Exam 1 - Part I – 28 questions – No Calculator Allowedtheriddles-brhs.weebly.com/.../5033946/abmcstudentexam1.pdfExam 1 - Part I – 28 questions – No Calculator Allowed 1. Find

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Exam 1 - Part I – 28 questions – No Calculator Allowed

1. Find

limx!0

6x5" 8x

3

9x3" 6x

5

A.

2

3 B.

!8

9 C.

4

3 D.

!8

3 E. nonexistent

2. Let f be a function such that

limx!4

f x( ) " f 4( )

x " 4= 2 . Which of the following must be true?

I. f is continuous at x = 4. II. f is differentiable at x = 4. III. The derivative of

! f is continuous at x = 4.

A. I only B. II only C. I and II only D. I and III only E. I, II and III

3. If

f x( ) = 2x + 1( ) x2! 3( )

4

, then " f x( ) = A.

2 x2! 3( )

3

x2

+ 4x !1( ) B.

4 2x +1( ) x 2 ! 3( )3

C.

8x 2x +1( ) x 2 ! 3( )3

D.

2 x2! 3( )

3

3x2

+ x ! 3( ) E.

2 x2! 3( )

3

9x2

+ 4x ! 3( )


4.

x4 ! x( )

2

" dx =

A.

x9

9+x3

3+ C B.

x9

9!x6

6+x3

3+ C C.

x9

9!x6

3+x3

3+ C

D.

x4! x

2( )3

3+ C E.

x4! x

2( )3

3 4x3! 2x( )

+ C

5.

d

dxtan t

4 !1( ) dt

1

x3

" =

A.

sec2x12!1( ) B.

tan x4!1( ) C.

tan x12!1( )

D.

3x2tan x

12!1( ) E.

12x11tan x

12!1( )

6. Newton the cat begins to walk along a ledge at time t = 0. His velocity at time t, 0 ≤ t ≤ 8, is given by the function whose graph is given in the figure to the right. What is Newton’s average speed from t = 0 to t = 8?

A. 0 B. 2 C. 3

D.

9

4 E. 5


7. The table below gives selected values of

v t( ) , of a particle moving along the x-axis. At time t = 0, the particle is at the origin. Which of the following could be the graph of the position,

x t( ) , of the particle for 0 ≤ t ≤ 4 ?

t 0 1 2 3 4

v t( ) 3 0 -1 1 3

A. B. C.

D. E.

8. The graph of a twice-differentiable function f is shown in the figure

to the right. Which of the following is true? A.

f !2( ) < " f !2( ) < " " f !2( ) B.

f !2( ) < " " f !2( ) < " f !2( ) C.

! ! f "2( ) < ! f "2( ) < f "2( ) D.

! f "2( ) < f "2( ) < ! ! f "2( ) E.

! ! f "2( ) < f "2( ) < ! f "2( )


9. What is the slope of the line tangent to the graph of

y =1

e2xx ! 2( )

at x = 1 ?

A.

!2

e B.

!1

e2

C. 0 D.

1

e2

E.

!2

e2

10. What are all the horizontal asymptotes of

f x( ) =6 + 3e

x

3! ex

in the xy-plane?

A.

y = 3 only B.

y = !3 only C.

y = 2 only D.

y = !3 and y = 0 E.

y = !3 and y = 2

11. If f is continuous for all real numbers x and

f x( )1

4

! dx = 10, then f x " 2( ) + 2x[ ]3

6

! dx =

A. 37 B. 39 C. 35 D. 57 E. 25

12. If

f x( ) = x3

+ x2

+ x + 1 and g x( ) = f!1x( ), what is the value of

! g 4( ) ?

A.

1

85 B. 1 C.

1

57 D.

1

6 E. 24


13. Find the y-intercept of the tangent line to

4 x + 2 y = x + y + 3 at the point (9, 4).

A. -2 B. 10 C.

5

2 D. 15 E.

4

9

14. The function f is continuous and non-linear for

!3 " x " 7 and

f !3( ) = 5 and f 7( ) = !5. If there is no value c, where

!3 < c < 7 , for which

! f c( ) = "1, which of the following statements must be true?

A.

For some k, where ! 3 < k < 7, " f k( ) < !1. B.

For some k, where ! 3 < k < 7, " f k( ) > !1. C.

For some k, where ! 3 < k < 7, " f k( ) = 0. D.

For ! 3 < k < 7, " f k( ) exists. E.

For some k, where ! 3 < k < 7, " f k( ) does not exist.

15. If

x2

+ 2y2

= 34 , find the behavior of the curve at (-4, 3). A. Increasing, concave up B. Increasing, concave down C. Decreasing, concave up D. Decreasing, concave down E. Decreasing, inflection point


16. A large block of ice in the shape of a cube is melting. All sides of the cube melt at the same rate. At the time that the block is s feet on each side, its surface area is decreasing at the rate of

24 ft2

hr . At what rate is the volume of the block decreasing at that time?

A.

12s ft3

hr B.

6s ft3

hr C.

4s ft3

hr D.

2s ft3

hr E.

s ft3

hr

17. A squirrel climbs a telephone pole and starts to walk along the

telephone wire. Its velocity v of the squirrel at time t, 0 ≤ t ≤ 6 is given by the function whose graph is to the right. At what value of t does the squirrel change direction?

A. 1 and 5 only B. 2 only C. 2 and 4 only D. 1, 3, and 5 only E. 1, 3, 4 and 5 only

18. The graph of

! f x( ), the derivative of f , is shown to the right. Which of the following statements is not true?

A. f is increasing on 2 ≤ x ≤ 3. B. f has a local minimum at x = 1. C. f has a local maximum at x = 0. D. f is differentiable at x = 3. E. f is concave down on -2 ≤ x ≤ 1.


19. The difference in maximum acceleration and minimum acceleration attained on the interval 0 ≤ t ≤ 3 by the particle whose velocity is given by

v t( ) = 2t3!12t

2+18t !1 is

A. 4 B. 6 C. 18 D. 21 E. 24

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

The trapezoidal approximation for

f x( ) dx

0

8

! found with 3 subintervals is 20k. What is the value of k?

x 0 3 5 8

f x( ) 5 k27 10

A. 4 B.

±4 C. 8 D. -8 E. No values of k

21.

sin2x

1! sin22x

" dx =

A.

2ln cos2x( ) + C B.

1

2ln cos2x( ) + C C.

1

2sec2x tan2x + C

D.

1

2sec2x + C E.

2tan2x + C

22. The line

x + y = k , where k is a constant, is tangent to the graph of

y = 2x3! 9x

2! x +1. What are the

only possible values of k?

A. 1 only B. 0 and - 29 C. 1 and -29 D. 0 and 3 E. 1 and -26


23. The graph of

y = ! f x( ) , the derivative of f , is shown in the figure to the right. If

f 0( ) = !1, then f 1( ) =

A. 1 B. 2 C. 3 D. 4 E. 5

24. The average value of

sin2x cos x on the interval

!

2,3!

2

"

# $

%

& ' is

A.

!2

3" B.

2

3! C. 0 D. -1 E. 1

25. The functions f and g are differentiable and

f g x( )( ) = x2 for all x. If

f 4( ) = 8, g 4( ) = 8, ! f 8( ) = "2, what is the value of ! g 4( )?

A.

!1

8 B.

!1

2 C.

!2 D.

!4 E. Insufficient data


26. Let f be a twice-differentiable function such that

f a( ) = b and f b( ) = a for two unknown constants a and b, a < b. Let

g x( ) = f f x( )( ) . The Mean Value Theorem applied to

! g on [a, b] guarantees a value k such that a < k < b such that

A.

! g k( ) = 0 B.

! ! g k( ) = 0 C.

! g k( ) =1 D.

! ! g k( ) = 1 E.

! g k( ) = b " a

27. The graph of

f x( ) = x2

+ 0.0001 ! 0.01 is shown in the graph to the right. Which of the following statements are true?

I. limx!0

f x( ) = 0.

II. f is continuous at x = 0.

III. f is differentiable at x = 0.

A. I only B. II only C. I and II only D. I, II, and III E. None are true

28. What is the area of the region enclosed by the graphs of

y = x ! 4x2 and y = !7x ?

A.

4

3 B.

16

3 C. 8 D.

68

3 E.

80

3


Exam 1 - Part II – 17 questions – Calculators Allowed

29. The slope field for the equation in the figure to the right could be

A.

dy

dx= x + y

2 B.

dy

dx= x ! y

2 C.

dy

dx= xy

D.

dy

dx= x + y E.

dy

dx= x

2! y

30. Let R be the region between the graphs of

y = 2sin x and y = cos x as shown in the figure to the right. The region R is the base of a solid with cross sections perpendicular to the x-axis as rectangles that are twice as high as wide. Find the volume of the solid.

A. 4.472 B. 7.854 C. 8.944 D. 15.708 E. 31.416

31. The first derivative of a function f is given by

! f x( ) = x " 2( )esin 2x . How many points of inflection does the graph of f have on the interval 0 < x < 2π?

A. Two B. Three C. Four D. Five E. Six


32. A conical tank is leaking water at the rate of

75in3min. At the same

time, water is being pumped into the tank at a constant rate. The tank’s height is 60 in while its top diameter is 20 inches. If the water level is rising at the rate of

5in min when the height of the water is 10 inches high, find the rate in which water is being pumped into the tank to the

nearest

in3min . (The volume of a cone is given by

V =1

3!r

2h )

A.

44 in3

min B.

175 in3

min C.

250 in3

min D.

119 in3

min E.

31 in3

min

33. At which points is the tangent line to the curve

8x2

+ 2y2

= 6xy +14 vertical? I. (-2, -3) II (3, 8) III. (4, 6) A. I only B. II only C. III only D. I and II only E. I and III only

34. For the function f ,

! f x( ) = 4 x " 3 and f 2( ) = 4. What is the approximation for

f 2.1( ) found by using the tangent line to the graph of f at x = 2.

A.

!2.6 B. 4.5 C. 4.8 D. 5.4 E. 9.4


35. Let

F x( ) be an antiderivative of

x3

+ x +1 . If

F 1( ) = !2.125, then F 4( ) =

A. -15.879 B. -11.629 C. 7.274 D. 15.879 E. 11.629

36. What is the area of the region in the first quadrant enclosed by the graph of

y = 2cos x,y = x, and the x-axis?

A. 0.816 B. 1.184 C. 1.529 D. 1.794 E. 1.999

37. A particle moves along the x-axis so that at any time t > 0, its acceleration is given by

a t( ) = cos 1! 2t( ) . If the velocity of the particle is -2 at time t = 0, then the speed of the particle at

t = 2 is

A. 0.613 B. 0.669 C. 1.331 D. 1.387 E. 1.796

38. A right triangle has legs a and b and hypotenuse c. The lengths of leg a and leg b are changing but at a certain instant, the area of the triangle is not changing. Which statement must be true?

A. a = b B.

da

dt=db

dt C.

da

dt= !

db

dt D.

ada

dt= !b

db

dt E.

adb

dt= !b

da

dt


39. A particle moves along a straight line with velocity given by

v t( ) = t ! 2.5cos2t . What is the acceleration

of the particle at t = 2 ?

A. 0.168 B. 0.238 C. 0.451 D. 0.584 E. 1.450

40. If the graph of

! f x( ) is shown to the right, which of the following could be the graph of

y = f x( )?

I. II. III. A. I only B. II only C. III only D. I and III only E. II and III only

41. The rate at which the gasoline is changing in the tank of a hybrid car is modeled by

f t( ) = t + .5sin t ! 2.5 gallons per hour, t hours after a 6-hour trip starts. At what time during the 6-hour trip was the gasoline in the tank going down most rapidly?

A. 0 B. 2.292 C. 3.228 D. 4.203 E. 6


42. The expression

1

75ln76

75+ ln

77

75+ ln

78

75+ ...+ ln2

!

" #

$

% & is a Riemann sum approximation for

A.

lnx

75

!

"

#

$ dx

1

2

% B.

lnx

75

!

"

#

$ dx

76

150

% C.

1

75ln x dx

76

150

! D.

ln x dx

1

2

! E.

1

75ln x dx

1

2

!

43. Let f be the function given by

f x( ) = cos t2

+ t( ) dt

0

x

! for " 2 # x # 2. Approximately, for what

percentage of values of x

for ! 2 " x " 2 is

f x( ) decreasing?

A. 30% B. 26% C. 44% D. 50% E. 59%


44. The rate of change of people waiting in line to buy tickets to a concert is given by

w t( ) = 100 t3! 4t

2! t + 7( ) for 0 " t " 4. 800 people are already waiting in line when the box office

opens at t = 0. Which of the following expressions give the change in people waiting in line when the line is getting shorter?

A.

! w t( ) dt

1.480

3.773

" B.

w t( ) dt

1.480

3.773

! C.

800 ! " w t( ) dt

1.480

3.773

#

D.

! w t( ) dt

0

2.786

" E.

w t( ) dt

0

2.786

!

45. A particle moves along the x-axis so that its velocity

v t( ) =12te!2t

! t +1. At t = 0, the particle is at position

x = 0.5. What is the total distance that the particle traveled from t = 0 to t = 3 ?

A. 1.448 B. 1.948 C. 2.911 D. 4.181 E. 4.681

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Exam 1 - Part I – 28 questions – No Calculator Allowedtheriddles-brhs.weebly.com/.../5033946/abmcstudentexam1.pdfExam 1 - Part I – 28 questions – No Calculator Allowed 1. Find