ap calculus ab practice test (1)

8/2/2019 AP Calculus AB Practice Test (1)

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AP calculus AB 1

CALCULUS AB

SECTION I, Part A

Time 55minutes

Number of questions 28

A calculator may not be used on this part of the examination.

Part A consists of 28 questions. In this section of the examination, as a correction for guessing,

one-fourth of the number of questions answered incorrectly will be subtracted from the number

of questions answered correctly.

Directions: Solve each of the following problems, using the available space for scratchwork.

After examining the form of the choices, decide which is the best of the choices given and fill

in the corresponding oval on the answer sheet. No credit will be given for anything written in

the test book. Do not spend too much time on any one problem.

In this test: Unless otherwise specified, the domain of a function ƒ is assumed to be the set of

all real numbers x for which ƒ( x) is a real number.

1. The function ƒ, whose graph consists of two line segments, is shown below. Which of the

following are true for ƒ on the open interval (8, 8)?

I. The domain of the derivative of ƒ is the open interval (

8, 8).II. ƒ is continuous on the open interval (8, 8).

III. The derivative of ƒ is positive on the open interval (3, 0) (3,6).

(A) I only

(B) II only

(C) III only

(D) II and III only

(E) I, II, and III

2.Find the average rate of change of y = x2

+ x +1 over the interval [0, 1].

(A)1

(B)1

(C)2

(D)None of the above

(E)Undecidable

3.What does the limit statement 1

2ln)1ln(

lim1

x

x

x represent?



AP calculus AB 2

(A)0 (B) ))1(ln( xdx

d

(C) f (1), if f ( x)=ln( x + 1 ) (D)1 (E) The limit does not exist

4.Find the derivative of the function 3

4

x y

(A)2

4 x (B)2

12

x(C)

2

12

x (D)

4

12

x(E)

4

12

x

5.Find415

1lim

2

2

1

x

x

x

(A)9 (B)8 (C)7 (D)1 (E)0

6.What is the x-intercept of the tangent line to the curve y = 4 x2

at the point(1, 3)？ (A)3 (B)0.3 (C)0.8 (D)2 (E) 2.5

Solution：Slope of the tangent line to y = 4 x2

at x =1 is )1( y =2. The point-slope formula

gives the tangent equation )1(23 x y . Setting y = 0, we get x =2.5.

7. What is x x

x x x

x cos103

sin2lim

(A)13

(B)1 (C)2 (D)3 (E)0

8.d

dx e

sin2 x=？

(A) cos2 xesin2 x

(B)cos2 xesin2 x

(C)2 cos2 xesin2 x

(D) 2esin2 x

(E) 2cos2 xesin2 x

9.If F ( x)=∫ 2

2 dt , then F (2)=？ (A)64

3 (B)64 (C)16

3 (D)16 (E)56

3 10.Let the graph of f is shown to the right, and g be defined as

x

dt t f xg10

)()( . Which of the following statements about f

and g is false？

(A) )3(g =0

(B)g has a local minimum at x = 3.

(C) g(10)=0



AP calculus AB 3

(D) )1( f does not exist.

(E) g has a local maximum at x =1.

11.Suppose for a real number x we defined [ x ] to be the greatest integer less than or equal to x.

Let f (t )= t 2t +1. Find )(lim

100 x f

x .

(A)2134 (B)2134 (C)9703 (D)4001 (E)6379

12.d

dx

3

2)(

xdt t f =？ (A) )3( f (B)2 f (2 x) (C) 2 f (2 x)

(D)2 )2( x f (E) 2 )2( x f

13.Which of the following defines a function f for which f ( x)= f ( x)？ (A) f ( x)= 2

x (B) f ( x)= sin x (C) f ( x)= cos x (D) f ( x)= log x (E) f ( x)= e x

14.For what value of k will x +k

xhave a relative maximum at x = 2？

(A) 4 (B) 2 (C)2 (D)4 (E)None of these.

15. Let )( x f =2 x+1 for all real x and let >0. For which of the following choices of is

|5)(| x f

whenever|2| x？

(A)ε

2(B) (C)

ε

+1(D)

ε+1

(E)3.

16. The function f is given by f ( x)= 1532248234 x x x x . All of these statements are true

EXCEPT

(A)1 and 3 are zeros of f .

(B) )2( f =0.

(C) )2( f =0.

(D)(2,1) is a point of inflection of f .

(E) (2,1) is a local minimum of f .

17. Determinedx

dyfor the curve defined by

33 y x =3 xy.

(A) x y

x

2

2

(B)2

2

y x

x

(C)

x y

x y

2

2

(D)1

1

y

x (E)

x y

y x

2

2

18.If 2)( x x f , which of the following could be the graph of y = f ( x)？



AP calculus AB 4

The correct answer is (E).

19. )(ln2 xe

dx

d =？

(A) x

e2

1(B)

xe

2

2(C)2 x (D)1 (E)2

20. If f ( x) e

1

x, then f ( x)

(A)e

1

x

x2

(B)e1

x (C)e

1

x

x

(D)e

1

x

x2

(E)1

x

e

1

x

1

21. If y sin x and y(n)

means “the nth derivative of y with respect to x ,” then the

smallest positive integer n for which y(n) y is

(A)2 (B)4 (C)5 (D)6 (E)8



AP calculus AB 5

22.∫||

−1is

(A)4 (B)2 (C)1 (D)0 (E)Nonexisten.

23. If x = 2et and y = t

21, then

=

(A)

t e

t (B)

t

e

t 2(C)

t

e

t

2(D)

t

e

t 2

4(E)

t e

24. The area of the region enclosed by the graph of y = x2+1 and the line y = 5 is

(A) 3

14(B)

3

16(C)

3

28 (D)

3

32(E)8π

25. dx x x )64(3

2

1 =？

(A)2 (B)4 (C)6 (D)36 (E)42

26.If ∫ ()

=a+2b, then ∫ (() 5)

=

(A) a+2b +5 (B)5a 5b (C) 7b 4a (D) 7b 5a (E) 7b 6a

27.The graph of y =3 x416 x

3+24 x

2+48 is concave down for

(A) x < 0 (B) x > 0 (C) x <2 or x > 3

2(D) x <

3

2or x >2 (E)

3

2< x <2

28.The set of all points ( et , t ), where t is a real number, is the graph of y =

(A) xe

1(B)

xe

1

(C) x xe

1

(D) xln

1 (E) xln

STOP

END OF PART A SECTION I

IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART

A ONLY. DO NOT GO ON TO SECTION II UNTIL YOU ARE TOLD TO DO SO.



AP calculus AB 6

Part B Sample Multiple-Choice Questions

A graphing calculator is required for some questions on this part of the examination.

Part B consists of 17 questions. In this section of the examination, as a correction for guessing,

one-fourth of the number of questions answered incorrectly will be subtracted from the number

of questions answered correctly. Following are the directions for Section I Part B and a

representative set of 10 questions.

Directions: Solve each of the following problems, using the available space for scratchwork.

After examining the form of the choices, decide which is the best of the choices given and fill

in the corresponding oval on the answer sheet. No credit will be given for anything written in

the test book. Do not spend too much time on any one problem.

In this test:

(1) The exact numerical value of the correct answer does not always appear among the choices

given. When this happens, select from among the choices the number that best approximatesthe exact numerical value.

(2) Unless otherwise specified, the domain of a function ƒ is assumed to be the set of all real

numbers x for which ƒ( x) is a real number.

29. For a certain function f ( x) you know that )3( f =2. Using this information, you saw, in a

Web Work problem, how you can calculate an approximate value for f (2.99) f (3).

Suppose you also know that f (3)=7.Then calculate an approximate value for f (2.99).

(A)6.15 (B)6.98 (C)6.52 (D)6.35 (E)6.21

30. If 32+ 2 xy +2 =2, then the value of

at x =1 is

(A)2 (B)0 (C)2 (D)4 (E)not defined.

31. ∫tan

cos

dx =

(A)0 (B)1 (C)e 1 (D) e (E) e + 1

32. If n is a non-negative integer, then 1

0 dx x

n = 1

0)1( dx x

nfor

(A)no n (B)n even, only (C)n odd, only (D)nonzero n only (E)all n.

33. If a, b, c, d , and e are real numbers and a≠0, then the polynomial equation

edxcxbxax 357 =0 has

(A)only one real root (B)at least one real root

(C)an odd number of nonreal roots (D)no real roots



AP calculus AB 7

(E)no positive real roots.

34. The function defined by f ( x)=√ 3cos x +3sin x has an amplitude of

(A)3√ 3 (B)√ 3 (C)2√ 3 (D) 3 + √ 3 (E)3√ 3

35. If xdx

dytan , then y =

(A) C x 2tan

2

1(B) C x 2

sec (C) C x |sec|ln

(D) C x |cos|ln (E) C x x cossec

36.

1

0

2 12 dx x x is

(A)1 (B)2

1 (C)

2

1(D)1 (E)none of the above.

37.∫ 3 =

(A) x

x3ln+C (B)

x

x

3

3ln+C (C) x x x ln +C

(D) x x x 3ln +C

(E) x x x 33ln3 +C

38.The radius of a sphere is increasing at a rate proportional to its radius. If the radius is 4

initially, and radius is 10 after two seconds, what will be the radius be after three seconds？ (A)62.50 (B)13.00 (C)15.81 (D)16.00 (E)25.00

39.If )( x f =ln x x + 2, at which of the following values of x does f have a relative minimum

value？ (A)5.146 (B)3.146 (C)1.000 (D0.159 (E)0.00

40. x

x tdt dx

d 5

2 cos =

(A)5cos5 x2cos2 x (B) 5sin5 x2sin2 x (C)cos5 xcos2 x

(D) sin5 xsin2 x (E)5sin5 x2cos2 x

41.If is an equation of the line normal to the graph of f at the point(1,4), then

)1( f =

(A)7 (B)7

1(C)

7

1 (D)

29

7 (E)7



AP calculus AB 8

42. A polynomial P( x) has a relative maximum at (2,4), a relative minimum at (1,1), a relative

maximum at(5,7), and no other critical points. How many zeros does have P( x)？ (A)One (B)Two (C)Three (D)Four (E)Five

43. If the graph of y = 423 bxax x has a point of inflection at(1,6), what is the value of

b？

(A)3 (B)0 (C)1 (D)3 (E)It cannot be determined from the

information given.

44.∫ √ 2 2

=

(A)

3

8(B)

3

16 (C)π (D)2π (E)4π

45. A person 2 meters tall walks directly away from a streetlight that is 8 meters above the

ground. If the person is walking at a constant rate and the person’s shadow is lengthening at the

rate of 9

4meter per second, at what rate, in meters per second, is the person walking?

(A)27

4(B)

9

4(C)

4

3 (D)

3

4(E)

9

16

STOP

END OF PART B SECTION I

IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART

B ONLY. DO NOT GO ON TO SECTION II UNTIL YOU ARE TOLD TO DO SO.



AP calculus AB 9

Section II

Free- Response Questions

Time: 1 hour and 30 minutes

Number of Problems: 6

Part A

Time: 45 minutes


You may use a calculator for any problem in this section.

1.

Let R be the region bounded by the graph of y=2√ and y = x, as shown in the figure above.

(a)Find the area of R.

(b)The region R is the base of a solid. For this solid, the cross sections perpendicular to the

x-axis are squares.Find the volume of this solid.(c)Write, but do not evaluate, an integralexpression for the volume of this solid generated when

R is rotated about the horizontal line y =4.

Solution：(a) A( R)= 4

0)2( dx x x =

3

8

(b)V (S)= 4

0

2)2( dx x x =

3

64

(c)V y = 4=π 4

0

22)24()4( dx x x =

3

32

2.Let ()=

x x

x

x

2

12

12

1for x > 0.

(1)Find the explicit form of ().

(2)Find )( x f =？

(3)Find f (99) )99( f =？



AP calculus AB 10

3.Consider the function whose graph is

(a)What is the value of 1

1)( dx x f .

(b)Where the function is concave up？

(c)Where the lim ( )o

x x f x

fails to exist？

Solution：(a)

1

1)( dx x f =

π

2.

(b) 2< x <3.

(c) x =3.



AP calculus AB 11

Section II

Part B

Time: 45 minutes


You may not use a calculator for any problem in this section.

During the timed portion for Section II, Part B, you may continue to work on the

problems in Part A without the use of a calculator.

4. Let f be the function given by )( x f =9

2 x

x.

(a)Find the domain of f .

(b)Write an equation for each vertical asymptote to the graph of f .

(c)Write an equation for each horizontal asymptote to the graph of f .

Solution：(a) 3|| x , x >3 or x <3.

(b) x =3, x = 3

(c) x =1, x = 1

5. Let f be a function that is everywhere differentiable and that has the following properties.

(i))()(

)()()(

h f x f

h f x f h x f

for all real numbers h and x.

(ii) 0)( x f for all real numbers x.

(iii) 1)0(

f .(a)Find the value of )0( f .

(b)Show that )( x f =)(

1

x f for all real numbers x.

(c)Using part (b), show that )()()( h f x f h x f for all real numbers h and x.

(d)Using the definition of the derivative to find )( x f in terms of )( x f .

Solution：(a)Let x=h=0, then )0( f =)0()0(

)0()0(

f f

f f

=1.

(b)Let h =0, 1)(

1)(

)0()(

)0()(

)0(

x f

x f

f x f

f x f

x f 1)()()()( x f x f x f x f

1)()( x f x f ∴ )( x f =)(

1

x f for all real numbers x.

(c)By(b), )( h x f =

)(

1

)(

1

)()(

h f x f

h f x f

= f ( x) f (h)for all real numbers h and x.

(d) )( x f =h

x f h x f

h

)()(lim

0

=h

x f h f x f

h

)()()(lim

0

= f ( x)h

h f

h

1)(lim

0

= f ( x) )()0( x f f



AP calculus AB 12

6. A particle, initially at rest, moves along the xaxis so that its acceleration at any time

t 0 is given by () 2 . The position of the particle when t =1 is x(1)=3.

(a)Find the value of t for which the particle is rest.

(b)Write an expression for the position () of the particle at any time t 0.

(c)Find the total distance traveled by the particle from t =0 to t =2.

Solution：(a) () ∫ () , let () 0, then 4(2 ) 0, t =0,1.

(b) () ∫ () 4 2 C , x(1)=3=12+C , C =4, () 4 2 ,for any t 0.

(c)(0)= 4, () 3, ()=12.Distance=1+9=10.

END OF EXAMINATION

ap calculus ab practice test (1)

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