AP CALCULUS BC – BC Multiple Choice Worksheet #48.5 NO CALCULATORS ON PROBLEMS 1 – 28.

1. 1

2

0( 1)x x dx+ =! A) 4

3 B) 9

7 C) 16

15 * D) 20

21 E) 4

21

2. If 4tx e= and sin 6 ,y t= then dy

dx =

*A) 4

3 cos6

2

te t!

B) 4

3cos6

2t

t

e! C)

43 cos

2

te t!

D) 4cos6t

e t! E) 6cos6t

3. The function f defined by 4 2( )f x x x= ! has a relative minimum at x =

A) 2 B) 1 * C) 2

2 D) 1

2 E) 0

4. 3

2 ln xdx e

dx= A) 3

6x * B) 45x C) 2

2 3x x+ D) 4 52x x+ E) 4

6x

5. If 5

2 6 21

( ) ( 2)4

xg x e x

!= + ! , then '(3)g = * A) 3 B) 5

2 C) 11

4 D) 1

4 E) 0

6. Find the slope of the line normal to the curve 4y x= ! + at the point where 0x =

A) 4! B) 1

4! C) 1

8! D) 1

4 * E) 4

7. Compute dydx

for the relation 2 2 32 3ln 3xy y x y+ = ! at the point (3,1)

A) 52

B) 34

C) 512

D) 18

* E) 16

8. 2

3 2

1(1 )

x

x

!

+" dx is A) 1

6! B) 1

24! C) 1

24 * D) 1

6 E) divergent

The function ( )h x is continuous and differentiable on the domain [ ]0,7 . The graph of '( )h x is shown. Use the graph for Questions 9, 10, and 11.

9. At what value of x does ( )h x have its absolute minimum? A) 0 * B) 1 C) 3 D) 5 E) 7

10. The point ( )5,2 is on the graph of ( )y h x= . An equation of the line tangent to ( )h x at ( )5,2 is

*A) 2 5y x! = ! B) 2y x= ! C) 2 2( 5)y x! = ! D) 5x = E) 2y =

11. How many inflection points does h have on the interval (0,7) ? A) 3 * B) 4 C) 5 D) 6 E) 7

AP CALCULUS BC – BC Multiple Choice Worksheet #48.5

12. The sum of the infinite geometric series 8 24 72 216...

25 125 625 3125! + ! + is

*A) 0.2 B) 0.6 C) 0.8 D) 1.0 E) 1.2 13. A particle moves along the x-axis so that its acceleration at any time t is ( ) 2 3a t t= ! . If the initial

velocity of the particle is 4! , at what time t in the time interval 0 5t! ! is the particle farthest left?

A) 0 B) 32

C) 3 * D) 4 E) 5

14. The graph of '( )f x is shown. It is tangent to the x-axis at the point c .Which of the following describes all

relative extrema of ( )f x on the open interval ( , )a b ?

A) One relative maximum and one relative minimum B) One relative maximum and two relative minima C) Three relative maxima and two relative minima D) Two relative maxima and two relative minima *E) Two relative maxima and one relative minimum 15. The length of the path described by the parametric equations 2sinx t= and 3cosy t= for 0 / 2t !" " is

given by *A)

/ 22

0

4 5sin tdt!

+" B) / 2

2 2

0

4cos 9sint tdt!

"# C) / 2

2 2

0

4sin 9cost tdt!

+"

D) 2

/ 2

20

9sin14cos

tdtt

!

+" E) none of the above

16. 2

9

3

lim3

x

x

e e

x!

"=

" A) 0 B)

9

3

e C) 3e9 * D) 6e9 E) !

17. Let f be the function defined by ( ) ln( 4)f x x= + . The third-degree Taylor polynomial for f centered about

x = -3 is

A) 2 3( 3) ( 3)

32 3

x xx

! !! ! ! B)

2 3( 3) ( 3)3

2 3

x xx

! !! ! + C)

2 3( 3) ( 3)3

2 3

x xx

+ +! + ! +

D) 2 3( 3) ( 3)

32 3

x xx

+ ++ + + *E)

2 3( 3) ( 3)3

2 3

x xx

+ ++ ! +

AP CALCULUS BC – BC Multiple Choice Worksheet #48.5

18. For what values of t does the curve defined by the parametric equations 3 24

3x t t= ! and

5 27y t t t= + ! have a vertical tangent?

A) 0 only * B) 0 and 12

C) 12

only D) 1 only E) 0, 12

, and 1

19. The graph of y = f(x) is shown. Let A and B be positive numbers that represent the area of each shaded

region. Evaluate 1 2

3 1( ) 3 ( )f x dx f x dx

!

!+" " in terms of A and B.

*A) –2A – B B) 2A + B C) 3A – B D) 3A + B E) -3A – B

20. What are all values of x for which the series 1

(2 1)

4

n

n

n

x

n

!

=

"#

i

converges?

A) 3 5

2 2x! " " * B) 3 5

2 2x! " < C) 3 5

2 2x! < " D) 3 5x! " < E) 3 5x! < "

21. The expression representing the area inside one leaf of the polar rose 3cos 2r != is given by

A) / 4

2

0

1 9sin 2 d!

" "+# B) / 2

2

0

1 9sin 2 d!

" "+# C) / 2

2

0

9cos 2 d!

" "#

D) / 4

2

0

19cos 2

2d

!

" "# * E) / 4

2

0

9cos 2 d!

" "#

22. The graph of f is shown. If ( ) ( )

x

ah x f t dt= ! , for what value of x does h(x) have its minimum?

A) a B) b * C) c D) d E) e

23. In the right triangle shown, ! is increasing at a constant rate of 2 radians per minute. In units per minute,

at what rate is x increasing when x = 12?

A) 2 B) 4 C) 5 * D) 10 E) 24

AP CALCULUS BC – BC Multiple Choice Worksheet #48.5

24. The Taylor series for cos x about x = 0 is 2 4 6

1 .2! 4! 6!

x x x! + ! +! If h is a function such that 3'( ) cosh x x= ,

then the coefficient of x7 in the Taylor series for h(x) about x = 0 is

*A) 1

14! B) 1

7!! C) 0 D) 1

7! E) 1

14

25. The closed interval [ ],w v is partitioned into k equal subintervals, each with width x! , by the numbers

0 1, ,...,

kx x x with w=

0 1 2 1... .

k kx x x x x v

!< < < < < = The

1

1lim

k

kj j

xx!"

=

#$ equals

A) v w! *B) ( )2 v w! C) 1 1

v w

! D) 1 12

v w

! "#$ %& '

E) ( )3/ 2 3/ 22

3v w!

26. ( )( )

6 8

3 2

xdx

x x

!=

! +" *A) 2ln 3 4ln 2x x C! + + + B) 2ln 3 2ln 2x x C! + + +

C) 2ln 3 4ln 2x x C+ + ! + D) 6 ln 3 8ln 2x x x C! + + + E) 4ln 3 2ln 2x x C! + + +

27. 2 cosx x! dx = A) 2sinx x C+ B)

2

2cos

2

xx C+ C) 2sin 2 cosx x x C! +

D) 2 sin 2cosx x x C! ! + *E) 2 sin 2cosx x x C+ + 28. Which of the following equations has the slope field shown?

A) 2

dyx

dx= B) 2

dyy

dx= * C) 2dy x

dx y= D) dy xy

dx= E) 2dy y

dx x=

AP CALCULUS BC – BC Multiple Choice Worksheet #48.5 CALCULATORS ALLOWED ON PROBLEMS 29 – 45. 29. Which of the follwing sequences converge?

I. 2

1

n

n

! "# $

+% & II. 3 7

8

n

n

+! "# $% &

III. 2

3

n

n n

e

e

! "# $

+% &

A) I only *B) II only C) I and II only D) II and III only E) None of them 30. When the region enclosed by the graphs of 2y x= and 2

6y x x= ! is revolved around the y-axis, the volume of the solid generated is given by

*A) ( )4

2 3

0

8 2x x! "# dx B) ( )16

2

0

2 4x x x! "# dx C) ( )4

2

0

2 4x x! "# dx

D) ( ) ( )4 2 22

0

6 2x x x! " #$ $% &' () dx E) ( )4

2

0

2 4x x x! "# dx

31. 0

1ln 1

limh

he

h!

" #+ +$ %

& ' is A) 1'( ) where ( )f e f x

x= B) 1

'( ) where ( ) lnf e f xx

! "= # $ %

& '

C) '(1) where ( ) lnx

f f xe

! "= # $

% & D) 1

' where ( ) ln( )f f x x ee

! "= +# $

% & * E) 1

' where ( ) lnf f x xe

! "=# $

% &

32. The position of an object oscillating on the x-axis is given by ( ) 2sin 4 cos 4x t t t= ! , where t is the time in

seconds. In the first 5 seconds, how many times is the velocity of the object equal to 0? A) 0 B) 4 C) 5 * D) 6 E) 7

33. Let h be the function defined by ( ) cos3 ln 4h x x x= + . What is the least value of x at which the graph of

h changes concavity? A) 1.555 * B) 0.621 C) 0.371 D) 0.096 E) 0.004 34. Let f be a continuous function on the closed interval [ 2,5]! . If ( 2) 3f ! = and (5) 7f = ! , then the

Intermediate Value Thorem guarantees that A) 7 ( ) 3f x! " " for all x between 2! and 5

B) 10'( )

7f c = ! for at least one c between 2! and 5

*C) ( ) 3f c = ! for at least one c between 2! and 5 D) ( ) 0f c = for at least one c between 7! and 3

E) ( )f x is continually decreasing between 2! and 5 35. Which of the following is the closest to the lowest value of x in the interval 0 6x! ! such that

2

0 2

( 3 )

x x

t t dt tdt! "# # ? A) 0 * B) 1.1075 C) 1.663 D) 1.745 E) 5.823

36. If 2cos cos

dyx y

dx= and

4y

!= when 0x = ,then *A) tan sin 1y x= + B) tan sin 1y x= ! +

C) 2sec sin 2y x= + D) ( )21

tan cos 12

y x= + E) 2tan sin

2y x= !

AP CALCULUS BC – BC Multiple Choice Worksheet #48.5

37. Using two terms of an appropriate MacLauren series, estimate 1

0

1 cos xdx

x

!" .

A) 196

*B) 2396

C) 14

D) 2596

E) undefined

38. Let f be a twice-differentiable function such that (2) 8f = and (4) 5f = . Which of the following must be

true for the function f on the interval 2 4x! ! ?

I. The average value of f is 132

II. The average rate of change of f is 3

2!

III. The average value of 'f is 3

2!

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

39. Find all values c that satisfy the Mean Value Theorem for the function 2

2( )

(1 )f x

x=

+on the interval

[ 1, 2]! . *A) 0.050 B) 0.050! C) 0.102 and 1.801 D) 0.050 and 2.449 E) None exist in the interval

40. The base of a solid is the region in the first quadrant enclosed by the graph of 2

9y x= ! and the coordinate axes. If every cross section of the solid perpendicular to the y-axis is a base of a rectangle and each height is three times the base, the volume of the solid is given by

A) 9

0

(9 )y dy!" B) 9

2 2

0

3 (9 )x dx!" * C) 9

0

3 (9 )y dy!" D) 3

2 2

0

3 (9 )x dx!" E) 9

0

3 (9 )y dy+!

41. Let 22

0( ) cos

x

f x tdt= ! . At how many points in the closed interval 0, !" #$ % does the instantaneous rate of

change of f equal the average rate of change of f on that interval? A) 1 B) 2 * C) 3 D) 4 E)5

42. If h is an antiderivative of 3

5( )

1

xg x

x=

+and h(1) = 2, then h(3) =

A) 4.407. * B) 2.555. C) 1.852. D) 0.555. E) -0.703. *43. A force of 12 pounds stretches a spring 3 inches beyond its natural length. Assuming Hooke’s Law

applies, how much work is done stretching the spring from its natural length to 6 inches beyond its natural length? A) 4 inch-pounds B) 8 inch-pounds C) 36 inch-pounds D) 54 inch-pounds * E) 72 inch-pounds

44. The length of the polar curve 1 2cos 2r != " is A) 6.925. B) 10.008. C) 13.365. D) 17.629. * E) 20.016.

45. If dy x ydx

= + and f(0) = 2, use Euler’s method with x! = 0.5 to approximate f(1).

A) 1.75 B) 3 C) 4 *D) 4.75 E) 5.75