AP Practice Exam (6)

8/10/2019 AP Practice Exam (6)

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Practice Questions for AP Calculus AB Exam: Section I – Version 2

Section I

This section contains 45 multiple-choice questions and contains two parts: Part A and Part B.

Part A has 28 questions that must be solved without a calculator. Part B has 17 questions,

including some questions that require the use of a graphing calculator.

Part A Multiple-Choice

You may not use a calculator on this portion of the exam.

Directions: Solve each problem in the provided space. Then choose the best option from among

the choices given. Be efficient with the use of your time.

Throughout this exam:

(1) The domain of each function f is the set of all real numbers for which ( ) f x is defined. If

the domain of a particular function differs from this, it will be specified in the problem.

(2) For trigonometric functions, the inverse may be represented with “ 1− ” or “arc”. For example,

the inverse cosine function may be represented as 1cos x− or arccos x .

1.3

cos4

x dx⎛ ⎞

=⎜ ⎟⎝ ⎠∫

(A+)4 3

sin3 4

x C ⎛ ⎞

+⎜ ⎟⎝ ⎠

(B)3

sin4

x C ⎛ ⎞

+⎜ ⎟⎝ ⎠

(C)

3 3sin

4 4 x C

⎛ ⎞+⎜ ⎟

⎝ ⎠

(D)3 3

sin4 4

x C ⎛ ⎞

− +⎜ ⎟⎝ ⎠

(E)4 3

cos3 4

x C ⎛ ⎞

+⎜ ⎟⎝ ⎠




2. Find2

2

4lim

2 x

x

x→

−

−.

(A) 2

(B) 1 (C) 0

(D+) 4

(E) Does not exist

3. For what value(s) of k, if any, is2

( )2

k x f x

k

⎧ −= ⎨

⎩

if 3

if 3

x

x

<

≥ continuous on ( ),−∞ ∞ ?

(A) 1, 3−

(B) 1

(C+) 1,3−

(D) 3

(E) Does not exist

4. If ( ) 3 7 1, then '( ) f x x x f x= − is

(A) 7 1 x x −

(B)21

2 7 1 x −

(C) 3 7 1 3 x x− +

(D+)21

3 7 12 7 1

x x

x− +

−

(E)7

3

7 1

x

x

+

−




5.12

1 12dx

x=

+∫

(A+) ln 1 12 x C + +

(B) 12 ln12 x x C + +

(C) ln 12 x C +

(D)1

ln 1 1212

x C + +

(E)1

ln 1 1212

x C − + +

6. A particle moves along the x-axis so that at any time 0t ≥ , its velocity is given by

( ) cos(4 )v t t = . If the position of the particle at time is 23

t xπ

= = , what is the particle’s

position at time 0t = ?

(A)5

3

(B)5

4

(C+)4

3

(D) 0

(E)3

2




7. The function f is continuous on the closed interval [0,8] and has the values given in the table

below.

x 0 2 4 6 8

f(x) 8 4 k 5 6

The trapezoidal approximation for

8

0

( ) f x dx∫

found with four subintervals of equal length is

52. What is the value of k ?

(A) 2

(B)5

(C) 7

(D+) 10

(E) 17

8. ( )( )2sin 3 7d

xdx

+ =

(A) ( )2cos 3 7 x +

(B) 2sin(3 7) x +

(C+) ( ) ( )6 cos 3 7 sin 3 7 x x− + +

(D) ( )23sin 3 7 x +

(E) 3 cos sin x x

9. A colony of bacteria starts with 400 and grows at a rate proportional to its size. After 2 hoursthere are 5000 bacteria. When will the population reach 20,000?

(A) 4.1235 hours

(B) 12.5000 hours

(C) 3.5355 hours

(D) 8 hours

(E+) 3.0977 hours




10. Given2

( )4

x f x

x=

+, for what values of x is the graph of f concave downwards?

(A) 0 4 x< <

(B) 4 x∞ < < −

(C+) 4 x− < < ∞

(D) 4 and 4 x x∞ < < − < < ∞

(E) 4 x = −

11. Let f be defined by ( ) 9 f x x= + for all real numbers x. For what values of x is the function

increasing?

(A) ( ), 9−∞ −

(B) ( ), 9−∞

(C) [ )9,0−

(D) ( )0,9

(E+) ( )9,− ∞

12. Find an equation for the line tangent to the graph of ( ) 2 1 f x x= + at the point where 4 x = .

(A) 2 1 y x= +

(B)1 7

6 3 y x= +

(C) 4 y =

(D+)1 5

3 3 y x= +

(E)

1

52 y x= − +






16. How many points of inflection does ( )4

( ) 6 2 f x x x= + .

(A) 0

(B+) 1

(C) 2

(D) 3

(E) 4

17. Find the area of the region bound by3 2( ) f x x x= − and

3( )g x x x= + .

(A+)1

6

(B) 12

(C)1

8

(D)1

12

(E)1

16

18. If

2 1

1

( ) ( ) and (5) 7, then '(2) x

F x f u du f F

+

= = =∫

(A) 4

(B+) 28

(C) 7

(D) −18

(E) 16




19. What is the equation of the line tangent to the curve3

sin y x x= − at the point ( )0,0 ?

(A) 2 1 y x= − +

(B) 3 4 y x= +

(C) y x=

(D) 0 y =

(E+) y x= −

20. A particle moves along the x-axis so that its velocity at any time 0t ≥ is given by2( ) 3 6v t t t = − . Which of the following expressions could represent the position ( ) x t of the

particle at any time 0t ≥ ?

(A+)

3 2

3 7t t − +

(B) 3 6t −

(C) 22t t −

(D) 3 23 6 3t t − −

(E) ( )3 2t t −

21. What is the slope of the tangent line to3 22 12 x xy y− − = that lies in the fourth quadrant at

the point where 2 x = ?

(A) 0

(B) 16

(C) −2

(D) 12

(E+) undefined




22. If ( ) ln 3 f x x= − then what is a derivative of the inverse of ( ) f x at 5 x = ?

(A) 5e−

(B+) 53 e−

(C) 43e

(D) ln 2

(E) undefined

23. The function3 2

3 2

4 7 21( )

6 5 17

x x f x

x x

− +=

+ +

has horizontal asymptote(s) at

(A)2 7

,3 5

y y= = −

(B)7

5 y = −

(C+)2

3 y =

(D) 0 y =

(E) no horizontal asymptotes




24. Find the derivative of

( )2

2

3 5( )

9

x f x

x

−=

−.

(A) 29 20 7 x x− + −

(B)( )2

3

4 9 x x −

(C+)

( )

2

32

9 20 7

9

x x

x

− + −

−

(D)

( )

2

32

15 20 7

9

x x

x

− −

−

(E)( )2

3

2 9 x −

25. Evaluate ( )4

0

5 9 6 for 0

xd

t t dt xdx

− + ≥∫ .

(A) 320 9t −

(B+) 45 9 6 x x− +

(C)

4

5 9t t −

(D) 320 9 x −

(E)5

4.5 6 x x x− +

26. Find3

3

27lim

3 x

x

x→

−

−

(A) does not exist

(B) 9

(C) 3

(D+) 27

(E) ∞




27. Which of the following statements is true about3 2( ) 4 12 7 f x x x= − + ?

(A) ( )is decreasing on ,1 f −∞

(B+) ( )is increasing on 2, f ∞

(C) ( )is increasing on 0,2 f

(D) ( )is decreasing on 2, f ∞

(E) is increasing for all real values f

28. For which points on the graph of f is '( ) f x negative?

(A) A and E

(B) A only

(C) C only

(D+) C, D, and E

(E) B only




Part B Multiple-Choice

This portion of the exam requires a graphing calculator for some questions.

This part of the exam contains 17 questions.

Directions: Solve each problem in the provided space. Then choose the best option from amongthe choices given. Be efficient with the use of your time.

Throughout this exam:

(1) Sometimes the exact numerical value of the solution is not given as a choice. In this event, pick the best numerical approximation from the given choices.

(2) The domain of each function f is the set of all real numbers for which ( ) f x is defined. If

the domain of a particular function differs from this, it will be specified in the problem.

(3) For trigonometric functions, the inverse may be represented with “ 1− ” or “arc”. For example,

the inverse cosine function may be represented as1

cos x−

or arccos x .

29. Find the position function, ( )P t , given ( ) 5a t = , (3) 4v = , and (2) 0P = where the

acceleration is given by ( )a t and the velocity is given by ( )v t .

(A)2( ) 2.5 11 12P t t t = − −

(B)2( ) 4 7 3P t t t = − +

(C)2

( ) 5 4P t t t = +

(D+)2( ) 2.5 11 12P t t t = − +

(E)2( ) 7 3P t t t = + +

30. Give a value of c that satisfies the conclusion of the Mean Value Theorem for Derivatives for

the function2( ) 3 4 5 f x x x= − + − on the interval [ ]1, 2 .

(A)2

3

(B+)3

2

(C)4

5

(D) 5−

(E)5

4




31. The area of a circle with radius r is2

V r π = . At the time when the area and radius are

changing at the same numerical amount, what is the area?

(A) 3.14159

(B) 0.31831

(C) 1.61803

(D+) 0.78539

(E) 0.76126

32. Given ( )1

( ) 3 1 x f x x= + find ( )0

lim x

f x→

.

(A+) 20.0855

(B) 1.0000

(C) 0.0498

(D) 20.0765

(E) 19.9955

33. Find the value of c on the interval [ ]0,π that satisfies the Mean Value Theorem for

Derivatives given ( ) 2 sin f x x x= − .

(A) 0.7854

(B) 1.4297

(C) 2.0000

(D+) 1.5708

(E) 3.1415

34. If '( ) 2 sin(2 ) for 0 4 f x x x x= < < then f has a local max at x ≈

(A) 3.990

(B) 1.9293

(C) 1.782

(D) 6.345

(E+) 1.571




35. A rock is thrown upwards (vertically) from the ground with an initial velocity of 48 feet per

second. If the acceleration due to gravity is2

32sec

ft − , how high will the rock go?

(A) 48 feet

(B) 40 feet

(C+) 36 feet

(D) 80 feet

(E) 32 feet

36. What is the average value of the function ( ) sin 3 ln f x x x= + on the interval [ ]1,5 ?

(A) 3.9704

(B+) 0.9926

(C) 2.1186

(D) 0.5297

(E) 1.5971

37. An object moves along the x-axis so that its velocity at any time 0t ≥ is given by3( ) 16v t t t = − . Find the total displacement of the particle from 0 to 6t t = = .

(A) 164

(B) 81

(C) 120

(D+) 36

(E) 40

38. Estimate the area under the curve of2

( ) cos ( ) 1 f x x= + from 2 to 4 x x= = . Use four sub-

intervals and right endpoints.

(A) 8.0993

(B) 6.9261

(C) 1.030

(D+) 3.4631

(E) 3.3361




39. The base of a solid is the region in the xy-plane enclosed by the curves

( ) sin( ), ( ) cos( ) f x x g x x= − = over the interval3

0,4

π ⎡ ⎤⎢ ⎥⎣ ⎦

. Cross sections of the solid

perpendicular to the x-axis are squares. Determine the volume of the solid.

(A) 2.3562

(B+) 2.8562

(C) 1.4142

(D) 0.7071

(E) 1.8657

40. The first derivative of the function f is given by4 2'( ) 5 40 f x x x= − . How many points of

inflection does the graph of f have?

(A) one

(B) two

(C+) three

(D) four

(E) zero

41. Find the average value of ( ) cos 2 sin f x x x= + on the interval 0,

2

π ⎡ ⎤⎢ ⎥

⎣ ⎦

.

(A) 0.8660

(B) 0.7071

(C) 0.0000

(D+) 0.6366

(E) 1.0000




42. If2( ) 7 x

f x xe−= , then the y-value of the horizontal tangent line is

(A) 0.5531

(B) 1.2807

(C) 9.5140

(D) 0.5000

(E+) 1.2876

43. Find the interval(s) on which the curve3( ) sin f x x x= − is concave up.

(A) ( )0.1247,− ∞

(B+) ( )0,∞

(C) ( ), 0−∞

(D) ( )0.0314, ∞

(E) ( )31.3742, 29.3371− − and ( )0.0314, ∞

44. The velocity of a particle moving along a line is ( )( ) cosv t t π π = meters per second. Find the

distance traveled in meters during the time interval 1 3t ≤ ≤ .

(A) 0

(B) 2

(C) π

(D+) 4

(E) 5




45. If

2 4

0

( ) ( )

x

F x f t dt

+

= ∫ and (13) 5 f = − then '(3)F =

(A) 30

(B) 6

(C) 5−

(D+) 30−

(E) 20−




Consolidated Answers

1 . A 16. B 31. D

2 . D 17. A 32. A

3 . C 18. B 33. D

4 . D 19. E 34. E

5 . A 20. A 35. C

6 . C 21. E 36. B

7 . D 22. B 37. D

8 . C 23. C 38. D

9 . E 24. C 39. B

10 . C 25. B 40. C

11 . E 26. D 41. D

12 . D 27. B 42. E

13 . B 28. D 43. B

14 . D 29. D 44. D

15 . B 30. B 45. D

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AP Practice Exam (6)