AP Calculus AB Practice Problems for Exam Review

Name _____________________ Score ______________

1. Find the slope of the line determined by the points )3,1(A and )7,4(B .

2. Write the standard (and then general) form of the equation of a line through

the points )1,2( and )2,2( .

3. Determine the slope of the line 1543 yx by putting the equation in slope-

intercept form.

4. Find the equation of the line going through the point

4,

2

2 and having a

slope of 4

. This is the equation of the line tangent to the curve xxf sin)( at

this point. Leave your values in exact form.

5. Factor completely: 6254 x

6. Factor: a. xx 39 2 b. xxx 844 23 c. 432 xx

7. Consider 21 xy .

a. sketch the graph of the equation

b. identify any intercepts

c. test for symmetry

8. Find the points of intersection between the two graphs:

A. 3xy B. 2522 yx

xy 102 yx

9. Specify a sequence of transformations that will yield the graph of 34)( xxg

when starting from the graph of 3)( xxf .

10. Specify a sequence of transformations that will yield the graph of

2)4()( 3 xxh when starting from the graph of 3)( xxf .

11. Given that 32

)( tth , find )64(h and )144(h . Graph )(th on your graphing

calculator (with an appropriate window) and locate both points on the graph.

Sketch the graph and the location of the points (from your calculator) below.

12. Let 3 2)( xxg and 2)( 3 xxf . Compute the compositions ))(( xgf and

))(( xfg . What do you notice?

13. Given that 93)( xxf and xxxg 3)( 2 :

a. determine ))(( xgf b. determine ))(( xfg

c. sketch the piecewise defined function

53)(

31)()(

xxf

xxgxh

14. Determine if the function 643 xxy has an inverse. Explain your

reasoning.

15. Solve for x: a. 813 x b. 25664 x c. 10242 x

16. Determine how much time is required for an investment to triple in value if

interest is earned at the rate of 5.75% compounded annually.

17. Given that 2

1

6sin

, determine the other value for , such that

2

1sin .

State your answer in radians only.

18. Given that 2

2

4

3cos

, determine the other value for , such that

2

2cos . State your answer in radians.

19. Sketch the graphs of

2sin)(

xxf and

2cos)(

xxf .

20. a. Compute the following for 2

1)(

xxf : )1(f , )5.1(f , )9.1(f , )99.1(f ,

)999.1(f , )2(f

b. Compute the following for 2

6)(

2

x

xxxf : )1(f , )5.1(f , )9.1(f ,

)99.1(f , )999.1(f , )2(f

21. Does xx

1lim

0 exist? Explain why or why not (use the definition of a limit).

22. Complete the table and use the result to estimate the limit.

2

22

2lim xx

x

x

x 1.9 1.99 1.999 2.001 2.01 2.1

f(x)

23. Use a graphing calculator to graph the following function and estimate the

limit (if it exists). What is the danger in looking solely at the graph of a

function to determine the limit? What is the domain of this function?

3

9)(

x

xxf ______)(lim

9

xf

x

24. Assume the 5)(lim3

xfx

and 2)(lim3

xgx

. Determine:

a. )()(3lim3

xgxfx

b.

)(

)(lim

2

3 xg

xf

x

25. Determine the limit:

1

1lim

21 x

x

x

26. Determine the following limits analytically:

A. 8

7lim

2

8

x

xxx

x B.

x

x

x

tanlim

0 C.

x

x

x

4sinlim

0

27. Determine: a.

1

1lim

1 xx b.

1

1lim

1 xx

28. Determine the following limits: a. 4

32lim

22 xx

b. 4

32lim

22 xx

29. Compute the following limits:

a.

4

43lim

2

x

xx

x b.

352

5lim

2 xx

x

x c.

6

3lim

x

x

x

30. a. Determine the discontinuities for xx

xxm

25

3)(

3 .

b. State the intervals on which m(x) is continuous.

31.

a. Does )4(f exist? Does )(lim4

xfx

exist? If so, what is its value? Does )(lim4

xfx

exist? If so, what is its value?

b. State all intervals where )(xf is continuous. What value should be assigned to

)4(f to make )(xf continuous at 4x ? List each x-value where )(xf is

discontinuous and determine the type of discontinuity that exists. What value

should be assigned to )3(f to make )(xf continuous at 3x .

32. a. Find the slope between the two points. 23)( 2 xxxf

Point #1 = )3(,3 f Point #2 = )4(,4 f

b. Find the slope between the two points. 23)( 2 xxxf

Point #1 = )3(,3 f Point #2 = )5.3(,5.3 f

c. Find the slope between the two points. 23)( 2 xxxf

Point #1 = )3(,3 f Point #2 = )1.3(,1.3 f

d. Find the slope between the two points. 23)( 2 xxxf

Point #1 = )3(,3 f Point #2 = )01.3(,01.3 f

33. Find the slope of the tangent line to the curve xxf )( at the point x = 1.

34. Find a general equation for the slope of the tangent line to x

xf1

)( at any x-

value. Then determine the specific slope at 2x and at 2x .

35. Consider the function 4

5)(

xxg .

a. Find a general equation for the slope of the tangent line to )(xg at any x-

value.

b. Then determine the specific slope at 9x and at 1x .

c. Also determine the equations of the tangent lines at these x-values.

36. Identify the function and the point at which we are determining the

derivative.

A. x

x

x

33

0

)5()5(lim B.

h

ee h

h

33

0lim

C. x

x

x

3tan

3tan

lim0

________)( xf ________)( xf ________)( xf

________c ________c ________c

37. Find the equation of the tangent line to the curve 38)( xxf at the point

1,

2

1.

38. Determine the equation of the tangent line to the curve x

y1

at 3x .

39. Determine the differentiability of the functions at 0x .

a.

0

03

21

2

xx

xxxy b.

0

013

2

xx

xxy

40. Determine the derivative at 5x and 5x , for 32)( xxxf .

41. Determine the derivative of the following function, using two different

methods. The first method is to apply the quotient rule. The second method

is to rewrite the denominator as a quantity to a negative power, then to use

the product rule. Show that you get the same result.

x

xx

xf2

13

)(

42. Given 29.41025)( ttts .

a. determine )(tv

b. determine )(ta

c. determine when and how high the object rises at its maximum

43. Particle Motion. Work in groups of two or three. The position (x-coordinate)

of a particle moving on a line 2y is given by 522132)( 23 ttttx , where t is

time in seconds.

(a) Describe the motion of the particle for 0t .

(b) When does the particle speed up? slow down?

(c) When does the particle change directions?

(d) When is the particle at rest?

(e) Describe the velocity and speed of the particle.

(f) When is the particle at the point (5, 2)?

44. Use NDER on the graphing calculator to make a table of values for the

derivative of xxf sin)( for several values of x. Write the values in exact form

(convert from the decimal values the calculator gives you). Sketch a graph of

)(xf below for x0 . On your graphing calculator and below, draw a

scatterplot of x versus )(xf . Determine a function (analytical) for )(xf .

x )(xf X )(xf

0

3

2

6

4

3

4

6

5

3

2

45. Write the equation of the tangent line at 4

x , for )cos()sin()( xxxf .

46. The position of a particle moving along a line is given by 790242)( 23 tttts

for 0t . For what values of t is the speed of the particle increasing?

(a) 3 < t < 4 only

(b) t > 4 only

(c) t > 5 only

(d) 0 < t < 3 and t > 5

(e) 3 < t < 4 and t > 5

47. What is the average rate of change of the function f given by xxxf 5)( 4 on

the closed interval ]3,0[ ?

(a) 8.5

(b) 8.7

(c) 22

(d) 33

(e) 66

48. Determine the derivative of )sin(cos xy .

49. Find the slope of the tangent line to the curve xxxf tan3)( 2 at

81

32,

9

42 .

50. Find dx

dy: 2

3

csc3 xy

51. Find y : 21

23

cossin5.0 yx

52. The slope of the tangent to the curve 6223 xyxy at (2, 1) is

(a) 2

3 (b) -1 (c)

14

5 (d)

14

3 (e) 0

53. Find the derivative of )3tan(cos)( 1 xxf .

54. Find )3(f , when )3csc)( 1 xxf

55. Find )(xf , when )(tan)( 1 xexf

56. Find )(xf : xexxf tantan)(

57. Find )(xf : a.

xxf

1ln2)( b. x

xxf 2

2log)(

58. Write a multiple choice item 59. Write a free response item

(think AP format!): (open-ended AP format):

1. 2. a.

a.

b. b.

c.

d. c.

60. If 742 23 yyxx , then when 1x , dx

dy

(a) 2

9 (b) 0

(c) 8 (d) 3

(e) 2

7

61. Let f and g be differentiable functions with the following properties:

I. 0)( xf for all x II. 2)5( g

If )(

)()(

xg

xfxh and

)(

)()(

xg

xfxh

, then )(xg

(a) )(

1

xf (b) )(xf

(c) )(xf (d) 0

(e) 2

62. A particle moves along the x-axis so that its position at time t is given by

127)( 2 tttx . For what value of t is the velocity of the particle zero?

(a) 2.5 (b) 3

(c) 3.5 (d) 4

(e) 4.5

63. If

433ln

303ln)(

xx

xxxf , then )(lim

3xf

x is

(a) 9ln (b) 27ln

(c) 3ln3 (d) 3ln3

(e) nonexistent

64. An equation of the line tangent to the graph of xxy cos3 at 0x is

(a) xy 2 (b) 12 xy

(c) 13 xy (d) 13 xy

(e) xy 4

65. If xexf 2cos)( , then )(xf

(a) xe2sin (b) xe2sin2

(c) xe2sin (d) xe2sin2

(e) xx ee 22 sin2

66. Use analytic methods to find the x-values for the absolute and relative

extrema for 53

)( xxf on the interval 32 x .

67. Determine where the absolute extrema occur on 5,3 for 53)( 2 xxxg

68. Determine where the absolute and relative (local) extrema occur for

852

132)( 23 xxxxf

69. Find the function with the given derivative whose graph passes through the

point P.

xxxf cos12)( )3,0(P

70. 53

)( xxf

a. Determine the interval(s) on which the function is increasing.

b. Determine the interval(s) on which the function is decreasing.

c. What do you notice about the point(s) at which the function changes

from increasing to decreasing (or vice versa)?

71. 10196)( 2 xxxg

a. Determine the interval(s) on which the function is increasing.

b. Determine the interval(s) on which the function is decreasing.

c. What do you notice about the point(s) at which the function changes

from increasing to decreasing (or vice versa)?

72. Determine and classify the local extrema of the function 10196)( 2 xxxg .

73. 25)( xxxf Draw sign charts to help you do the following:

a. Use analytic calculus methods to locate the x-values of the local extrema.

b. Determine the intervals on which the function is increasing.

c. Determine the intervals on which the function is decreasing.

74. x

xh2

)( Draw sign charts to help you do the following:

a. Use analytic calculus methods to locate the x-values of the local extrema.

b. Determine the intervals on which the function is increasing.

c. Determine the intervals on which the function is decreasing.

75. Make a sign chart to find the critical numbers of f and determine the intervals

of increasing and decreasing. Then determine at which x-values the function

has local extrema, then classify using the Fundamental Theorem of Calculus.

2

3)(

x

xxf

76. The graph of f , the derivative of f, is shown above for 52 x . On what

intervals is f increasing?

a. ]1,2[ only

b. ]3,2[

c. ]5,3[ only

d. ]5.1,0[ and ]5,3[

e. ]1,2[ , ]2,1[ , and ]5,4[

77. The first derivative of the function f is defined by xxxf 3sin)( for 20 x .

On what intervals is f increasing?

a. 445.11 x only

b. 691.11 x

c. 875.1445.1 x

d. 445.1577.0 x and 2875.1 x

e. 10 x and 2691.1 x

78. The derivative of the function f is given by 22 cos)( xxxf . How many points

of inflection does the graph of f have on the open interval )2,2( ?

(a) One (b) Two (c) Three

(d) Four (e) Five

79. On the interval 5,5 , f is continuous and differentiable. If 2)3)(12)(1()( xxxxf , briefly explain the following conclusions.

a. There is a local maximum on f at 2

1x .

b. There is a horizontal tangent but no extrema at 3x .

c. If 7)2( f , then .7)3( f

Complete each of the following statements for a continuous function f:

80. When f is positive, it means that f is _________________________.

81. When f is negative, it means that f is _________________________.

82. When f is changing from negative to positive, it means that f is at a

_________________________ value.

83. When f is changing from positive to negative, it means that f is at a

_________________________ value.

84. We find the critical values of a function f and put them on the ____________.

We find the critical values of a function f by determining when the derivative

function is equal to ________ and when it is __________. The numerator set

equal to zero tells us when the derivative is ___________. The denominator

set equal to zero tells us when the derivative is __________.

85. Let g be a function defined and continuous on the closed interval ba, . If g

has a local minimum at c where bca , which of the following statements

must be true?

I. If )(cg exists, then .0)( cg

II. )()( bgcg

III. g is monotonic on ba,

(A) I only (B) II only (C) III only

(D) I and II only (E) I and III only

Let f be the function given by x

xxf

ln)( for all 0x . The derivative of f is given by

2

ln1)(

x

xxf

. [No calculator is allowed for this problem.]

86. Write an equation for the line tangent to the graph of f at 2ex .

87. Find the x-coordinate of the critical point of f. Determine whether this point

is a relative minimum, a relative maximum, or neither for the function f.

Justify your answer.

88. The graph of the function f has exactly one point of inflection. Find the x-

coordinate of this point.

89. Find )(lim0

xfx

.