AP Calculus AB Exam 3Multiple Choice SectionName:_____________

2. An equation of the line tangent to the graph of f( x ) = x ( 1 – 2x)3 at the point ( 1,-1) is

12 (E) 87 (D)

32 (C) 56 (B) 67 (A)

xyxy

xyxyxy

1. Find the value of the derivative of f(x) = x3 – x2 at x = 1.

(A) -1 (B) 1 (C) 0 (D) 6 (E) It does not exist

3. If f ( x ) = 2 + |x – 3| for all x, then the value of the derivative f (x) at x = 3 is

(A) -1 (B) 0 (C) 1 (D) 2 (E) nonexistent

1 (E) 6 (D) 12 (C) ln3 (B) ln2 (A)

.4

2312123 33

xxeex xx

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dx

d

5. If the position of an object dropped from the top of a cliff is given as s(t) = -16t2 + 225 where s is measured in feet and t is measured in seconds, find the velocity of this object after it has been falling for one seconds.

A) –32 ft/s B) –64 ft/s C) –128 ft/s D) –225 ft/sec E) –16 ft/sec

8. If f ( x ) = x + sin x, then f ( x ) =

(A) 1 + cos x (B) 1 cos x (C) cos x

(D) sin x + x cos x (E) sin x + x cos x

6. A particle moves along the x-axis so that at any time t > 0 its position is given by x( t) = t3 - 3t2 – 9t + 1. For what values of t is the particle at rest?

(A) It is never at rest (B) t =1 only (C) t =3 only (D) t = 5 only (e) t = 1 and 3

7. At what point on the graph y = x2 is the tangent line parallel to the line 2x – 4y=3?

(A) (,-) (B) (,) (C) (1,-) (D) (1,) (E) (2,2)

10. If is differentiable at x = 2, then a + b =

2 ,2 , )(

2

xbaxxxxf

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

2

3 (E)

4

3 (D)

4

6 (C)

4

3 (B)

4

6 (A)

then ,4

3 If 9.

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xxx

x

x

x

x

x

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12. Suppose that f is an odd function and that f (c) exists. Which of the following must necessarily be equal to f (-c)?

these.of None (E) )(

1 (D)

)(

1 (C) )( (B) )( (A)

cf

cf cf cf

The function f given by f ( x ) = x3 + 12x 24 is

(A) increasing for x < -2, decreasing for -2 < x < 2, increasing for x>2(B) decreasing for x < 0, increasing for x > 0(C) increasing for all x(D) decreasing for all x (E) decreasing for x < -2, increasing for -2 < x < 2, decreasing for x>2

An equation of the line tangent to the graph of at the point (1,5) is

1332 (E) 6613 (D)

6413 (C) 1813 (B) 813 (A)

yxyx

yxyxyx

23

32

x

xy

5 (E) 3 (D) 1 (C) 1 (B) 3 (A)

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fx

xxxf

13.

16.

15.

A particle moves along a line so that at time t, where 0 < t < , its position is given by s ( t ) = -2cos t t 2 + 10. What is the velocity when the acceleration is zero?

42

2 (E)

62

1 (D)

3

3 (C) 62

1 (B)

33 (A)

14.

h

xxhxhx

h

hhhxhx

h

hhxx

h

xxxhx

h

hxx

h

hh

hh

)()()(lim (E)

)()()(lim (D)

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)()(

lim (B) lim (A)

22

0

22

0

22

0

22

0

2

0

?

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218

21

0 h

hh

given.n informatio thefrom determined becannot It (E)

exist.not doeslimit The (D) 1 (C) 2

1 (B) 0 (A)

18. Which of the following will calculate the derivative of f(x) if f(x) = x2 + x?

AP Calculus AB Exam 3Calculator Active SectionName:_____________

20. Find any point(s) on where there is a horizontal tangent linexxxf 2)( 3

(A) (0.816, 1.089) & (-0.816, -1.089) (B) (1.414,0) & (-1.414,0) only

(C) (1.414,0) , (-1.414,0) & (0,0) (D) (0.816, 1.089) only

(E) (0.816, 0) & (-0.816, 0)

22. At x = 3, the function given by

(A) Undefined(B) Continuous but not differentiable(C) Differentiable but not continuous(D) Neither continuous nor differentiable(E) Both continuous and differentiable

3,963,)(

2

xxxxxf

21. If f is a differentiable function, then f (a) is given by which of the following?

h

xfhxfax

afxfh

afhaf

ax

ax

h

)()(lim III.

)()(lim II.

)()(lim I.

0

(A) I only (B) II only (C) I and II only (D) I and III only (E) I, II and III

23 If the second derivative of f is given by f = 2x – cos x, which of the following could be f(x)?

(A) x3 + cos x – x + 1

(B)x3 cos x – x + 1

(C) x3 + cos x – x + 1

(D) x2 sin x + 1

(E) x2 + sin x + 1

24.

25. If the line 3x – 4y = 0 is tangent in the first quadrant to the curve y = x 3 + k , then k is

2

1 (E)

8

1 (D) 0 (C)

4

1 (B)

2

1 (A)

27. The position function of a particle that is moving along the x-axis for 0 < t < 6 is given as

a. Find the average velocity for t = 1 to t = 3

b. Find any the interval(s) where the particle is traveling to the right.

Justify your answer(s)

c. Find the total distance traveled by the particle for 0 < t < 6

0for 862

5

3

1)( 23 tttttx

Let f be a function such that Which of the following must be true?

I. f is continuous at x = 2. II. f is differentiable at x = 2. III. The derivative of f is 5.

(A) I only (B) II only (C) III only (D) I and II only (E) I, II and III

.5)2()2(

lim0

h

fhfh

26.

Free Response