8/9/2019 APexams69 AB

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90 MinutesNo Calculator

 Note: In this examination, ln x denotes the natural logarithm of  x (that is, logarithm to the base  e).

1. Which of the following defines a function   f    for which ( ) ( )? f x f x

  (A) 2( ) f x x   (B) ( ) sin f x x   (C) ( ) cos f x x

  (D) ( ) log f x x   (E) ( )   x f x e

2. ln 2 0 if and only if   x

  (A) 3 x   (B) 0 3 x   (C) 2 3 x

  (D) 2 x   (E) 3 x

3. If 

2 5 7( ) , for 2,

2

(2)

 x x f x x

 x

 f k 

  and if   f    is continuous at 2 x , then   k 

  (A) 0 (B)1

6

  (C)1

3

  (D) 1 (E)7

5

4.8

0 1

dx

  (A) 1 (B)3

2  (C) 2 (D) 4 (E) 6

5. If  2 23 2 2, x xy y  then the value of   dy

dx

 at 1 x  is

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

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6. What is

8 8

0

1 18 8

2 2lim ?h

h

h

  (A) 0 (B)1

2  (C) 1 (D) The limit does not exist.

  (E) It cannot be determined from the information given.

7. For what value of  k  will  k 

 x x

 have a relative maximum at 2? x

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

8. If ( ) 2 p x x x k   and if the remainder is 12 when ( ) p x  is divided by 1, x  then   k 

  (A) 2 (B) 3 (C) 6 (D) 11 (E) 13

9. When the area in square units of an expanding circle is increasing twice as fast as its radius in

linear units, the radius is

  (A)1

4  (B)

1

4  (C)

1  (D) 1 (E)

10. The set of all points ( , )t e t  , where t  is a real number, is the graph of   y

  (A)1 xe

  (B)

1

 xe   (C)

1

 x x e   (D)1

ln  (E) ln x

11. The point on the curve 2 2 0 x y  that is nearest the point1

0,2

 occurs where y is

  (A) 12

  (B) 0 (C) 12

  (D) 1  (E) none of the above

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12. If 4

( )1

 f x x

 and ( ) 2 , g x x  then the solution set of ( ) ( ) f g x g f x  is

  (A) 13

  (B) 2 (C) 3 (D) 1,2 (E) 1 ,23

13. The region bounded by the x-axis and the part of the graph of cos y x  between2

 x  and

2 x  is separated into two regions by the line   k . If the area of the region for 

2k   is

three times the area of the region for ,2

k x  then k  =

  (A)1arcsin4

  (B) 1arcsin3

  (C)6

  (D)4

  (E)3

14. If the function   f    is defined by 5 1( ) 1, then f x x f   , the inverse function of   f   , is defined by

1( ) f x

  (A)5

1

1 x  (B)

5

1

1  (C)

51

  (D)5

1 (E)5

1 x

15. If ( ) f x  and ( ) g x  exist and ( ) ( ) f x g x  for all real  x, then the graph of ( ) y f x  and the graph

of ( ) y g x

  (A) intersect exactly once.

  (B) intersect no more than once.

  (C) do not intersect.

  (D) could intersect more than once.

  (E) have a common tangent at each point of intersection.

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16. If   y is a function of  x such that 0 y  for all x and 0 y  for all x, which of the following could

 be part of the graph of ( )? y f x

17. The graph of  4 55 y x x  has a point of inflection at

(A) (0,0)  only (B) (3,162)  only (C) (4,256)  only

(D) (0,0) and (3,162) (E) (0,0) and (4,256)

18. If ( ) 2 3 f x x  for all x, then the value of the derivative ( ) f x  at 3 x  is

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

19. A point moves on the x-axis in such a way that its velocity at time  t  0t   is given byln

.t 

vt 

  At what value of  t  does v attain its maximum?

  (A) 1 (B)

1

2e   (C)   e   (D)

3

2e

  (E) There is no maximum value for   v.

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20. An equation for a tangent to the graph of arcsin2

 x y  at the origin is

  (A) 2 0 x y   (B) 0 x y   (C) 0 x   (D) 0 y   (E) 2 0 x y

21. At 0 x , which of the following is true of the function   f    defined by 2 2( )   x f x x e  ?

  (A)   f    is increasing.

  (B)   f    is decreasing.

  (C)   f    is discontinuous.

  (D)   f    has a relative minimum.

  (E)   f    has a relative maximum.

22. 2ln   xd 

edx

  (A)2

1 xe

  (B)2

2 xe

  (C) 2 x   (D) 1 (E) 2

23. The area of the region bounded by the curve 2 x y e , the x-axis, the y-axis, and the line 2 x  is

equal to

  (A)4

2

ee   (B)

4

12

e  (C)

4 1

2 2

e

  (D) 42e e   (E) 42 2e

24. If sin , 0 , y x e x  what is  dy

dx in terms of  x ?

  (A) tan x   (B) cot (C) cot x   (D) tan x   (E) csc x

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25. A region in the plane is bounded by the graph of 1

 y , the x-axis, the line , x m  and the line

2 , x m 0m . The area of this region

  (A) is independent of  m .

  (B) increases as m increases.

  (C) decreases as m increases.

  (D) decreases as m increases when1

2m ; increases as m increases when

1

2m .

  (E) increases as m increases when1

2m ; decreases as m increases when

1

2m .

26.

1 2

0 2 1 x x dx  is

  (A) 1

  (B)1

2

  (C)1

2

  (D) 1

  (E) none of the above

27. If  tan ,dy

 xdx

 then   y

  (A) 21

tan2

 x C   (B) 2sec   x C    (C) ln sec x C 

  (D) ln cos   C   (E) sec tan x x C 

28. The function defined by ( ) 3 cos 3sin f x x x  has an amplitude of 

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

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29.2

4

cos

sin

 xdx

 x

  (A) ln 2 (B) ln 4   (C) ln 3 (D)

3

ln 2   (E) ln e

30. If a function   f    is continuous for all x and if   f    has a relative maximum at ( 1,4) and a relative

minimum at (3, 2) , which of the following statements must be true?

  (A) The graph of   f    has a point of inflection somewhere between 1 x  and 3. x

  (B) ( 1) 0 f  

  (C) The graph of   f    has a horizontal asymptote.

  (D) The graph of   f    has a horizontal tangent line at 3 x .

  (E) The graph of   f    intersects both axes.

31. If  ( ) ( ) f x f x  and (1) 1, f    then ( ) f x

  (A) 2 21

2

 xe   (B) 1 xe   (C) 1   xe   (D)   xe   (E)   xe

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

7 5 3 0ax bx cx dx e  has

  (A) only one real root.

  (B) at least one real root.

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

  (E) no positive real roots.

33. What is the average (mean) value of  3 23t t   over the interval 1 2t  ?

  (A)

11

4   (B)

7

2   (C) 8 (D)

33

4   (E) 16

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34. Which of the following is an equation of a curve that intersects at right angles every curve of the

family1

 y k  x

 (where k  takes all real values)?

  (A)   y x   (B) 2 y x   (C) 313

 y x   (D) 313

 y x   (E) ln y x

35. At 0t   a particle starts at rest and moves along a line in such a way that at time t  its acceleration

is 224t   feet per second per second. Through how many feet does the particle move during the first

2 seconds?

  (A) 32 (B) 48 (C) 64 (D) 96 (E) 192

36. The approximate value of 4 sin at 0.12 y x x , obtained from the tangent to the graph at0, x  is

  (A) 2.00 (B) 2.03 (C) 2.06 (D) 2.12 (E) 2.24

37. Which is the best of the following polynomial approximations to cos 2 x  near 0 x ?

  (A) 12

 x  (B) 1   x   (C)

2

12

  (D) 21 2 x   (E) 21 2 x x

38.3

2

 x

 xdx

e

  (A)31

ln3

 xe C   (B)

3

3

 xeC   (C)

3

1

3   xC 

e

  (D)31

ln3

 xe C   (E)3

3

3   x

 xC 

e

39. If 1

tan , , and ln , y u u v v xv

 what is the value of   dy

dx at ?e

  (A) 0 (B)1

e  (C) 1 (D)

2

e  (E) 2sec   e

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40. If  n is a non-negative integer, then1 1

0 01

  nn dx x dx  for 

  (A) no n  (B)   n even, only (C)   n odd, only

  (D) nonzero n, only (E) all n

41. If 

2

2

( ) 8 for 2 2,

( ) elsewhere ,

 f x x x

 f x x  then

3

1( ) f x dx  is a number between

  (A) 0 and 8 (B) 8 and 16 (C) 16 and 24 (D) 24 and 32 (E) 32 and 40

42. What are all values of  k  for which the graph of  3 23 y x x k  will have three distinct

 x-intercepts?

  (A) All 0k    (B) All 4k    (C) 0, 4k    (D) 0 4k    (E) All   k 

43. sin 2 3 x dx

  (A)1

cos 2 32

 x C   (B) cos 2 3 x C   (C) cos 2 3   C 

  (D)1

cos 2 32

 x C   (E)1

cos 2 35

 x C 

44. The fundamental period of the function defined by 2( ) 3 2cos3

 x f x  is

  (A) 1 (B) 2 (C) 3 (D) 5 (E) 6

45. If ( ) ( )d 

 f x g xdx

 and 2( ) ( ) ,d 

 g x f xdx

 then2

3

2( )

d  f x

dx

  (A) 6 f x   (B) 3 g x   (C) 2 33 x g x

  (D) 4 6 39 6 x f x x g x   (E) 6 3 f x g x