unit 4 practice problems (print) name - mr guillen's

AP Calculus AB Test Booklet

Unit 4 Practice Problems (Print) Name

Copyright © 2017. The College Board. These materials are part of a College Board program. Use or distribution of these materials online or in printbeyond your school’s participation in the program is prohibited. of 39

1.

The figure above shows the region A, which is bounded by the x- and y-axes, the graph of for x > 0, and the vertical line x = k. If k increases at a rate of units per second,

how fast is the area of region A increasing when ?

A 0

B 3/4

C

D

E

2. A cup of tea is cooling in a room that has a constant temperature of 70 degrees Fahrenheit If the initial temperature of the tea, at time minutes, is and the temperature of

the tea changes at the rate degrees Fahrenheit per minute, what is thetemperature, to the nearest degree, of the tea after 4 minutes?


Unit 4 Practice Problems (Print)


A

B

C

D

E

3.

The graph of the function f shown above consists of two line segments and a semicircle. Let g be

defined by What is the value of




A 0

B

C

D

E

Ruth rode her bicycle on a straight trail. She recorded her velocity v(t) , in miles per hour, for selectedvalues of t over the interval 0≤t≤2.4 , as shown in the table above, For 00 .

4. Using correct units, interpret the meaning of in the context of the problem.

Approximate using a midpoint Riemann sum with three subintervals of equal length

and values from the table.

Please respond on separate paper, following directions from your teacher.

5. Let f be a differentiable function such that and for all x. Of the following,which is not a possible value for f (2)?




A -10

B -5

C 0

D 1

E 2

The continuous function f is defined on the interval−5≤x≤8 . The graph of f, which consists of four line

segments, is shown in the figure above. Let g be the function given by .

6. Find g′(x) in terms of f(x) . For each of g″(4) and g″(−2) , find the value or state that it does notexist.





7. On what intervals, if any, is the graph of g concave down? Give a reason for your answer.


8. Find and .


The function f is defined on the interval , where c > 0 and f(c) = 0. The graph of f, whichconsists of three line segments and a quarter of a circle with center ( -3,0) and radius 2, is shown in thefigure above.

9.




For −5 ≤ x ≤ c, let g be the function defined by . Find the x-coordinate of

each point of inflection of the graph of g. Justify your answer.


10.

For −5 ≤ x ≤ c, let g be the function defined by g(x) = . Find the x-coordinate

of each point of inflection of the graph of g. Justify your answer.


11. Which of the following is an equation for the line tangent to the graph of at the

point where x = -1 ?

A y - 3 = -3e(x + 1)

B y - 3 = -e(x + 1)

C y - 3 = 0

D y - 3 = -1/e(x + 1)

E y - 3 = 3e(x + 1)




12.Let g be the function defined by On which of the following intervals is g

decreasing?

A

B

C

D

E

13. For

A

B

C

D

E

14. For ,




A

B

C

D

E

15.If for all real numbers x > 0, then F'(x) =

A

B

C x

D 2x2

E

16. For ,




A

B

C

D

E

17. If , for what value of b is a minimum?

A 0

B 0.452

C 1.145

D 1.550

E 2

18. Which of the following is an antiderivative of




A

B

C

D

E

19. Let h be the function defined by . Which of the following is an equation for the

line tangent to the graph of h at the point where

A y = 1/2

B

C

D

E




Let f be the continuous function defined on [-4, 3] whose graph, consisting of three line segments and a

semicircle centered at the origin, is given above. Let g be the function given by .

20. For each of g'(-3) and g''(-3), find the value or state that it does not exist.





21.

The graph of the function f in the figure above consists of four line segments. Let g be the function

defined by Which of the following is an equation of the line tangent to the

graph of g at

A

B

C

D

E




The graph of the continuous function f, consisting of three line segments and a semicircle, is shown

above. Let g be the function given by .

22. Find g(-6) and g(3).


23. Find g'(0).





24.

The graph of the differentiable function is shown in the figure above. Let be the function defined

by . Which of the following correctly orders , , and ?

A

B

C

D

E




The graph of the continuous function f, consisting of three line segments and a semicircle, is shown

above. Let g be the function given by .

25. Find g'(0).


26.

The graph of the piecewise linear function f is shown above. Let h be the function given by

On which of the following intervals is h increasing?




A [-1, 3]

B 0, 5]

C [2, 5] only

D [2, 9]

E [3, 9] only

27.

The graph of a differentiable function f is shown above. If which of the

following is true?




A

B

C

D

E

28.

The figure above shows the graph of the function f. If and the shaded region has

an area of 2, what is the value of g(2)?




A -3

B -2

C 0

D 1

E 2

29. What are all values of x for which is equal to 0?

A -2 only

B 0 only

C 2 only

D -2 and 2 only

E -2, 0, and 2

30. What are all values of x for which is equal to 0 ?




A -2 only

B 0 only

C 2 only

D -2 and 2 only

E -2, 0, and 2

31. Let f and g be continuous functions such that and

What is the value of

A 3

B 7

C 11

D 15

E 19




32.

The graph of a function f is shown above. What is the value of

A 6

B 8

C 10

D 14

E 18




33.

The graph of f is shown above for What is the value of

A -1

B 0

C 2

D 6

E 12




34.

The graph of the function f consists of two line segments, as shown in the figure above. The value

of is

A

B

C

D

E nonexistent




35.

The graph of the piecewise linear function f is shown above. What is the value of

A 7.5

B 9.5

C 27.5

D 47

E 48.5




36.

The graph of the function f, which has a domain of [0, 7] , is shown in the figure above. The graphconsists of a quarter circle of radius 3 and a segment with slope -1. Let b be a positive number

such that . What is the value of b ?

A 3.760

B 5.548

C 5.659

D 6.760

E There is no such value of b.




37.

The graph of the derivative of f, is shown in the figure above. If which of thefollowing could be the value of

A 15

B 20

C 25

D 35

E 40




38. Let and be continuous functions such that , , and

. What is the value of ?

A −23

B −19

C

D 19

E 23

39. T h e f u n c t i o n f i s c o n t i n u o u s o n t h e c l o s e d i n t e r v a l [ 1 , 7 ] . I f

A -148

B 10

C 12

D 20

E 148

40.




A

B

C

D

E

41. If is an antiderivative for , then

A −0.281

B −0.102

C 0.102

D 0.260

E 0.282

42.




A -1

B

C

D

E 1

43.

A

B

C

D

E

44. If -10e-t/2 and y(0) = 20, what is the value of y(6)?




A 20e-6

B 20e-3

C 20e-2

D 10e-3

E 5e-3

45. If f'(x) = cos(x2) and f(3) = 7, then f(2) =

A 0.241

B 5.831

C 6.416

D 6.759

E 7.241




46.

The graph of the derivative of a function f, consists of two line segments and a semicircle, asshown in the figure above. If then

A

B

C

D

E




47.

The derivative of the function f is continuous on the closed interval [0,4]. Values of f and f ' for

selected values of x are given in the table above. If , then f(4) =

A 0

B 3

C 5

D 10

E 13

48.

The derivative of the function f is continuous on the closed interval [0,4]. Values of f and f' for

selected values of x are given int he table above. If , then f(4) =




A 0

B 3

C 5

D 10

E 13

49. Let f be the function given by . If four subintervals of equal length are used, what is the

value of the right Riemann sum approximation for ?

A 20

B 40

C 60

D 80

E 120

50. Let be the function given by If four subintervals of equal length are used, what is the

value of the right Riemann sum approximation for ?




A 20

B 40

C 60

D 80

E 120

The function f is twice differentiable for x > 0 with f'(1) =15 and f''(1) = 20. = Values of f', the derivative of f,are given for selected values of x in the table above.

51. Use a midpoint Riemann sum with two subintervals of equal length and values from the table to

approximate .Use the approximation for to estimate the value of f(1.4).

Show the computations that lead to your answer.





52.

The table above gives selected values for a continuous function f. If f is increasing over the closed

interval [ 0,3], which of the following could be the value of

A 50

B 62

C 77

D 100

E 154

53.

The function f is continuous on the closed interval [0, 6] and has values as shown in the table

above. Using the intervals [0,2], [2,4], and [4,6], what is the approximation of obtained

from a midpoint Riemann sum?




A 0

B 3

C 4

D 6

E 8

The twice-differentiable function W models the volume of water in a reservoir at time t, where W(t) ismeasured in gigaliters (GL) and t is measured in days. The table above gives values of W'(t) sampled atvarious times during the time interval days. At time t =30, the reservoir contains 125 gigalitersof water.

54. Use a left Riemann sum, with the three subintervals indicated by the data in the table, to

approximate . Use this approximation to estimate the volume of water W(t), in

gigaliters, in the reservoir at time t = 0. Show the computations that lead to your answer.





55.

Water is flowing into a tank at the rate r(t), where r(t) is measured in gallons per minute and t ismeasured in minutes. The tank contains 15 gallons of water at time t = 0. Values of r(t) for selectedvalues of t are given in the table above. Using a trapezoidal sum with the three intervals indicatedby the table, what is the approximation of the number of gallons of water in the tank at time t = 9 ?

A 52

B 57

C 67

D 77

E 79

56.

A tank contains 50 liters of oil at time hours. Oil is being pumped into the tank at a rate where is measured in liters per hour, and t is measured in hours. Selected values of aregiven in the table above. Using a right Riemann sum with three subintervals and data from thetable, what is the approximation of the number of liters of oil that are in the tank at time hours?




A 64.9

B 68.2

C 114.9

D 116.6

E 118.2

57.

The table above gives selected values for a continuous function f. If f is increasing over the closed

interval which of the following could be the value of

A 50

B 62

C 77

D 100

E 154




58. x 0 25 30 50

f(x) 4 6 8 12 The values of a continuous function f for selected values of x are given in the table above. What is

the value of the left Riemann sum approximation to using the subintervals [0, 25] [25,

30 ,] and [30, 50 ] ?

A 290

B 360

C 380

D 390

E 430

Kathleen skates on a straight track. She starts from rest at the starting line at time t = 0. For seconds, Kathleen’s velocity k, measured in feet per second, is differentiable and increasing. Values ofk(t) at various times t are given in the table above.

59. Use a right Riemann sum with the four subintervals indicated by the data in the table to

approximate Indicate units of measure. Is this approximation an overestimate or

an underestimate for the value of ? Explain your reasoning.



Copyright © 2017. The College Board. These materials are part of a College Board program. Use or distribution of these materials online or in print beyond your school’sparticipation in the program is prohibited. of 39


unit 4 practice problems (print) name - mr guillen's

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