BC Practice Test 3

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 cylindrical tank, shown in the figure above, is partially full of water at time t = 0, when more water begins flowing in at a constant rate. The tank becomes half full when t = 4 and is completely full when t = 12. Let h represent the height of the water at time t. During which interval is  increasing?

(A) 0 < t < 4
(B) 0 < t < 8
(C) 0 < t < 12
(D) 4 < t < 12

2. is

(A) 1
(B) –1
(C). 0
(D) nonexistent

3. 

(A) = 1
(B) = 3
(C). = 4
(D) diverges

4. An equation of the tangent to the curve 2x² – y⁴ = 1 at the point (–1,1) is

(A) x + y = 0
(B) x + y = 2
(C) x – y = 0
(D) x – y = –2

5. The nth term of the Taylor series expansion about x = 0 of the function  is

(A) (2x)ⁿ
(B) 2xⁿ
(C) (–1)^{n – 1}(2x)^{n – 1}
(D) (–1)ⁿ (2x)^{n – 1}

6. 

7. Suppose the function f is defined as

Which of the following is true?

(A) The function f is both continuous and differentiable at x = 2.
(B) The function f is continuous but not differentiable at x = 2.
(C) The function f is differentiable but not continuous at x = 2.
(D) The function f is neither continuous nor differentiable at x = 2.

8. Which of the following series converge?

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

9. A particular solution of the differential equation whose slope field is shown above contains point P. This solution may also contain which other point?

(A) A
(B) B
(C) C
(D) D

10. 

11. As the tides change, the water level in a bay varies sinusoidally. At high tide today at 8 A.M., the water level was 15 feet; at low tide, 6 hours later at 2 P.M., it was 3 feet. How fast, in feet per hour, was the water level dropping at noon today?

12. Let . Then f(3) =

(A) –3π
(B) –1
(C) 1
(D) 3π

13. is equal to

14. 

15. Consider the polar curve r = 2 cos(3θ). What is the slope of the line tangent to the curve when ?

16. The area bounded by the parabola y = x² and the lines y = 1 and y = 9 equals

17. The first-quadrant region bounded by , y = 0, x = q, where 0 < q < 1, and x = 1 is rotated about the x-axis. The volume obtained as q → 0+

18. A curve is given parametrically by the equations x = 3 – 2 sin t and y = 2 cos t – 1. The length of the curve from t = 0 to t = π is

(A)
(B) π
(C) 2π
(D) 4π

19. Suppose the graph of f is both increasing and concave up on a ≤ x ≤ b. If  is approximated using various sums with the same number of subintervals, and if L, R, M, and T denote, respectively, left Riemann Sum, right Riemann Sum, midpoint Riemann Sum, and trapezoidal sum, then it follows that

(A) R ≤ T ≤ M ≤ L
(B) L ≤ T ≤ M ≤ R
(C) R ≤ M ≤ T ≤ L
(D) L ≤ M ≤ T ≤ R

20. Which of the following statements about the graph of  is (are) true?

I. The graph has no horizontal asymptote.
II. The line x = 2 is a vertical asymptote.
III. The line y = x + 2 is an oblique asymptote.

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

21. The only function that does not satisfy the Mean Value Theorem on the interval specified is

22. 

23. The velocity vector of a particle moving in the plane is given by 〈e^{4t + 2},sin(9t + 1)〉 for t ≥ 0. What is the acceleration vector of the particle?
(A)〈e^{4t + 2},9 cos(9t + 1)〉
(B) 
(C)〈4e^{4t + 2},9 cos(9t + 1)〉
(D)〈4e^{4t + 2},–9 cos(9t + 1)

24. The graph of function f shown above consists of three quarter-circles. Which of the following is (are) equivalent to 

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

25. The base of a solid is the first-quadrant region bounded by , and each cross section perpendicular to the x-axis is a semicircle with a diameter in the xy-plane. The volume of the solid is

26. To determine whether the series  converges or diverges, we will use the Limit Comparison Test. Which of the following series should we use?

 

27. The area inside the circle r = 3 sin θ and outside the cardioid r = 1 + sin θ, shaded in the figure above, is given by

28. Let 

Which of the following statements is (are) true?

I. f is defined at x = 6.
II.   exists.
III. f is continuous at x = 6.

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

29. The table below shows values of f″(x) for various values of x:

The function f could be

(A) a linear function
(B) a quadratic function
(C) a cubic function
(D) an exponential function

30. Choose the Riemann Sum whose limit is the integral 

 

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. Where, in the first quadrant, does the rose r = sin 3θ have a vertical tangent?

(A) θ = 0.393
(B) θ = 0.468
(C) θ = 0.598
(D) θ = 0.659

32. A cup of coffee placed on a table cools at a rate of  per minute, where H represents the temperature of the coffee and t is time in minutes. If the coffee was at 120°F initially, what will its temperature be, to the nearest degree, 10 minutes later

(A) 73°F
(B) 95°F
(C) 100°F
(D) 105°F

33. Consider the power series . It is known that at x = –1, the series converges conditionally. Of the following, which is true about the convergence of the power series at x = 10?

(A) There is not enough information.
(B) At x = 10, the series diverges.
(C) At x = 10, the series converges conditionally.
(D) At x = 10, the series converges absolutely.

34. The density of the population of a city is 10,000 people per square mile at the beginning of a certain year. We can model the time, in years (t), it will take until the population reaches a certain density, in people per square mile (D), by using the function t = g(D). What are the units of g′(D)?

(A) years
(B) years per people per square mile
(C) people per square mile per year
(D) people per square mile

Questions 35 and 36. The graph below shows the velocity of an object during the interval 0 ≤ t ≤ 9.

 

35. The object attains its greatest speed at t =

(A) 3
(B) 5
(C) 6
(D) 8

36. The object was at the origin at t = 3. It returned to the origin

(A) during 6 < t < 7
(B) at t = 7
(C) during 7 < t < 8
(D) at t = 8

37. An object in motion in the plane has acceleration vector a(t) = 〈sin t,e^–t〉 for 0 ≤ t ≤ 5. It is at rest when t = 0. What is the maximum speed it attains?

(A) 1.022
(B) 1.414
(C) 2.217
(D) 3.162

38. If  is replaced by u, then  is equivalent to

39. The set of all x for which the power series  converges is

(A) {–3,3}
(B) |x| < 3
(C) –3 ≤ x < 3
(D) –3 < x ≤ 3

40. The graph above shows region R, which is bounded above by the polar graph of r = 3 + 2 sin(3θ) and below by the polar graph of r = 5 sin θ. The intersection of the graphs occur at θ = 0.893 and θ = 2.249 as indicated by the dashed line segments. What is the area of region R?

(A) 3.301
(B) 4.155
(C) 6.141
(D) 8.667

41. The definite integral  represents the length of a curve of an increasing function f(x). If one end of the curve is at the point (1,2), then an equation describing the curve is

(A) y = 3 ln x + 2
(B) y = 3 ln x
(C)
(D) 

42. The Maclaurin series for the function f(x) is given . If the kth-degree Maclaurin polynomial is used to approximate the values of the function in the interval of convergence, then . If we desire an error of less than 0.01 when approximating f(2) with P_k(2), what is the least degree, k, we would need so that the Alternating Series Error Bound guarantees | f(2) – P_k(2)| < 0.01?

(A) 31
(B) 33
(C) 35
(D) 37

43. Which statement is true?

(A) If f′(c) = 0, then f has a local maximum or minimum at (c, f(c)).
(B) If f″(c) = 0, then the graph of f has an inflection point at (c, f(c)).
(C) If f is differentiable at x = c, then f is continuous at x = c.
(D) If f is continuous on (a,b), then f maintains a maximum value on (a,b).

44. The graph of f′ is shown above. Which statement(s) about f must be true for a < x < b?

I. f is increasing.
II. f is continuous.
III. f is differentiable.

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

45. Given  and g′(x) < 0 for all x, select the table that could represent g on the interval [–1,4].

(A)

(B)

(C)

(D)

 

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. The Boston Red Sox play in Fenway Park, notorious for its Green Monster, a wall 37 feet tall and 315 feet from home plate at the left-field foul line. Suppose a batter hits a ball 2 feet above home plate, driving the ball down the left-field line at an initial angle of 30° above the horizontal, with initial velocity of 120 feet per second. (Since Fenway is near sea level, assume that the acceleration due to gravity is –32.172 ft/sec².)

(a) Write the parametric equations for the location of the ball t seconds after it has been hit.
(b) At what elevation does the ball hit the wall?
(c) How fast is the ball traveling when it hits the wall?

 

2. The table shows the depth of water, W, in a river, as measured at 4-hour intervals during a daylong flood. Assume that W is a differentiable function of time t.

(a) Find the approximate value of W′(16). Indicate units of measure.
(b) Estimate the average depth of the water, in feet, over the time interval 0 ≤ t ≤ 24 hours by using a trapezoidal approximation with subintervals of length Δt = 4 hours.
(c) Scientists studying the flooding believe they can model the depth of the water with the function , where F(t) represents the depth of the water, in feet, after t hours. Find F′(16) and explain the meaning of your answer, with appropriate units, in terms of the river depth.
(d) Use the function F to find the average depth of the water, in feet, over the time interval 0 ≤ t ≤ 24 hours.

 

Part B

TIME: 30 MINUTES
2 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. The region R is bounded by the curves f(x) = cos(πx) – 1 and g(x) = x(2 – x), as shown in the figure.

(a) Find the area of R.
(b) A solid has base R, and each cross section perpendicular to the x-axis is an isosceles right triangle whose hypotenuse lies in R. Set up, but do not evaluate, an integral for the volume of this solid.
(c) Set up, but do not evaluate, an integral for the volume of the solid formed when R is rotated around the line y = 3.

 

4. Two autos, P and Q, start from the same point and race along a straight road for 10 seconds. The velocity of P is given by  feet per second. The velocity of Q is shown in the graph.

(a) At what time is P’s actual acceleration (in ft/sec²) equal to its average acceleration for the entire race?
(b) What is Q’s acceleration (in ft/sec²) then?
(c) At the end of the race, which auto was ahead? Explain.

 

5. Given that a function f is continuous and differentiable throughout its domain and that f(5) = 2, f′(5) = –2, f″(5) = –1, and f′″(5) = 6.

(a) Write a Taylor polynomial of degree 3 that approximates f around x = 5.
(b) Use your answer to estimate f(5.1).
(c) Let g(x) = f(2x + 5). Write a cubic Maclaurin polynomial approximation for g.

 

6. Let f be the function that contains the point (–1,8) and satisfies the differential equation .

(a) Write an equation of the line tangent to f at x = – 1.
(b) Using your answer to part (a), estimate f(0).
(c) Using Euler’s method with a step size of 0.5, estimate f(0).
(d) Estimate f(0) using an integral.