y2+4.

50. ___________________________

51. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form B 249 52. Evaluate:∫∫ex − 3 0

2

−1 0

(

) dydx

52. ______________________

53. Solve the differential equation y ′ := 6 x 3 y.

53. ______________________

54. Solve the initial value problem y :′ = sin x + 2 x; y(0 ) = 2.

54. ______________________

4y − 8 y 55. For the differential equation′ :=y2 (a) Find the equilibrium value(s).

55. (a) _____________________

(b) Access the stability of each equilibrium value.

55. (b) _____________________

(c) Determine the point(s) of inflection; and

55. (c) _____________________

(d)

Sketch sample solutions of the differential equation.

56. A tank initially contains 100 lb of salt dissolved in 200 gal of water. Saltwater containing 4 lb of salt per gal enters the tank at the rate of 5 gal per min. The mixture (kept uniform by stirring) is removed at the same rate. How many pounds of salt are in the tank after an hour?

57. Solve: y′′+3 y′ − 10 y = 0.

55. (d) _____________________

56. ______________________

57. ______________________

58. Solve the first-order linear system:

x′ = 10x−4y,y ′ = − x +8y

58. ______________________

59. Two cards are dealt without replacement from a well shuffled deck of cards. Find the probability that the first card is a heart or a club.

59. ______________________

60. Testing for a disease. A diagnostic test for a certain disease has a sensitivity of 0.98 and a specificity of 0.96.

The probability that a patient has the disease is estimated as 0.2. The diagnostic test is given and returns negative. What is the posterior probability that the patient does not have the disease?

60. ______________________

250 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form B 61.

The probability density function Xis f(x) x2 125 = 3of over [0,5]. Compute each of the following. (a) E ( X ) (b) SDX (

62.

63.

61. (a) _________________________

)

61. (b) _________________________

Find P ( 9.15≤x≤10) given that the distribution for xis Normal 62. ___________________________ ( 10,0.5).

Multiple-Choice Test. A multiple-choice test offers four possible answers for every question, of which only one is the correct answer. A student guesses on five of the questions. What is the probability that she gets exactly 63. ____________________________ two of the guessed questions correct?

________________________________________________________________ 64.

4 x3 − y. f Find ∂ for f (x,y)= xy + 2 ∂y

65.

P(t)be the number of yeast cells in Yeast GrowthLet .

64. ___________________________

a laboratory experiment. Suppose that P(t)follows a logistic model of inhibited growth, given by

dP

⎛ = kP 1 ⎝⎜− dt

P ⎞

.

⎟ 600⎠ = 10and P5( ) = 375.

Find kif P(0)

65. ___________________________

66. Use your grapher to graphf ( x) = 8 x3 − 3 x5 . Then

sketch the graph.

66.

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Final Exam, Form C 1.Write an equation of the line with slope −3 and containing the 2 point⎛⎜ 3 ,−1⎞⎟ . ⎝ ⎠

____________________ 1. ___

.

2. For f(x)=4x2−x, find the simplified difference quotient 2. ______________________ 3. Solve for the missing side, x.

7.81 32

x

3.______________________

4. Find the amplitude, period and mid-line. Also, find the maximum and minimum.

y=

1 cos(2πt)+ 3 2

4. ______________________

2

⎧ x2+ 2, forx ≠ −1 5. (a) Graph:f () =⎪−⎨ for x = −1 ⎪⎩−1,

5. (a) 5

y

4 3 2 1 −5

−4

−3

−2

−1

−1 −2

x 1

2

3

4

5

−3 −4 −5

(b) Find limf ( x) .

5. (b) ____________________

(c) Find f (− 1).

5. (c) ____________________

(d) Is f continuous at −1 ?

5. (d) ____________________

x→−1

252 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form C Find the limit, if it exists.

6. lim(−x3+3x−5) x→ 2

6 7. lim x→0x

8. lim

7.___________________________

x+2

8.___________________________

x→−2 x2 4 −

9. lim

1

9.___________________________

x

x→∞

6.___________________________

Differentiate.

10. y=x4− 3 x2 + 5

10. ___________________________

11. y=x34

11. ___________________________

12.

13.

14.

f(x)= sinx3()

12. ___________________________

f(x) = ( x + 4 )(x − 2 )

13. ___________________________

f ( x ) =x2 3− 4

14. ___________________________

x − 3x +1) =ln(

15. ___________________________

5

15. y

x +5

2

) x() 16. f ( x ) =tan ( 2xsec

16. ___________________________

x2

17. y = e − 4 x

17. ___________________________

18. y = cos )) x ( tan(3

18. ___________________________

d2y . dx2

19. ___________________________

d 20. Differentiate the following function implicitly to find. y 2 xy 4 − 3=4 x 2 d x 21. Find the equation of the tangent line to the graph of y=x 2 − 5 + 3exat the point (0,−2).

20. ___________________________

19. For y = 2 x6 − 4 x3 , find

21. __________________________

CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form C 253 Find the relative extrema of the function. Find the relative extrema of trigonometric

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

22.

f(x)=x3+6x2+9x+2

22.

10

y

8 6 4 2 −10

−8

−6

−4

−2

x 2

−2

4

6

8

10

−4 −6 −8 −10

23.

f ( x) = − 2 x2 − 8 + x4

23.

10

y

8 6 4 2 −10

−8

−6

−4

−2

x 2

−2

4

6

8

10

−4 −6 −8 −10

24.

f ( x) =

12x x 2+4

24.

10

y

8 6 4 2 −10

−8

−6

−4

−2

−2 −4 −6 −8 −10

x 2

4

6

8

10

254 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form C 25.

f ( x) = sin2x + cosx

25.

2.0

y

0 1.5 0

x

1.0 0 −0.

π/2

π

3π/2

2π

0.5 50 0 −1.

00 −1.

50

Find absolute maximum and minimum values of the function, if they exist, over the −2. indicated interval. Where no interval is specified, use the real line. List your answers 00 in terms of ordered pairs.

f(x)=−x2+2x+1 26.

27.

2 f(x)= x3−x2−4x+6;[−3,3]

26. ___________________________ 27. ___________________________ 28. ___________________________

3

28. y=sin(x)−cos(x);[0,2π] 29. 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 is the maximum volume?

29. ___________________________

30. Use linearization to approximate 170.

30. ___________________________

31. Use Newton’s method and1=3 x to find an approximate

solution of f(x)=4x2−2. (0.707107)

31. ___________________________

32. Population Growth. A town is experiencing a growth pattern of 8% per year in its population P; that is dP

=0.08P, dt where P is the population of the town and t is the time, in years, from 2005. (a) Given that the town’s population in 2005 was 400, 0=400 find the solution of the equation assuming P and k=0.08.

(b)

What will the population be in 2015?

(c) What is the doubling time of the population?

32. (a) _________________________ 32. (b) _________________________

32. (c) _________________________

CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form C 255 Evaluate.

33. ∫ 6 x 4 dx

33. ______________________

) dx

34. ______________________

35. ∫ 2 xcos (2

35. ______________________

ecott csc2tdt

36. ______________________

6) lnxdx 37. ∫ ( x +

37. ______________________

−2 dx 38. ∫ x

38. ______________________

4 xdx 39. ∫

39. ______________________

34. ∫3 x + 4 ex 2

0

(

x ) dx

∫

36.

9

0

y = x3 − 4xover the 40. Find the area under the graph of

interval [−2,0].

41. Find the volume generated by revolving about the x-axis (x)=ex10 the regions bounded fby , x= 2, and x = 5.

40. ______________________

41. ______________________

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

∞ 1 dx 42. ∫ 5x

42. ______________________

43. ∫x−4dx

43. ______________________

0

∞

1

44. Compute.

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

−1 ⎤ 0⎥ ⎥ −3 ⎥⎦

44. ______________________

256 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form C 45. Find the determinant of the matrix and determine if the matrix is invertible.

⎡0 2 1 ⎤ ⎢3 2⎥ ⎢ −1 ⎥ ⎢4 0 1 ⎥ ⎣ ⎦

45. ___________________________

46. Find the eigenvalues and corresponding eigenvectors.

⎡ 4 −2⎤ ⎢2 7 ⎥ ⎣ ⎦

46. ___________________________

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

⎡ 0.7 2 ⎤ G=⎢ ⎣0.5 0.4⎥ ⎦ The population vector for year 1 is ⎡hatchlings⎤ ⎡200⎤ ⎥. ⎥=⎢ p=⎢ ⎣ adults ⎦ ⎣170⎦

(a) Estimate the population vector for year 2.

47. (a) _________________________

(b) Estimate the population vector for year 3.

47. (b) _________________________

Given f ( x , y=) 2 y2 + xy3− 2 e

sinx) ( , find each of the following.

48.

∂f ∂y

48. ___________________________

49.

∂ f ∂y ∂x

49. ___________________________

2

50. Find the Jacobian of f ( x , y=) 2 yx ( x + 3) and

sine ( x) .

g ( x, y) =

2y

51. Find the relative maximum and minimum values of f (x,y ) =5x2+6y2+2.

50. ___________________________

51. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form C 257 2 ex + 1 52. Evaluate:∫ 2∫ 4

(0 1

) dydx

52. ______________________

53. Solve the differential equation y ′ := 4 x 5 y.

53. ______________________

54. Solve the initial value problem′:=ycosx − 2;x y(0 ) = 4.

54. ______________________

55.

For the differential equation′ :=y10 y − 5 y2 (a) Find the equilibrium value(s).

55. (a) _____________________

(b) Access the stability of each equilibrium value.

55. (b) _____________________

(c) Determine the point(s) of inflection; and

55. (c) _____________________

(d)

Sketch sample solutions of the differential equation.

56. A tank initially contains 150 lb of salt dissolved in 400 gal of water. Saltwater containing 5 lb of salt per gal enters the tank at the rate of 5 gal per min. The mixture (kept uniform by stirring) is removed at the same rate. How many pounds of salt are in the tank after an hour?

57. Solve: y′′+ 11y ′ − 12 y = 0.

55. (d) _____________________

56. ______________________

57. ______________________

58. Solve the first-order linear system:

x′ = −5 x − 5 y , y′ = 4 x + 7 y

58. ______________________

59. Two cards are dealt without replacement from a well shuffled deck of cards. Find the probability that the first card is an ace or a jack.

59. ______________________

60. Testing for a disease. A diagnostic test for a certain disease has a sensitivity of 0.82 and a specificity of 0.88.

The probability that a patient has the disease is estimated as 0.25. The diagnostic test is given and returns positive. What is the posterior probability that the patient has the disease?

60. ______________________

258 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form C 61.

3 The probability density function = xof4 Xis f(x) over [0,2]. Compute each of the following.

(a) E ( X )

61. (a) _________________________

(b) SDX (

62.

63.

)

61. (b) _________________________

Find P ( 9≤x≤12.6) given that the distribution for xis Normal 62. ___________________________ ( 12,1.5).

Multiple-Choice Test. A multiple-choice test offers five possible answers for every question, of which only one is the correct answer. A student guesses on six of the questions. What is the probability that she gets exactly three of the guessed questions correct? 63. ____________________________

________________________________________________________________ 64.

Differentiate ) : f(x=x2ln(xex).

65.

Yeast Growth. Let Ptbe ( ) the number of yeast cells

64. ___________________________

in a laboratory experiment. Suppose that P(t)follows a logistic model of inhibited growth, given by

dP dt

⎛ = kP1⎜

P ⎞.

−600⎟

Find kif P(0) ⎠ = 15 and P (5)=400.

65. ___________________________

66. Use your grapher to approximate the integral:

∫x ∞

−∞

2

3 +2

dx.

66. ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Final Exam, Form D 1.Write an equation of the line with slope −3 and containing the 1 point⎛⎜ 3,2 ⎞⎟ . ⎝ ⎠

____________________ 1. ___

.

2. For f(x)=3x2−x, find the simplified difference quotient 2. ______________________ 3. Solve for the missing side, x.

38

3.18

x

3.______________________

4. Find the amplitude, period and mid-line. Also, find the maximum and minimum.

y =3 sin (4 t ) − 2 x⎧⎪4 ) − x2, for x≠1 5. (a) Graph: f(=⎨ for x=1 ⎪⎩1,

4. ______________________

5. (a) 5

y

4 3 2 1 −5

−4

−3

−2

−1

−1 −2

x 1

2

3

4

5

−3 −4 −5

(b) Find lim f ( x) .

5. (b) ____________________

(c) Find f (1 ) .

5. (c) ____________________

1? (d) Is f continuous at

5. (d) ____________________

x→1

260 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form D Find the limit, if it exists.

6.

lim(2+xx2+ 1)

6.___________________________

x→−1

6x 7. lim x→ 2 x− 2

7.___________________________

x+3 8. lim 2 − x − x→−3 x 12

8.___________________________

5 9. lim− x3

9.___________________________

x→∞

Differentiate.

10. y=x8 − 3x 2 + 6

10. ___________________________

11. y=x34

11. ___________________________

12.

f(x) = cosx)(

12. ___________________________

) = ( x + 3x) ( 4 5 − ) ) = 5 x − 2 (

13. f ( x 14. f x

13. ___________________________

14. ___________________________

3−x2

= ln ( x− 7 x +1)

15. ___________________________

16. f ( x ) =cot ( x ) sin)(2 x

16. ___________________________

17. y = ex 2− x−3

17. ___________________________

15. y

18. y

2

x ))( =sin( cos 2

18. ___________________________

d2y . dx2

19. ___________________________

d 20. Differentiate the following function implicitly to find. y 2 xy 2 d 8 x+ = x 21. Find the equation of the tangent line to the graph of y=lnx+x 2 + 4at the point (1,5).

20. ___________________________

19. For y = 5 x4 +16 x , find

21. __________________________

CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form D 261 Find the relative extrema of the function. Find the relative extrema of trigonometric

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

22.

f(x)=x3−6x2+9x−8

22.

10

y

8 6 4 2 −10

−8

−6

−4

−2

x 2

−2

4

6

8

10

−4 −6 −8 −10

23.

f ( x) = 2 x2 + 5 − x4

23.

10

y

8 6 4 2 −10

−8

−6

−4

−2

x 2

−2

4

6

8

10

−4 −6 −8 −10

24.

12x f ( x) = − x 2+ 9

24.

10

y

8 6 4 2 −10

−8

−6

−4

−2

−2 −4 −6 −8 −10

x 2

4

6

8

10

262 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form D 25.

f ( x) = sin x + cos2( x)

25.

2.0

y

0 1.5 0

x

1.0 0 −0.

π/2

π

3π/2

2π

0.5 50 0 −1.

00 −1.

50

Find absolute maximum and minimum values of the function, if they exist, over the −2. indicated interval. Where no interval is specified, use the real line. List your answers 00 in terms of ordered pairs.

26.

27.

f(x)=−4x2+5x−3 2 f(x)= x3−4x2−10x+7;[1,10]

26. ___________________________ 27. ___________________________ 28. ___________________________

3

28. y=sin(2x)−cos(x);[0,2π] 29. 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?

29. ___________________________

30. Use linearization to approximate63.

30. ___________________________

31. Use Newton’s method and1=3 x to find an approximate

solution of f(x)=4x2−2. (0.707107)

31. ___________________________

32. Population Growth. A bacteria culture is experiencing a growth pattern of 15% per hour in its population P; that is dP

=0.15P, dt where P is the population of the culture and t is the time, in hours, from noon. . (a) Given that the town’s population at noon was 8000, 0=8000 find the solution of the equation assuming P and k=0.15.

(b)

What will the population be at 5 pm?

(c) What is the doubling time of the population?

32. (a) _________________________ 32. (b) _________________________

32. (c) _________________________

CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form D 263 Evaluate.

33. ∫ 4 x 3 dx 34. ∫4 ex +

(

2

1

33. ______________________

6 x ) dx

34. ______________________

x ) dx x sin (3 35. ∫

35. ______________________

esint costdt 36. ∫

36. ______________________

2) lnxdx 37. ∫ ( x +

37. ______________________

−8 dx 38. ∫ x

38. ______________________

4 3 xdx 39. ∫

39. ______________________

8

0

y = x3 + 2xover the 40. Find the area under the graph of

interval [0,2].

40. ______________________

41. Find the volume generated by revolving about the x-axis (x)=ex4, x= 0, and x = 8. the regions bounded fby

41. ______________________

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

42. ∫10x dx

42. ______________________

43. ∫x−8dx

43. ______________________

∞

1

1

∞

1

44. Compute.

⎡ −3 ⎡−1 −8 0⎤ ⎢ ⎢ 4 0 2⎥⎢1 ⎣ ⎦ ⎣⎢ −2

0⎤ ⎥ 7⎥ 4⎥⎦

44. ______________________

264 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form D 45. Find the determinant of the matrix and determine if the matrix is invertible. ⎡0 2 −1⎤ ⎢3 1⎥ ⎢ −2 ⎥ ⎢3 2 1⎥ ⎣ ⎦

45. ___________________________

46. Find the eigenvalues and corresponding eigenvectors.

⎡ 4 2⎤ ⎢ ⎥ ⎣ −2 7 ⎦

46. ___________________________

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

⎡1.2 G=⎢ ⎣ 0.8

1. ⎤ 9 ⎥ 0. ⎦ 6

The population vector for year 1 is

⎡hatchlings⎤ ⎡225⎤ ⎥. ⎥=⎢ p=⎢ ⎣ adults ⎦ ⎣185⎦

(a) Estimate the population vector for year 2.

47. (a) _________________________

(b) Estimate the population vector for year 3.

47. (b) _________________________

Given f ( x , y=) 2xy + y2 + 3e

cosy) ( , find each of the following.

48.

∂f ∂y

48. ___________________________

49.

∂ f ∂y ∂x

49. ___________________________

2

( −1)and

50. Find the Jacobian of f ( x , y=)3 x y2

cosexy ( 2. )

g ( x, y) =

51. Find the relative maximum and minimum values of f (x,y ) = −2x2−5y2+6.

50. ___________________________

51. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form D 265 52. Evaluate:∫∫2 y2 + ey 1

x

−1 0

(

) dydx

52. ______________________

y′=:−x3y.

53. Solve the differential equation

53. ______________________

54. Solve the initial value problem′:=ysin x + 3 x; y(0 ) = 1.

54. ______________________

55.

For the differential equation′ :=y3y − 6y 2 (a) Find the equilibrium value(s).

55. (a) _____________________

(b) Access the stability of each equilibrium value.

55. (b) _____________________

(c) Determine the point(s) of inflection; and

55. (c) _____________________

(d)

Sketch sample solutions of the differential equation.

56. A tank initially contains 200 lb of salt dissolved in 500 gal of water. Saltwater containing 4 lb of salt per gal enters the tank at the rate of 5 gal per min. The mixture (kept uniform by stirring) is removed at the same rate. How many pounds of salt are in the tank after 2 hours?

57. Solve: y′′−2 y′ − 15 y = 0.

55. (d) _____________________

56. ______________________

57. ______________________

58. Solve the first-order linear system:

x′=

3 2

x−

3 y,y′=3x−4y 2

58. ______________________

59. Two cards are dealt without replacement from a well shuffled deck of cards. Find the probability that the first card is a five or a queen.

59. ______________________

60. Testing for a disease. A diagnostic test for a certain disease has a sensitivity of 0.82 and a specificity of 0.88.

The probability that a patient has the disease is estimated as 0.25. The diagnostic test is given and returns negative. What is the posterior probability that the patient does not have the disease?

60. ______________________

266 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form D 61.

The probability density function Xis f(x) x3 625 = 4of over [0,5]. Compute each of the following. (a) E ( X ) (b) SDX (

62.

63.

61. (a) _________________________

)

61. (b) _________________________

Find P ( 3.1≤x≤3.5) given that the distribution for xis Normal 62. ___________________________ ( 3,0.25).

Multiple-Choice Test. A multiple-choice test offers five possible answers for every question, of which only one is the correct answer. A student guesses on six of the questions. What is the probability that she gets exactly 63. ____________________________ two of the guessed questions correct?

________________________________________________________________ 64.

65.

Differentiate: f(x)=exln(xex).

64. ___________________________

Yeast Growth. Let P(t)be the number of yeast cells in a laboratory experiment. Suppose that P(t)follows a logistic model of inhibited growth, given by

P ⎞ ⎛ = kP 1 − ⎟. ⎝⎜ dt 700⎠ dP

Find kif P(0) = 20 and P (8)=500.

65. ___________________________

66. Use your grapher to approximate the integral:

∫2 + x dx. ∞

−∞

4

2

66. ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Final Exam, Form E 1 1.Write an equation of the line with slope − 2 and containing the point(3, − 4 ).

____________________ 1. ___

2. For f ( x)= 3x2+4, find the simplified difference quotient.

2. ______________________

3. Solve for the missing side, x.

48

6.27

x

3.______________________ 4. Find the amplitude, period and mid-line. Also, find the maximum and minimum.

y=2cos(2πt)+3 x⎧⎪x2 ) − 4, for x ≠ 2 5. (a) Graph: f(=⎨ for x=2 ⎪⎩4,

4. ______________________

5. (a) 5

y

4 3 2 1 −5

−4

−3

−2

−1

−1 −2

x 1

2

3

4

5

−3 −4 −5

(b) Find limf ( x) .

5. (b) ____________________

(c) Find f (2 ) .

5. (c) ____________________

2? (d) Is f continuous at

5. (d) ____________________

x→ 2

268 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form E Find the limit, if it exists.

6.

lim(6x4−3x+8)

x→−1

6.___________________________

10 7. lim x→0x

7.___________________________

x −3 8. lim 2 x→−3 x+ x −12

8.___________________________

4 9. lim x 4 x

9.___________________________

→∞

Differentiate.

10. y=3x3+

x +18

10. ___________________________

11. y=x35

11. ___________________________

f(x)= sinx3()

12. ___________________________

f(x) = ( x − 2)(x + 3)4

13. ___________________________

f (x) 3=2x − 1

14. ___________________________

x − 2 x) =ln(

15. ___________________________

16. f ( x ) =tan( x) sin ( x )

16. ___________________________

17. y = ex4−1

17. ___________________________

18. y = sin sin (2 ))x

18. ___________________________

12.

13.

14.

15. y

x +6 4

(

d2y . dx2

19. ___________________________

d 20. Differentiate the following function implicitly to find. y x 2+ 4x=xy d x 21. Find the equation of the tangent line to the graph of y=4lnx+x 2 − 2at the point (1,−1).

20. ___________________________

19. For y = 13x 3 + 8 x , find

21. __________________________

CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form E 269 Find the relative extrema of the function. Find the relative extrema of trigonometric

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

22.

f(x)=−x3−6x2−9x+5

22.

10

y

8 6 4 2 −10

−8

−6

−4

−2

x 2

−2

4

6

8

10

−4 −6 −8 −10

23.

2 4 f ( x) = 8 −x8 +x

23.

10

y

8 6 4 2 −10

−8

−6

−4

−2

x 2

−2

4

6

8

10

−4 −6 −8 −10

24.

f ( x) =

6x 3 + x2

24.

10

y

8 6 4 2 −10

−8

−6

−4

−2

−2 −4 −6 −8 −10

x 2

4

6

8

10

270 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form E 25.

1 f ( x) = sin⎛⎜ 2 x ⎞⎟ − cosx() ⎝ ⎠

25.

2.0

y

0 1.5 0

x

1.0 0 −0.

π/2

π

3π/2

2π

0.5 50 0 −1.

00 −1.

Find absolute maximum and minimum values of the function, if they exist 50 , over the indicated interval. Where no interval is specified, use the real line. List your answers −2. in terms of ordered pairs.

26.

27.

f(x)=2x2+ 8x − 10 f(x)=

28.

y=

1 3

00

26. ___________________________ 27. ___________________________

x3 − 2x2 + 3x+2;[−2,2]

⎛ 1 x⎟cos ⎞ (x);[0,2π] + ⎝2 ⎠

28. ___________________________

29. 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?

29. ___________________________

30. Use linearization to approximate 63.

30. ___________________________

1=2xto find an approximate 31. Use Newton’s method and

solution of f(x)=8x2−5.

31. ___________________________

32. Population Growth. A bacteria culture is experiencing a growth pattern of 10% per hour in its population P; that is dP

=0.1P, dt where P is the population of the culture and t is the time, in hours, from noon. . (a) Given that the town’s population at noon was 10000, 0=10000 find the solution of the equation assuming P and k=0.1.

32. (a) _________________________

(b) What will the population be at 3 pm?

32. (b) _________________________

(c) What is the doubling time of the population?

32. (c) _________________________

CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form E 271 Evaluate.

33. ∫ 3 x 4 dx

33. ______________________

) dx

34. ______________________

( x ) dx 35. ∫ 4 xcos

35. ______________________

sin2t costdt 36. ∫

36. ______________________

x −(8) lnxdx 37. ∫

37. ______________________

2 38. ∫ x dx

38. ______________________

16 2 39. ∫ 0 dx x

39. ______________________

34. ∫8 ex + 3 x2

(

2

1

y = 5 x − x3 over the 40. Find the area under the graph of

interval [0,2].

41. Find the volume generated by revolving about the x-axis the regions bounded f(x)=e−x by , x= 1, and x = 3.

40. ______________________

41. ______________________

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

42. ∫x−19dx

42. ______________________

exdx 43. ∫ 3

43. ______________________

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

44. ______________________

∞

1

∞

1

3

272 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form E 45. Find the determinant of the matrix and determine if the matrix is invertible. ⎡ 1 −1 0 ⎤ ⎢1 0 −1⎥ ⎢ ⎥ ⎢ −6 2 3 ⎥ ⎣ ⎦

45. ___________________________

46. Find the eigenvalues and corresponding eigenvectors.

⎡ −3 −5⎤ ⎢6 8 ⎥⎦ ⎣

46. ___________________________

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

⎡ 0.8 3 ⎤ G=⎢ ⎥ ⎣0 .6 0.4⎦ The population vector for year 1 is

⎡hatchlings⎤ ⎡200⎤ ⎥. ⎥=⎢ p=⎢ ⎣ adults ⎦ ⎣175⎦

(a) Estimate the population vector for year 2.

47. (a) _________________________

(b) Estimate the population vector for year 3.

47. (b) _________________________

y + 2 x − 6 esin y),(find each of the following.

Given f (x, y) = x 48.

∂f ∂y

48. ___________________________

49.

∂ f ∂y ∂x

49. ___________________________

2

50. Find the Jacobian of f ( x , y=) 5 x 2 y2 and

sinxe (22 ) .

g ( x,y ) =

yx

51. Find the relative maximum and minimum values of f (x,y ) =2x2−y2+6.

50. ___________________________

51. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form E 273

∫ ( x ) dydx

52. Evaluate:∫ 3ey +

52. ______________________

53. Solve the differential equation y ′ := −8 x 7 y.

53. ______________________

2− x;y(0 ) = −1.

54. Solve the initial value problem′:=ycos (x)

54. ______________________

1

1

−1 0

55.

For the differential equation′ :=yy2 − 5 y (a) Find the equilibrium value(s).

55. (a) _____________________

(b) Access the stability of each equilibrium value.

55. (b) _____________________

(c) Determine the point(s) of inflection; and

55. (c) _____________________

(d)

Sketch sample solutions of the differential equation.

56. A tank initially contains 75 lb of salt dissolved in 250 gal of water. Saltwater containing 3 lb of salt per gal enters the tank at the rate of 4 gal per min. The mixture (kept uniform by stirring) is removed at the same rate. How many pounds of salt are in the tank after an hour?

57. Solve: y′′+9 y′ + 20 y = 0.

55. (d) _____________________

56. ______________________

57. ______________________

58. Solve the first-order linear system:

x′ = 6 x − 2 y , y′ = 7 x − 3 y

58. ______________________

59. Two cards are dealt without replacement from a well shuffled deck of cards. Find the probability that the first card is a diamond or a heart.

59. ______________________

60. Testing for a disease. A diagnostic test for a certain disease has a sensitivity of 0.76 and a specificity of 0.84. The probability that a patient has the disease is estimated as 0.15. The diagnostic test is given and returns positive. What is the posterior probability that the patient has the disease?

60. ______________________

274 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form E 61.

The probability density function Xis f(x) x4 32 = 5of over [0,2]. Compute each of the following. (a) E ( X ) (b) SDX (

62.

63.

61. (a) _________________________

)

61. (b) _________________________

Find P ( 18≤x≤21) given that the distribution for xis Normal 62. ___________________________ ( 20,2.5).

Multiple-Choice Test. A multiple-choice test offers four possible answers for every question, of which only one is the correct answer. A student guesses on four of the questions. What is the probability that she gets exactly three of the guessed questions correct? 63. ____________________________

________________________________________________________________ 64.

65.

f 2 y 3 x3 Find ∂ for f(x,y)= − . x+ 2y ∂y

64. ___________________________

Yeast Growth. Let P(t)be the number of yeast cells in a laboratory experiment. Suppose that P(t)follows a logistic model of inhibited growth, given by

dP dt

= kP

P ⎞ ⎟. − 500⎠ ⎜1 ⎝ ⎛

Find kif P(0) = 10 and P (8)=275.

65. ___________________________

66. Use your grapher to approximate the area between the following curves:

y = x 3− 3 x 2 − 4 x , y = x2− 4x, x = 0, x= 4.

66. ___________________________

CALCULUS FOR THE LIFE SCIENCES

Name: _________________

Final Exam, Form F 1.Write an equation of the line with slope −4 and containing the point(−3,5).

____________________ 1. ___

.

2. For f ( x)=2x2+3x, find the simplified difference quotient 2. ______________________ 3. Solve for the missing side, x.

27

2.87

x

3.______________________

4. Find the amplitude, period and mid-line. Also, find the maximum and minimum.

y =2 sin (3 t ) − 5 ⎧− x2 + 2, forx ≠ 1 5. (a) Graph: f () =⎪⎨ ⎪⎩0, for x = 1

4. ______________________

5. (a) 5

y

4 3 2 1 −5

−4

−3

−2

−1

−1 −2

x 1

2

3

4

5

−3 −4 −5

(b) Find lim f ( x) .

5. (b) ____________________

(c) Find f (1 ) .

5. (c) ____________________

(d) Is f continuous at 1?

5. (d) ____________________

x→1

276 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form F Find the limit, if it exists.

6.

lim(4x3−x2+x)

x→−1

12 7. lim x→−3 x + 3

7.___________________________

x +5 x− 10 x→−5 x + 3

8. lim

6.___________________________

8.___________________________

2

8 9. lim x

9.___________________________

x→∞

Differentiate.

10. y=5x4 − 2 x 3+6

10. ___________________________

11. y=x43

11. ___________________________

12.

f(x) = cosx()

12. ___________________________

13. f ( x) = ( x − 5 ) ( x)+ 1 4

( )=3x+2

14. f x

13. ___________________________

14. ___________________________

x2−8

5x) (x3−

15. ___________________________

x) ( 16. f ( x ) =sec ( x ) cot

16. ___________________________

17. y = ex 5 −x

17. ___________________________

18. y = cos ( cos )) (3x

18. ___________________________

15. y =ln

d2y . dx2

19. ___________________________

d 20. Differentiate the following function implicitly to find. y d 4 − x=2xy x 21. Find the equation of the tangent line to the graph of y=x4−lnx + 1at the point (1,2).

20. ___________________________

19. For y = 4 x4 + 3 x2 , find

21. __________________________

CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form F 277 Find the relative extrema of the function. Find the relative extrema of trigonometric

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

22.

f(x)=−2x3+12x2−18x+6

22.

10

y

8 6 4 2 −10

−8

−6

−4

−2

x 2

−2

4

6

8

10

−4 −6 −8 −10

23.

f ( x) = 8 x2 − x4 − 8

23.

10

y

8 6 4 2 −10

−8

−6

−4

−2

x 2

−2

4

6

8

10

−4 −6 −8 −10

24.

f ( x) =

20 x 5 + x2

24.

10

y

8 6 4 2 −10

−8

−6

−4

−2

−2 −4 −6 −8 −10

x 2

4

6

8

10

278 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form F 25.

f ( x) = sin2( x) + 2 cosx)(

25.

2.0

y

0 1.5 0

x

1.0 0 −0.

π/2

π

3π/2

2π

0.5 50 0 −1.

00 −1.

Find absolute maximum and minimum values of the function, if they exist 50 , over the indicated interval. Where no interval is specified, use the real line. List your answers −2. in terms of ordered pairs.

00

f(x)=6x2+3x−2 26.

27. f(x)= 28.

26. ___________________________ 27. ___________________________

2 x3−x2−4x+5;[−3,3]

28. ___________________________

3

f (x)=sin2 (x−) 2cos(x); [0, 2π]

29. 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?

82.

30. Use linearization to approximate

29. ___________________________

30. ___________________________

31. Use Newton’s method and1=2 x to find an approximate

solution of f(x)=6x2−5.

31. ___________________________

32. Population Growth. A bacteria culture is experiencing a growth pattern of 12% per hour in its population P; that is dP

=0.12P, dt where P is the population of the culture and t is the time, in hours, from noon. . (a) Given that the town’s population at noon was 40000, 0=40000 find the solution of the equation assuming P and k=0.12.

(b)

What will the population be at 6 pm?

(c) What is the doubling time of the population?

32. (a) _________________________ 32. (b) _________________________ 32. (c) _________________________

CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form F 279 Evaluate.

33. ∫ 8 x 4 dx

33. ______________________

) dx

34. ______________________

x ) dx x sin (2 35. ∫

35. ______________________

34. ∫2 ex + 2 2

0

(

∫

sint cos2tdt

36. ______________________

x −(6) lnxdx 37. ∫

37. ______________________

10 x dx 38. ∫

38. ______________________

2 xdx 39. ∫

39. ______________________

36.

9

4

y = 5 x − x2 over the 40. Find the area under the graph of

interval [2,3].

41. Find the volume generated by revolving about the x-axis (x)=e3x, x= 0, and x = 6. the regions bounded fby

40. ______________________

41. ______________________

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

1

42. ∫ x dx

42. ______________________

43. ∫x−3dx

43. ______________________

∞

0

∞

1

44. Compute. ⎡9 ⎢0 ⎣

−1⎤ −3 1 ⎤ 4 ⎡ 2 −1⎥⎦ ⎥ ⎢ ⎦ ⎣

44. ______________________

280 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form F 45. Find the determinant of the matrix and determine if the matrix is invertible.

⎡1 3 3 ⎤ ⎢1 4 3 ⎥ ⎥ ⎢ ⎢⎣1 3 4 ⎥ ⎦

45. ___________________________

46. Find the eigenvalues and corresponding eigenvectors.

⎡ −3 5 ⎤ ⎢ 6 8⎥ ⎣− ⎦

46. ___________________________

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

⎡1.2 2 ⎤ 0.3⎥ G=⎢ ⎦ ⎣ 0.5 The population vector for year 1 is

⎡hatchlings⎤ ⎡275⎤ ⎥. ⎥=⎢ p=⎢ ⎣ adults ⎦ ⎣220⎦ (a) Estimate the population vector for year 2.

47. (a) _________________________

(b) Estimate the population vector for year 3.

47. (b) _________________________

Given f ( x , y=)

3x cosx() 2 +e + y , find each of the following. y

48.

∂f ∂y

48. ___________________________

49.

∂ f ∂y ∂x

49. ___________________________

2

50. Find the Jacobian of f ( x , y=)3xy4 and

g ( x , y= )cos ye(3x ).

51. Find the relative maximum and minimum values of f (x,y)= −x2−2y2+8.

50. ___________________________

51. ___________________________

CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form F 281 0 2 1 y2dxdy 52. Evaluate:∫ −1∫ x +1

52. ______________________

53. Solve the differential equation y ′ := 3 x 4 y.

53. ______________________

3.

54. Solve the initial value problem′:=ysinx() − 4 x; y(0 ) = −

54. ______________________

55.

For the differential equation′ :=y3y2−6 y (a) Find the equilibrium value(s).

55. (a) _____________________

(b) Access the stability of each equilibrium value.

55. (b) _____________________

(c) Determine the point(s) of inflection; and

55. (c) _____________________

(d)

Sketch sample solutions of the differential equation.

56. A tank initially contains 125 lb of salt dissolved in 500 gal of water. Saltwater containing 4 lb of salt per gal enters the tank at the rate of 5 gal per min. The mixture (kept uniform by stirring) is removed at the same rate. How many pounds of salt are in the tank after two hours?

57. Solve: y′′+ y′ − 20 y = 0.

55. (d) _____________________

56. ______________________

57. ______________________

58. Solve the first-order linear system:

x′ = 6 x − 2 y , y′ = 7 x − 3 y

58. ______________________

59. Two cards are dealt without replacement from a well shuffled deck of cards. Find the probability that the first card is a two or a nine.

59. ______________________

60. Testing for a disease. A diagnostic test for a certain disease has a sensitivity of 0.76 and a specificity of 0.84. The probability that a patient has the disease is estimated as 0.15. The diagnostic test is given and returns negative. What is the posterior probability that the patient does not have the disease?

60. ______________________

282 CALCULUS FOR THE LIFE SCIENCES, Final Exam, Form F 61.

3 The probability density function = xof4 Xis f(x) over [0,2]. Compute each of the following.

(a) E ( X ) (b) SDX ( 62.

63.

61. (a) _________________________

)

61. (b) _________________________

Find (P4.6≤x≤6.2) given that the distribution for xis Normal 62. ___________________________ ( 5,1.6).

Multiple-Choice Test. A multiple-choice test offers four possible answers for every question, of which only one is the correct answer. A student guesses on four of the questions. What is the probability that she gets exactly three of the guessed questions correct? 63. ____________________________

________________________________________________________________ 64.

65.

Findlim

x→−3

x 4 − 5 x 2 − 36 x+ 3

.

64. ___________________________

Yeast Growth. Let P(t)be the number of yeast cells in a laboratory experiment. Suppose that P(t)follows a logistic model of inhibited growth, given by P ⎞ ⎛ = kP 1 800 ⎠⎟. ⎝⎜− dt

dP

Find kif P(0) = 25 and P (10)=300.

65. ___________________________

66. Use your grapher to approximate the area between the following curves:

y = x 3− 3 x 2 − 5 x , y = x − 2 x 2, x = 0, x= 3.

66. ___________________________

CHAPTER 1 ANSWERS TO CHAPTER TEST FORMS

CHAPTER 1, FORM A

14.

y

10 8 6

1. (a) 125 (b) 250 mg/dl

4 2 −10

−8

−6

−4

−2

(b) x 3+3 x 2 h+ 3 x 2 h+ h − 4 15. − 16.

10

3 2

1

2 17. Undefined

5. 3

−$200/yr

18. 6.64

3 mph

n, 19. π +π2π +nπ 3 3

1 I= V 3

20.

9. (a) 2.2 (b) x − 4 ≤ x ≤ 3 (c) 2 (d) y − 5 ≤ y ≤ 4

7π 6

n, + 2π

11π 2ππ + n, +2nπ 6 2

21. amplitude: 3;period π:;midline: y= −4; maximum− : 1; minimum− :7 22. amplitude: 1;period y := −2; π : ;midline

maximum: 2; minimum: 0

10. −2± 2 3 23.

11. x−45

y = 4 sin (2)x −3

24.

1 3

8

−8

2 4. y = 3 x − 8 2

12.

6

−10

1 : (0, 3. m = − 2 ; y-intercept − 5)

8.

4

−6

3

7.

2

−2 −4

2. (a) − 12

6.

x

5

y2

y

4 3 2

13.

5

1

y

4

f(x)=2/x

−5

3

−4

−3

−2

2

−4

−3

−2

−1

−1 −2

−1 −2

x 1

−3

1 −5

−1

x 1

2

3

4

−4 −5

5

−3 −4 −5

25. (a) g = 0.25x − 486.75 (b) 10.25 gal

26. − 1 ± 1 i 2 2

2

3

4

5

283

284

CHAPTER 1 ANSWERS TO CHAPTER TEST FORMS

27.

CHAPTER 1, FORM B

f ( x) = x3 − 4 x2 + 2 x − 3

1. (a) Approximately 310 mg (b) 0.5 hr and 4.4 hr

15

2. (a) − 12 −5

5

28. − 6,6

8x−9376, or 29. (a) g= 25 15 (b) 11.093 gal

(b) 2 x3 + 6 x 2 a+ 6 xa 2 + 2 a3 + 4

3. m= − 4; y - intercept : (0,5) y 2 4. = x− 4 5

7

5.

g = 0.32 x− 625.06

4 1 /student bag 4

6.

−26 m/s 3 8. F = 50 d 7.

9. (a) −4 (b) (−∞∞) , (c) −2,3

(d) [ −6.25, ∞)

10. 1±

5

11. y−23 12.

1 4

p3

13.

5

y

4 3

f(x)=3/x

2 1

−5

−4

−3

−2

−1

−1 −2 −3 −4 −5

x 1

2

3

4

5

CHAPTER 1 ANSWERS TO CHAPTER TEST FORMS

14.

27.

y

10 8

f ( x) = 5 x3 − 6 x 2 + x − 2

6

10

4 2 −10

−8

−6

−4

−2

x 2

−2 −4

4

6

8

10

−6

−5

−8

5

−10

2 2

15.

-10

16. − 1 2 17. Undefined

28. − 11.18,11.18 29. (a) p = −84.33 x+169796.33

18. 4.37 19.

(b) $1382.67

π nππ,5 6

+

20. 2π + 3

+nπ

6

2nππ4, 3

+2nπ,π+2nπ

1 2 21. amplitude: ;period: ;midline: y= 3 ; 2 3 maximum: 3.5; minimum: 2.5 2; 22. amplitude: 2;period: 6π;midline = :−y maximum: 0; minimum: −4

23.

y = 2 cos⎛⎜ 2π x ⎞⎟ + 1 ⎝ 3 ⎠

24.

5

y

4 3 2 1 −5

−4

−3

−2

−1

−1 −2

x 1

−3 −4 −5

25. (a) y = −60 x +121,210 (b) $1,150 26.

23 2 ± i 3

2

3

4

5

285