Calculus AB and Calculus BC
BC Practice Examination 3
SECTION I
Part A TIME: 55 MINUTES
The use of calculators is not permitted for this part of the examination.
There are 28 questions in Part A, for which 55 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. (where [x] is the greatest integer in x) is
(A) 1
(B) 2
(C) 3
(D) ∞
(E) nonexistent
2.
(A) 1
(B) −1
(C) 0
(D) ∞
(E) none of these
3.
(A)
(B) = 1.
(C) = 3.
(D) = 4.
(E) diverges.
4. The equation of the tangent to the curve 2x^{2} − y^{4} = 1 at the point (−1, 1) is
(A) y = −x
(B) y = 2 − x
(C) 4y + 5x + 1 = 0
(D) x − 2y + 3 = 0
(E) x − 4y + 5 = 0
5. The nth term of the Taylor series expansion about x = 0 of the function
(A) (2x)^{n}
(B) 2x^{n}^{ − 1}
(C)
(D) (−1)^{n}^{ − 1}(2x)^{n}^{ − 1}
(E) (−1)^{n}(2x)^{n}^{ − 1}
6. When the method of partial fractions is used to decompose one of the fractions obtained is
(A)
(B)
(C)
(D)
(E)
7. A relative maximum value of the function is is
(A) 1
(B) e
(C)
(D)
(E) none of these
8. When a series is used to approximate the value of the integral, to two decimal places, is
(A) −0.09
(B) 0.29
(C) 0.35
(D) 0.81
(E) 1.35
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
(E) E
10. Let Which of the following statements is (are) true?
I. The domain of F is x ≠ ±1.
II. F(2) > 0.
III. The graph of F is concave upward.
(A) none
(B) I only
(C) II only
(D) III only
(E) II and III only
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?
(A) 3
(B)
(C)
(D)
(E)
12. Let sin πx. Then f (3) =
(A) − 3π
(B) −1
(C) 0
(D) 1
(E) 3π
13. is equal to
(A) ln(1 + e^{2u}) + C
(B)
(C)
(D) tan^{−1}e^{u} + C
(E)
14. Given f (x) = log_{10}x and log_{10}(102) 2.0086, which is closest to f ′(100)?
(A) 0.0043
(B) 0.0086
(C) 0.01
(D) 1.0043
(E) 2
15. If G(2) = 5 and then an estimate of G(2.2) using a tangent-line approximation is
(A) 5.4
(B) 5.5
(C) 5.8
(D) 8.8
(E) 13.8
16. The area bounded by the parabola y = x^{2} and the lines y = 1 and y = 9 equals
(A) 8
(B)
(C)
(D) 32
(E)
17. The first-quadrant region bounded by y = 0, x = q (0 < q < 1), and x = 1 is rotated about the x-axis. The volume obtained as q →0^{+} equals
(A)
(B)
(C) 2π
(D) 4π
(E) none of these
18. A curve is given parametrically by the equations
x = 3 − 2sint and y = 2cos t − 1.
The length of the arc from t = 0 to t = π is
(A)
(B) π
(C) 2 + π
(D) 2π
(E) 4π
19. Suppose the graph of f is both increasing and concave up on a ≤ x ≤ b. Then, using the same number of subdivisions, and with L, R, M, and T denoting, respectively, left, right, midpoint, and trapezoid sums, it follows that
(A) R ≤ T ≤ M ≤ L
(B) L ≤ T ≤ M ≤ R
(C) R ≤ M ≤ T ≤ L
(D) L ≤ M ≤ T ≤ R
(E) none of these
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 II only
(D) I and III only
(E) all three
21. The only function that does not satisfy the Mean Value Theorem on the interval specified is
(A) f (x) = x^{2} − 2x on [−3, 1]
(B)
(C)
(D)
(E)
22.
(A) −3e − 1
(B) −e
(C) e − 2
(D) 3e
(E) 4e − 1
23. 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) none
(B) 0 < t < 4
(C) 0 < t < 8
(D) 0 < t < 12
(E) 4 < t < 12
24. The graph of function f shown above consists of three quarter-circles.
Which of the following is (are) equivalent to
I.
II.
III.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) all three
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
(A)
(B)
(C)
(D)
(E)
26. The average value of f (x) = 3 + |x| on the interval [−2, 4] is
(A)
(B)
(C)
(D)
(E) 6
27. The area inside the circle r = 3 sin θ and outside the cardioid r = 1 + sin θ is given by
(A)
(B)
(C)
(D)
(E) none of these
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
(E) none of the statements
Part B TIME: 50 MINUTES
Some questions in this part of the examination require the use of a graphing calculator. There are 17 questions in Part B, for which 50 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.
29. Two objects in motion from t = 0 to t = 3 seconds have positions x_{1}(t) = cos (t^{2} + 1) and respectively. How many times during the 3 seconds do the objects have the same velocity?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
30. The table below shows values of f ″(x) for various values of x:
x |
−1 |
0 |
1 |
2 |
3 |
f ″(x) |
−4 |
−1 |
2 |
5 |
8 |
The function f could be
(A) a linear function
(B) a quadratic function
(C) a cubic function
(D) a fourth-degree function
(E) an exponential function
31. Where, in the first quadrant, does the rose r = sin 3θ have a vertical tangent?
(A) nowhere
(B) θ = 0.39
(C) θ = 0.47
(D) θ = 0.52
(E) θ = 0.60
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 10 minutes later?
(A) 73°F
(B) 95°F
(C) 100°F
(D) 118°F
(E) 143°F
33. An investment of $4000 grows at the rate of 320e^{0.08t} dollars per year after t years. Its value after 10 years is approximately
(A) $4902
(B) $8902
(C) $7122
(D) $12,902
(E) none of these
34. The sketch shows the graphs of f (x) = x^{2} − 4x − 5 and the line x = k. The regions labeled A and B have equal areas if k =
(A) 5
(B) 7.766
(C) 7.899
(D) 8
(E) 11
Questions 35 and 36. The graph shows the velocity of an object during the interval 0 ≤ t ≤ 9.
35. The object attains its greatest speed at t =
(A) 2 sec
(B) 3 sec
(C) 5 sec
(D) 6 sec
(E) 8 sec
36. The object was at the origin at t = 3. It returned to the origin
(A) at t = 5 sec
(B) at t = 6 sec
(C) during 6 < t < 7 sec
(D) at t = 7 sec
(E) during 7 < t < 8 sec
37. An object in motion in the plane has acceleration vector 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) 2.958
(E) 3.162
38. If is replaced by u, then is equivalent to
(A)
(B)
(C)
(D)
(E)
39. The set of all x for which the power series converges is
(A) {−3,3}
(B) |x| < 3
(C) |x| > 3
(D) − 3 x < 3
(E) − 3 < x 3
40. A particle moves along a line with acceleration a = 6t. If, when t = 0, v = 1, then the total distance traveled between t = 0 and t = 3 equals
(A) 30
(B) 28
(C) 27
(D) 26
(E) none of these
41. The definite integral represents the length of an arc. If one end of the arc is at the point (1,2), then an equation describing the curve is
(A) y = 3 ln x + 2
(B) y = x + 3 ln x + 1
(C)
(D)
(E)
42. Suppose f (3) = 2, f ′(3) = 5, and f ″(3) = −2. Then at x = 3 is equal to
(A) −20
(B) 10
(C) 20
(D) 38
(E) 42
43. Which statement is true?
(A) If f (x) is continuous at x = c, then f ′(c) exists.
(B) If f ′(c) = 0, then f has a local maximum or minimum at (c,f (c)).
(C) If f ′(c) = 0, then the graph of f has an inflection point at (c,f (c)).
(D) If f is differentiable at x = c, then f is continuous at x = c.
(E) 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 statements 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 and III only
(E) all three
45. After a bomb explodes, pieces can be found scattered around the center of the blast. The density of bomb fragments lying x meters from ground zero is given by fragments per square meter. How many fragments will be found within 20 meters of the point where the bomb exploded?
(A) 13
(B) 278
(C) 556
(D) 712
(E) 4383
SECTION II
Part A TIME: 30 MINUTES
2 PROBLEMS
A graphing calculator is required for some of these problems.
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/sec^{2}.)
(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 day-long flood. Assume that W is a differentiable function of time t.
t (hr) |
0 |
4 |
8 |
12 |
16 |
20 |
24 |
W(t) (ft) |
32 |
36 |
38 |
37 |
35 |
33 |
32 |
(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: 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. 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.
(b) 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/sec^{2}) equal to its average acceleration for the entire race?
(b) What is Q’s acceleration (in ft/sec^{2}) 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 the 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.