BC Practice Test 2

Section I

Part A

TIME: 60 MINUTES

The use of calculators is not permitted for this part of the examination.
There are 30 questions in Part A, for which 60 minutes are allowed. Because there is no deduction for wrong answers, you should answer every question, even if you need to guess.

DIRECTIONS: Choose the best answer for each question.

1. A function f(x) equals  for all x except x = 1. If f(1) = k, for what value of k would the function be continuous at x = 1?

(A) 0
(B) 1
(C) 2
(D) No such k exists.

2. is

(A) 2
(B) 0
(C) 
(D) nonexistent

3. The first four terms of the Taylor series about x = 0 of are

4. Using the line tangent to  at x = 0, an estimate of f(0.06) is

(A) 0.02
(B) 2.98
(C) 3.01
(d) 3.02

5. What is the radius of convergence for the Maclaurin series for ?

(A) 
(B) 1
(C) 5
(D) 8

6. The motion of a particle in a plane is given by the pair of equations x = cos 2t, y = sin 2t. The magnitude of its acceleration at any time t equals

(A) 1
(B) 2
(C) 4
(D) 16

7. Let 

The interval of convergence of f’(x) is

(A) 0 ≤ x ≤ 2
(B) 0 ≤ x < 2
(C) 0 < x ≤ 2
(D) 0 < x < 2

8. A point moves along the curve y = x² + 1 so that the x-coordinate is increasing at the constant rate of  units per second. The rate, in units per second, at which the distance from the origin is changing when the point has coordinates (1,2) is equal to

9. 

(A) = 0
(B) 
(C) = 1
(D) does not exist

10. 

11. 

(A) e – 1
(B) e + 1
(C) 1
(D) –1

12. Consider the series . To what value does the series converge when ?

13. 

(A) ln |x²(x – 3)| + C
(B) –ln |x²(x – 3)| + C
(C) 
(D) 

14. Given f′ as graphed, which could be a graph of f?

 

I

II

III

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

15. The first woman officially timed in a marathon was Violet Piercy of Great Britain in 1926. Her record of 3:40:22 stood until 1963, mostly because of a lack of women competitors. Soon after, times began dropping rapidly, but lately they have been declining at a much slower rate. Let M(t) be the curve that best represents winning marathon times in year t. Which of the following is (are) positive for t > 1963?

I. M(t)
II. M′(t)
III. M″(t)

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

16. The graph of f is shown above. Let  and . Which of the following is true?

(A) G(x) = H(x)
(B) G ′(x) = H ′(x + 2)
(C) G(x) = H(x + 2)
(D) G(x) = H(x) + 3

17. Consider the series . Which of the following is true?

(A) The series converges by the nth Term Test.
(B) The series diverges by the nth Term Test.
(C) The series converges by the Limit Comparison Test with .
(D) The series diverges by the Limit Comparison Test with .

18. Which function could be a particular solution of the differential equation whose slope field is shown above?

19. A particular solution of the differential equation  passes through the point (2,1). Using Euler’s method with Δx = 0.1, estimate its y-value at x = 2.2.
(A) 1.30
(B) 1.34
(C) 1.60
(D) 1.64

Questions 20 and 21. Use the graph below, consisting of two line segments and a quarter-circle. The graph shows the velocity of an object during a 6-second interval.

20. For how many values of t in the interval 0 < t < 6 is the acceleration undefined?
(A) none
(B) one
(C) two
(D) three

21. During what time interval (in seconds) is the speed increasing?

(A) 0 < t < 3
(B) 3 < t < 5
(C) 5 < t < 6
(D) never

22. If  and y = 3 when x = 1, then
(A) 
(B) y = x + 2
(C) 
(D) y = 3x

23. A solid is cut out of a sphere of radius 2 by two parallel planes each 1 unit from the center. The volume of this solid is

24. Which one of the following improper integrals converges?

25. The function f(x) = x⁵ + 3x – 2 passes through the point (1,2). Let f^-1 denote the inverse of f. Then (f^–1)′(2) equals

26. Find the domain of the particular solution of  that passes through the origin.

27. Which of the following statements is (are) true about the graph of y = ln (4 + x²)?

I. It is symmetric to the y-axis.
II. It has a local minimum at x = 0.
III. It has inflection points at x = ±2.

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

28. 

29. Choose the integral that is the limit of the Riemann Sum:

30. Which infinite series converge(s)?

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

 

Part B

TIME: 45 MINUTES

Some questions in this part of the examination require the use of a graphing calculator. There are 15 questions in Part B, for which 45 minutes are allowed. Because there is no deduction for wrong answers, you should answer every question, even if you need to guess.

DIRECTIONS: Choose the best answer for each question. If the exact numerical value of the correct answer is not listed as a choice, select the choice that is closest to the exact numerical answer.

31. Find the area bounded by the spiral r = ln θ on the interval π ≤ θ ≤ 2π.

(A) 2.405
(B) 3.743
(C) 4.810
(D) 7.487

32. Write an equation for the line tangent to the curve defined by F(t) = 〈t² + 1,2^t〉 at the point where y = 4.

(A) y – 4 = (ln 2)(x – 2)
(B) y – 4 = (4 ln 2)(x – 2)
(C) y – 4 = (ln 2)(x – 5)
(D) y – 4 = (4 ln 2)(x – 5)

33. Bacteria in a culture increase at a rate proportional to the number present. An initial population of 200 triples in 10 hours. If this pattern of increase continues unabated, then the approximate number of bacteria after 1 full day is

(A) 1056
(B) 1440
(C) 2793
(D) 3240

34. When the substitution x = 2t – 1 is used, the definite integral  may be expressed in the form , where {k,a,b} =

35. The curve defined by x³ + xy – y² = 10 has a vertical tangent line when x =

(A) 1.037
(B) 1.087
(C) 2.074
(D) 2.096

Questions 36 and 37. Use the graph of f shown on [0,7]. Let 

36. G’(1) is

(A) 1
(B) 2
(C) 3
(D) 6

37. G has a local maximum at x =

(A) 1
(B) 
(C) 2
(D) 8

38. You are given two thrice-differentiable functions, f(x) and g(x). The table above gives values for f(x) and g(x) and their first and second derivatives at x = 2. Find .

(A) 0
(B) 1
(C) 6
(D) nonexistent

39. Using the left rectangular method and four subintervals of equal width, estimate  where f is the function graphed below.

(A) 4
(B) 8
(C) 12
(D) 16

40. Given the function f(x) = 2 cos(3x – 1) – 0.5e^0.5x, find all values of x that satisfy the result of the Mean Value Theorem for the function f on the interval [–3,–1]. NOTE: The derivative of f is f′(x) = –6 sin(3x – 1) – 0.25e^0.5x.

 

(A) –2.800 and –1.772
(B) –2.242 and –1.296
(C) –2.812 and –1.760
(D) –2.843 and –1.729

41. The base of a solid is the region bounded by x² = 4y and the line y = 2, and each plane section perpendicular to the y-axis is a square. The volume of the solid is

(A) 8
(B) 16
(C) 32
(D) 64

42. An object initially at rest at (3,3) moves with acceleration a(t) = 〈2,e^–t〉. Where is the object at t = 2?

(A) (4,–0.865)
(B) (4,1.135)
(C) (7,2.135)
(D) (7,4.135)

43. Find the length of the curve. y = ln x between the points where  and. y = 1.
(A) 0.531
(B) 0.858
(C) 1.182
(D) 1.356

44. Using the first two terms in the Maclaurin series for. y = cos. x yields accuracy to within 0.001 over the interval |x| <. k when. k =

 

(A) 0.394
(B) 0.707
(C) 0.786
(D) 0.788

45. Consider the power series . It is known that at x = 4, the series converges conditionally. Of the following, which is true about the convergence of the power series at x = 1?

(A) There is not enough information.
(B) At x = 1, the series diverges.
(C). At x = 1, the series converges conditionally.
(D) At x = 1, the series converges absolutely.

 

Section II

Part A

TIME: 30 MINUTES
2 PROBLEMS

A graphing calculator is required for some of these problems. See instructions on page 8.

1. Let function f be continuous and decreasing, with values as shown in the table:

(a) Use a trapezoidal sum to estimate the area between f and the x-axis on the interval 2.5 ≤ x ≤ 5.0.
(b) Find the average rate of change of f on the interval 2.5 ≤ x ≤ 5.0.
(c) Estimate the instantaneous rate of change of f at x = 2.5.
(d) If g(x) = f^–1(x), estimate the slope of g at x = 4.

 

2. An object starts at point (1,3) and moves along the parabola y = x² + 2 for 0 ≤ t ≤ 2, with the horizontal component of its velocity given by .

(a) Find the object’s position at t = 2.
(b) Find the object’s speed at t = 2.
(c) Find the distance the object traveled during this interval.

 

Part B

TIME: 60 MINUTES
4 PROBLEMS

No calculator is allowed for any of these problems.

If you finish Part B before time has expired, you may return to work on Part A, but you may not use a calculator.

3. Given a function f such that f(3) = 1 and .

(a) Write the first four nonzero terms and the general term of the Taylor series for f around x = 3.
(b) Find the radius of convergence of the Taylor series.
(c) Show that the third-degree Taylor polynomial approximates f(4) to within 0.01.

 

4. The curve  divides a first-quadrant rectangle into regions A and B, as shown in the figure above.

(a) Region A is the base of a solid. Cross sections of this solid perpendicular to the x-axis are rectangles. The height of each rectangle is 5 times the length of its base in region A. Find the volume of this solid.
(b) The other region, B, is rotated around the y-axis to form a different solid. Set up, but do not evaluate, an integral for the volume of this solid.

 

5. A bungee jumper has reached a point in her exciting plunge where the taut cord is 100 feet long with a 1/2-inch radius and is stretching. She is still 80 feet above the ground and is now falling at 40 feet per second. You are observing her jump from a spot on the ground 60 feet from the potential point of impact, as shown in the diagram below.

(a) Assuming the cord to be a cylinder with volume remaining constant as the cord stretches, at what rate is its radius changing when the radius is 1/2 inch?
(b) From your observation point, at what rate is the angle of elevation to the jumper changing when the radius is 1/2 inch?

 

6. The figure above shows the graph of f, whose domain is the closed interval [–2,6]. Let .

(a) Find F (–2) and F (6).
(b) For what value(s) of x does f(x) = 0?
(c) On what interval(s) is F increasing?
(d) Find the maximum value and the minimum value of F.
(e) At what value(s) of x does the graph of F have points of inflection? Justify your answer.