BC Exam 1 - Part I – 28 questions – No Calculator Allowed - Solutions€¦ · x+1 x−1 2 +C C. x2 2 −2lnx2−1+C D. x2+(lnx−1+lnx+1) 2 +C E. ln x+3 x+1 5 +C 10. The region

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

1. Find

�

limx→0

6x5 − 8x3

9x3 − 6x5

A.

�

23

B.

�

−89

C.

�

43

D.

�

−83

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( ) x 2 − 3( )4, then ′ f x( ) =

A.

�

2 x 2 − 3( )3 x 2 + 4x −1( ) B.

�

4 2x +1( ) x 2 − 3( )3 C.

�

8x 2x +1( ) x 2 − 3( )3 D.

�

2 x 2 − 3( )3 3x 2 + x − 3( ) E.

�

2 x 2 − 3( )3 9x 2 + 4x − 3( )


4.

�

x 4 − x( )2∫ dx =

A.

�

x 9

9+ x 3

3+ C B.

�

x 9

9− x

6

6+ x 3

3+ C C.

�

x 9

9− x

6

3+ x 3

3+ C

D.

�

x 4 − x 2( )33

+ C E.

�

x 4 − x 2( )33 4x 3 − 2x( ) + C

5. Find

�

limx→0

ex + cos x − x − 2x 4 − x 3

A.

�

− 130

B.

�

− 124

C.

�

− 16

D. 0 E. nonexistent

6. For what values of p does the infinite series

�

nn2p−1 + 2n=1

∞

∑ converge?

A.

�

p > 0 B.

�

p >1 C.

�

p ≥1 D.

�

p > 12

E.

�

p > 32


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.

�

x 3

x 2 −1dx =∫

A.

�

x 2

2+ 2ln x 2 −1 + C B.

�

x 2 + ln x +1x −1

2+ C C.

�

x 2

2− 2ln x 2 −1 + C

D.

�

x2 + ln x −1 + ln x + 1( )2

+ C E.

�

ln x + 3x +1

5

+ C

10. The region bounded by the graph of

�

y = 4x

, the line x = 4 and the x - axis is rotated about the x-axis.

Find the volume of the solid. A. π B. 2π C. 4π D. 16π E. infinite

11. If the length of a curve from

�

x = 2 to x = 8 is given by

�

1+ 81x42

8

∫ dx , and the curve passes through the

point (-1, 4), which of the following could be the equation for the curve? A.

�

y =13− 9x 2 B.

�

y = 4 − 3x 3 C.

�

y = 7 + 3x 3 D.

�

y = −1− 3x 3 E.

�

y = 9x 2 − 5


12. Let

�

f x( ) be the power series for

�

sin x , centered at x = 0. Which of the following is a power series?

I.

�

f x 2( ) . II.

�

f x( ) III.

�

f ex( )

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

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.

�

52

D. 15 E.

�

49

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. Let f be a function having derivatives for all orders of real numbers. The first three derivatives of

�

f at x = 0 are given in the table below. Use the third-degree Taylor polynomial at x = 0 to approximate

�

f 12

⎛ ⎝ ⎜

⎞ ⎠ ⎟ .

�

x f x( ) ′ f x( ) ′ ′ f x( ) ′ ′ ′ f x( )0 5 2 −8 24

A.

�

12

B.

�

112

C. 7 D.

�

173

E.

�

232

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 ft 3 hr B.

�

6s ft 3 hr C.

�

4s ft 3 hr D.

�

2s ft 3 hr E.

�

s ft 3 hr

17. The position of an object moving in the xy-plane with position function

�

r t( ) = 1+ sin t,t + cos t , t ≥ 0. What is the maximum speed attained by the object? A. 1 B.

�

2 C. 2 D. 4 E.

�

2 2


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. A power series is used to approximate

�

e−x3dx

0

1

∫ with a maximum error of 0.01. What is the minimum

number of terms needed to obtain this approximation? A. 2 B. 3 C. 4 D. 5 E. 6

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( ) dx0

8

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

�

x 0 3 5 8f x( ) 5 k 2 7 10

A. 4 B.

�

±4 C. 8 D. -8 E. No values of k


21. The Maclaurin series for a certain function f converges to

�

f x( ) for all x in the interval of convergence. The nth derivative of f at x = -1 is given by

�

f n( ) −1( ) =−1( )n+1 n + 1( )!

1− 2n( )2 for n ≥ 2

If the graph of f has a horizontal tangent at (-1, -4), describe the behavior of the graph of f at x = -1.

A. relative maximum B. relative minimum C. cusp point D. inflection point E. none of these

22. The line

�

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

�

y = 2x 3 − 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 shaded region between the graph of

�

y = 2tan−1 x and the x-axis for 0 ≤ x ≤ 1 as shown in the figure is the base of a solid whose cross-sections

perpendicular to the x-axis are squares. Find the volume of the solid.

A.

�

π4

+ ln2 −1 B.

�

π + e − ln2

C.

�

π − ln2 D.

�

π4− ln2 E.

�

π2− ln2


24. The average value of

�

sin2 x cos x on the interval

�

π2, 3π2

⎡ ⎣ ⎢

⎤ ⎦ ⎥ is

A.

�

−23π

B.

�

23π

C. 0 D. -1 E. 1

25. The functions f and g are differentiable and

�

f g x( )( ) = x 2 for all x. If

�

f 4( ) = 8, g 4( ) = 8, ′ f 8( ) = −2, what is the value of ′ g 4( )?

A.

�

−18

B.

�

−12

C.

�

−2 D.

�

−4 E. Insufficient data

26. 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


27. The graph of

�

f x( ) = x 2 + 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. The hyperbolic sine function is defined as

�

sinh x = 12ex − e−x( ). Give the general term for the Maclaurin

series for

�

sinh x .

A.

�

x 2n+1

2n +1( )! B.

�

−1( )n x 2n+1

2n +1( )! C.

�

x 2n

2n( )! D.

�

−1( )n+1x 2n

2n( )! E.

�

−1( )n x 2n2n( )!


BC Exam 1 - Part II – 17 questions – Calculators Allowed - Solutions

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

A.

�

dydx

= x + y 2 B.

�

dydx

= x − y 2 C.

�

dydx

= xy

D.

�

dydx

= x + y E.

�

dydx

= x 2 − y

30. Consider the differential equation

�

dydx

= cos x with initial condition

�

f 0( ) = 0. Find the difference

between the exact value of

�

f π2

⎛ ⎝ ⎜

⎞ ⎠ ⎟ and an Euler approximation of

�

f π2

⎛ ⎝ ⎜

⎞ ⎠ ⎟ using two equal steps.

A. 0 B. 0.230 C. 0.341 D. 0.555 E. 0.707


31. The function f is defined by the power series

�

f x( ) =1+ x − 7( ) + x − 7( )2 + ...+ x − 7( )n + ... for all real numbers x for which the series converges. What is the range of

�

f x( ) within the interval of convergence?

A.

�

12,∞

⎡ ⎣ ⎢

⎞ ⎠ ⎟ B.

�

12,∞

⎛ ⎝ ⎜

⎞ ⎠ ⎟ C.

�

0,∞[ ) D.

�

0,∞( ) E.

�

−∞,∞( )

32. An object moving along a curve in the xy-plane has position

�

x t( ),y t( )( ) at time t with

�

dxdt

= t2 + 3t + 1 and dydt

= et2 −1 for t ≥ 0.

At time t = 0, the object is at position (-6, -7). Find the position of the object at t = 2.

A. (4.667, 13.053) B. (-3.683, 12.718) C. (2.317, 19.718) D. (4.427, 6.053) E. (-1.573, -0.947)


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

�

8x 2 + 2y 2 = 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. The Maclaurin series for a certain function f converges to

�

f x( ) for all x in the interval of convergence.

The nth derivative of f at x = 0 is given by

�

f n( ) 0( ) =n + 1( )!

0.5( )n n3 for n ≥ 0. Find the radius of

convergence for f, if it exists, about x = 0.

A.

�

12

B. 1 C. 2 D. All reals E. does not converge

35. Let

�

F x( ) be an antiderivative of

�

x 3 + 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. The graph of the polar curve

�

r = 2 − 4sinθ is a limaçon with two loops as shown in the figure to the right. Find the area between the two loops.

A. 25.688 B. 35.187 C. 35.525 D. 37.361 E. 37.699

38. Find the Lagrange error in calculating

�

f −0.1( ) for the third degree Taylor polynomial for

�

f x( ) = xex about x = 0.

A.

�

124

B.

�

−0.1( )36

C.

�

−0.1( )324

D.

�

−0.1( )46

E.

�

−0.1( )424


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. A particle moving along the polar curve

�

r =1− sinθ has position

�

x t( ),y t( )( ) at time with

�

θ = 0 when t = 0. The particle moves along the curve such that

�

drdt

= drdθ

. Describe the motion of the

particle at

�

t = π6.

I. Getting closer to the x-axis II. Getting closer to the y-axis III. Getting closer to the origin A. I only B. II only C. I and II only D. II and III only E. I, II and III

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

�

175

ln 7675

+ ln 7775

+ ln 7875

+ ...+ ln2⎛ ⎝ ⎜

⎞ ⎠ ⎟ is a Riemann sum approximation for

A.

�

ln x75

⎛ ⎝

⎞ ⎠ dx

1

2

∫ B.

�

ln x75

⎛ ⎝

⎞ ⎠ dx

76

150

∫ C.

�

175

ln x dx76

150

∫ D.

�

ln x dx1

2

∫ E.

�

175

ln x dx1

2

∫

43. Let f be the function given by

�

f x( ) = cos t2 + t( ) dt0

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. A curve C is defined by the parametric equations

�

x = t 2 − t − 4 and y = t 3 − 7t − 2 . Which of the following is the equation of the line tangent to the graph of C at the point (2, 4) ?

A.

�

y = 6 − x B.

�

x − 4y +14 = 0 C.

�

5x − 3y + 2 = 0 D.

�

y = 4x − 4 E. No tangent line at (2, 4)


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

�

w t( ) = 100 t 3 − 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( ) dt1.480

3.773

∫ B.

�

w t( ) dt1.480

3.773

∫ C.

�

800 − ′ w t( ) dt1.480

3.773

∫

D.

�

′ w t( ) dt0

2.786

∫ E.

�

w t( ) dt0

2.786

∫

Documents

BC Exam 1 - Part I – 28 questions – No Calculator Allowed - Solutions€¦ · x+1 x−1 2 +C C. x2 2 −2lnx2−1+C D. x2+(lnx−1+lnx+1) 2 +C E. ln x+3 x+1 5 +C 10. The region