ap calculus ab semester one exam review...

AP Calculus AB Semester One Exam Review 2019-2020

2019-2020 AP Calculus AB Semester One Exam Review (edited 6/20/2019)

Calculator Section – Problems 1-23

1. For 3 2 34 3 28x xy y , find

dx

dy at the point (3, 4) on the curve.

A. 5

2 B.

4

3 C.

5

12 D.

7

6 E.

2

3

2. Given 3 24

; 43

V r S r for a sphere, and the fact that the radius of the sphere is decreasing at 1/2 inches

per minute, find the rate at which the volume is changing, in cubic inches per minute for a sphere at the time when the

radius is 6 inches.

3. from AP 2002 AB 5

A container has the shape of an open right circular cone, as shown. The height of the container is 10 cm and the diameter of the opening is 10 cm. Water in the container is evaporating so that its depth h is changing at a constant rate

of 10

3 cm/hr. hrV 2

3

1( ). Find the rate of change of the volume of water in the

container, with respect to time, when h = 5 cm. Indicate units of measure.

4. The first derivative of the function f is given by 2sin 1

'9

xf x

x . How many critical values does f have on the

open interval 0, 10 ?

5. A 25 ft ladder is sliding down a wall so that the base is moving along the ground at 3 ft/hour. When the ladder is 10

feet high (on the wall), how fast in ft/hour is the top of the ladder moving along the wall?

A. -3 ft/hour B. -6.874 ft/hr C. -21.869 ft/hr D. 21.869 ft/hr E. 6.874 ft/hr



x 1 2 3 4 5 6 f(x) 3 4 2 5 6 6.2

6. Use the table at the right to answer the following.

If f is a continuous and differentiable function, then approximate the value of '(3)f .

A. 4.5 B. 4 C. 2 D. 1/2 E. 1/4

7. Given4 3' ( ) 2sin (2 ) 3sin (4 )f x x x , over the open interval (0, 0.7), at what value of x is the tangent line to f

horizontal?

A. 0.390 B. 0.555 C. 0.368 D. 0.567 E. 0.195

8. Given4 3' ( ) 2sin (2 ) 3sin (4 )f x x x , at which x-value over the interval (0, 2] does the graph of f have a

relative minimum?

A. 1.938 B. 1.146 C. 0.368 D. 1.571 E. 0.567

9. TRUE or FALSE, given the graph of f to the right.

a. 1

lim ( )x

f x

exists. ____

b. 2

lim ( )x

f x

exists. ____

c. f is continuous over the interval [-1, 4]. ______

d. f is differentiable over the interval [-1, 4]. ______

e. f is differentiable over the interval [-3, -1] ______



10. A rectangle with one side on the x-axis has its upper vertices on the graph of cosy x as shown in the figure

below. What is the maximum area of the rectangle?

A. 0.860 B. 0.982 C. 1.122 D. 3.426 E. 6. 577

11. The function f is continuous and differentiable over the interval [-4, 6]. If ( 4) 3f and (6) 23f then tell

(true or false) which statements must be true.

a. Over [-4, 6], the function f has a maximum value. ____

b. Over [-4, 6] there exists a value c so that ( ) 0f c . _____

c. Over [-4, 6] there exists a value c so that ( ) 10f c . _____

d. Over [-4, 6] there exists a value c so that '( ) 0f c . _____

e. Over [-4, 6] there exists a value c so that '( ) 2f c . _____

f. Over [-4, 6], f is increasing.___

g. Over [-4, 6] '( )f c exists for every c in [-4, 6]. _____

12. What is the slope of the line tangent to 2 2cos (3 )y x x at the point on the curve when x = 0.861 ?



13.

The graph of

2

3y x is shown above, on the interval [ -5, 5] . Which is true?

A. The graph is continuous and differentiable over the interval.

B. The graph is continuous but not differentiable over the interval.

C. The graph is not continuous but is differentiable over the interval.

D. The graph is not continuous, nor differentiable over the interval.

14. The graph of a twice-differentiable function f is shown below.

Which of the following is true?

A. (1) '(1) "(1)f f f B. (1) "(1) '(1)f f f C. "(1) (1) '(1)f f f

D. '(1) (1) "(1)f f f E. "(1) '(1) (1)f f f

15. Given4 3( ) 2sin (2 ) 3sin (4 )f x x x . Find the interval of x-coordinates for which the graph of f is concave up

on the interval [0, 0.78].



16.

Consider the average rate of change of the graph of f above, between the points shown. Between which two points is

the average rate of change the least?

A. Points A and B B. Points A and D C. Points B and D D. Points B and C E. Points A and C

17.

The graph of f is shown in the figure above. Which of the following could be a graph of the derivative of f ?

18. A particle moves along the x-axis so that at time t ≥ 0 its position is given by ttx cos)( . What is the velocity of

the particle at the first instance the particle is at the origin?



19. Two people are walking away from each other, one heading north at 3 miles per hour and the other heading east at

4 miles per hour. How fast are the two people moving away from each other after walking for two hours?

20. The function f is continuous on the closed interval [2, 4] and twice differentiable on the open interval (2, 4). If

'(3) 2 " 0f and f on the open interval (2, 4), which of the following could be a table of values for f ?

A. B. C. D. E.

21. Given )4(sin3)2(sin2)(' 34 xxxf , at which x-coordinate(s) below does the graph of )(xf change concavity

over the interval (0, 2)?

A. 1.938 B. 1.146 C. 0.667 D. 1.571 E. 0.567

22. At which interval is the graph of )(xf (for )(' xf defined in #22) concave up over the interval (0, 0.8)?

A. (0, 0.8) B. (0.368, 0.785) C. (0, 0.368) D. (0.368, 0.8)

23. If 3 2( ) 2 15 24 1v t t t t is the velocity function for a particle moving along the x-axis, for time t seconds,

find each of the following:

a. when the particle changes direction for 0t .

b. for what intervals of time ( 0t ) the particle is moving to the left.

c. for what intervals of time ( 0t ) the particle has increasing speed.

d. for what intervals of time ( 0t ) the acceleration of the particle is positive.



Non-Calculator Section – Problems 24-52

24. If 3cos (4 1)y x then

dy

dx=

A. 212cos (4 1)x B.

212sin (4 1)x C. 23cos (4 1)x

D. 212cos (4 1)sin(4 1)x x E.

23cos (4 1)sin(4 1)x x

25. Which is an equation of the tangent line at x = 2, for the graph of26 12 1y x x ?

A. 19 1y x B. 5 1y x C. 12 23y x D. 5 7y x E. 12 8y x

26. Evaluate each of the following limits:

a.

2

2

5lim

2 500n

n

n n

b. 2

2

1

935lim

x

xx

x

c. 21

95lim

x

x

x

d. h

xhx

h

44

0

5)(5lim

e. x

xx

x

5

52

5lim

f.

xx

xx

x 4

12lim

2

2

4

g. h

h

h

)sin()sin(lim

0

h. 2

0

1lim

x

x

e x

x

i. 0

3sin(4 )lim

x

x

x



27. Let

2,3

2,2

4

)(

2

x

xx

x

xg

Which of the following statements are TRUE?

I) )(lim2

xgx

exists II) g(2) exists III) g is continuous at x = 2

A) Only I B) Only II C) I and II only D) None of them E) All of them

28. A function ( )f x equals

2

1

x x

x

for all x except x = 1. For the function to be continuous at x = 1, the value of

)1(f must be

A. 0 B. 1 C. 2 D. E. none of these

29. The graph of

2

2

2

4

xy

x

has

A. two horizontal asymptotes B. two horizontal and one vertical asymptotes C. two vertical but no horizontal asymptotes D. one horizontal and one vertical asymptote E. one horizontal and two vertical asymptotes

30. Find the inflection POINT(S) of the graph of each of the following:

a. 3 22 6y x x

b. 42 8y x x



31. Shown to the right: the graph of f over [-5, 5] X [-3,3]. Find each of the following:

a. 3

lim ( )x

f x

=___ b. f(-3)=____

c. 0

lim ( )x

f x

=___ d. f(0)=_____

e. 2

lim ( )x

f x

=____ f. 2

lim ( )x

f x

______

g. 4

lim ( )x

f x

=____ h. 4

lim ( )x

f x

_____

i. 4

lim ( )x

f x

____ j. )3(f _____

k. Estimate the x-coordinates where 1)( xf . l. 5

lim ( )x

f x

= ______

32. 4)1(",3)1(',8)1( fff for a twice-differentiable function f. Write an equation of the line tangent to f

at x=1.

A. 2x y

B. 3 5x y

C. 3 11x y

D. 3 7x y

E. 3 1x y

33. Given 2( ) 3 2 1f x x x , and

1( ) ( )g x f x , then find the value of '(2)g .

A. 9 B. 1/10 C. 4 D. 1/4 E. 1/2

34. Consider the implicitly defined function, 122 yxyx .

a) Which of the following point(s) lie on the curve defined by the function?

(2, 3) (1, 1) (−1, 2) (1, 0)

b) Determine the tangent line for each of the above points that lie on the curve.



35. If 3

1

2 )63()( xxxf , then )0('f ?

36. What is )(' xf if x

xxf

sin)(

3

?

37. At any time t, the position of a particle, in feet, moving along an axis is ttt

ts 323

)( 23

. What is the

acceleration, in ft/sec2 of the particle at t = 4 seconds?

38. Given xexf 5)( , find )('' xf .

39. If g is a continuous function on the closed interval [-4, 4] and g(-4) = 2 and g(4) = 7, which of the following

statements is necessarily TRUE?

A. The absolute maximum value for g on [-4, 4] is 7 B. If -4 < x < 4, g(x) = 5 for at least one value of x C. g(x) = 0 for at least one value of x on the closed interval [-4, 4] D. g is increasing for all x on [-4, 4] E. The absolute minimum value of g on [-4, 4] is 2

40. Give an expression for dx

dy at the point (x, y) for each function below.

a. 43 xy b. xey 5

c. )65ln( xy d. x

xxy1

154

e. xexy 42 f. cos(4 )y x

g. sin

xy

x h.

2

3 1

xy

x



41. Which is a line which can be used to approximate values of 3y x near 8x ?

A. 3 2x y

B. 3 14x y

C. 12 32x y

D. 12 8x y

E. 12 16x y

Use this graph of ( )y f x to answer questions #42-44. Increments on both axes are one.

42. Over the interval [-2, 2], how many values on f appear to satisfy the conclusion of the Mean Value Theorem?

A. 0 B. 1 C. 2 D. 3 E. 4

43. What is the average rate of change of f over the interval [-2, 2] ?

A. 1 B. 2

3 C. 2 D.

2

5 E. 0

44. Consider the interval [-2, k]. For what value of k is the average rate of change of f equal to 0?

45. Use the Intermediate Value Theorem to show that 3( )f x x x has a value of 9 at some point on the interval

[1, 2].



46. Find the value of 2

2

d y

dx for each function below, at the value x c given.

a. 3

24y x at 1x b. cos(3 )y x at 3

x

c.

21

2x

y e at 1x

47. A local minimum value of the function

xey

x is at x = ?

48. If ( ) 2f x x then '(2)f

A. 1

4 B.

1

2 C.

2

2 D. 1 E. 2

49. Find the value(s) of c that satisfy the conclusion of the Mean Value Theorem for Derivatives for 1)( xxf on

5,2 .

50. Find the global extrema of the function 510)( 2 xxxf on the interval 8,3 .

51. Suppose 2 2' ( ) ( 4) ( 3)f x x x x . Which of the following is (are) true?

I. 𝑓 has a local maximum at 𝑥 = 4.

II. 𝑓 has a local minimum at 𝑥 = −3.

III. 𝑓 has neither a local maximum nor a local minimum at 𝑥 = 0.

52. A position function 3 2( ) 2 15 24 1p t t t t gives the position of a particle moving left and right along the

x-axis, for t seconds. Answer each of the following:

a. What is the initial position of the particle?

b. In what direction is the particle moving at 2t seconds?

c. For what interval(s) for 0t is the particle moving to the left?

d. What is the acceleration of the particle at t=2 seconds?

ap calculus ab semester one exam review...

Documents