Multiple Choice
16 questions
A graphing calculator is required for some of the questions on this portion of the exam.
Time limit: 40 minutes (60 for extended time)

Mark the letter of the best choice on the answer sheet provided.

1.

Which of the following are important topics in the AP Calculus AB curriculum?

 

 

 

(A) functions, graphs, and limits
(B) the differential calculus (rates of change)
(C) the integral calculus (accumulations of rates of change)
(D) applications of differentiation and integration
(E) all of the above

 

 

2.

Let f (x) be a continuous function on [a, b]. Which of the following must be true?

 

 

 

I.    f is differentiable on (a, b)
II.   f is differentiable on [a, b]
III. ∫ab f (x) dx exists

 

 

 

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

 

 

3.

The Intermediate Value Theorem states that if q is a real number strictly between p and r, and if f (a) = p, and if f (b) = r, then

 

 

 

(A) there exists a value x in (a, b) such that f (x) = q
(B) there exists a value x in (a, b) such that f (x) = q, provided f is continuous on [a, b]
(C) there exists a value x in [a, b] such that f (x) = q
(D) there exists a value x in [a, b] such that f (x) = q, provided f is continuous on [a, b]
(E) for any value x in [a, b], f (x) = q, provided f is continuous on [a, b]

 

 

4.

A pizza (in the conventional round shape, not rectangular) is magically growing in diameter at a rate of 1 inch per minute. At the instant when the pizza has a diameter of 15 inches, its area is growing at a rate of . . .

 

 

 

(A) 5p square inches per minute
(B) 7.5p square inches per minute
(C) 15p square inches per minute
(D) 30p square inches per minute
(E) none of these

 

 

 

5.

Let y be a function of x, and let k be a constant. The solution of y ′ = ky that satisfies the initial condition y(0) = 21.8 is . . .

 

 

 

(A) y = 21.8ekx
(B) y = ky2/2 + 21.8
(C) y = kx2/2 + 21.8
(D) y = kx2/2 + C, where C is another constant
(E) none of these

 

 

6.

Let G′(x) = tan–1(x2), and suppose that G(0) = 1.3. Compute G(2) to 3 decimal places.

 

 

 

(A) 0.118
(B) 1.418
(C) 2.718
(D) 4.018
(E) none of these

 

 

7.

Let f (x) = cos(ex – e–x) – 27sin2 e2x. Let Q(x) be an antiderivative of f such that Q(–2) = 2.108. Compute Q(0) to 3 decimal places.

 

 

 

(A) –3.388
(B) –3.398
(C) –3.408
(D) –3.418
(E) none of these

 

 

8.

Which of the following must be true if the point (c, d) is an inflection point of the function y = f (x)?

 

 

 

I.    f ″(c) = 0
II.   f ″(x) changes sign at x = c
III. f is continuous at x = c
IV. f ′(c) exists

 

 

 

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

 

 

9.

Let y be a differentiable function of x. At a local maximum, which of the following statements must be true?

 

 

 

(A) y′ = 0, and y″ > 0
(B) y′ = 0, and y″ < 0
(C) y′ = 0, and y″ exists
(D) y′ = 0, but y″ need not exist
(E) y′ = 0, and y′ is positive both to the left and to the right of the x value of the local max.

 

 

 

10.

If f (x) is differentiable on Β, and if f ′(c) = 0, and if f ″(c) > 0, then what can we say about the point (c, f (c))?

 

 

 

(A) local minimum
(B) local maximum
(C) stationary point (a.k.a. plateau point)
(D) point of inflection
(E) no conclusion possible

 

 

11.

Let P(x) be a polynomial of odd degree. How many global maxima does P(x) have?

 

 

 

(A) none
(B) one
(C) two
(D) three
(E) insufficient information to determine

 

 

12.

Consider the function y = x1/3 on the interval [0, 4]. The conclusion of the Mean Value Theorem is satisfied when x approximately equals . . .

 

 

 

(A) 0.470
(B) 0.570
(C) 0.670
(D) 0.770
(E) none of these

 

 

13.

Function f is continuous on all of Β except for a vertical asymptote at x = c and a step discontinuity at x = d. What can we conclude about f ′(x)?

 

 

 

(A) f ′(x) exists for all x in Β
(B) f ′(x) fails to exist for exactly one value of x
(C) f ′(x) fails to exist for exactly two values of x
(D) f ′(x) fails to exist for at least two values of x
(E) f ′(x) does not exist anywhere

 

 

14.

Let h(x) = (x ln x – x) / (ln 2). Compute h′(x).

 

 

 

(A) ln x
(B) ln 2 + ln x
(C) log2 x
(D) log2 x + ln 2
(E) log2 x – ln 2

 

15.

Compute the area bounded by the parabola y = 10 – x2 and the x-axis, to the nearest square unit.

 

 

 

(A) 39
(B) 40
(C) 41
(D) 42
(E) none of these

 

 

16.

A particle has acceleration (in m/sec2) given by a(t) = 2t. If initial velocity and position are known to be v(0) = 1 and s(0) = 3, respectively, compute the particle’s position at time t = 1.5 seconds.

 

 

 

(A) 5.125 m
(B) 5.625 m
(C) 6.125 m
(D) 6.625 m
(E) none of these