AP BC Calc Final Exam

Let the integral from 0 to x of f(t)dt = xsin(πx). Then f(3) =

(A) -3π

When the method of partial fractions is used to decompose (2x^2 - x + 4 / x^3 - 3x^2 + 2x), one of the fractions obtained is

(A) -5/(x-1)

Given f(x) = log base 10 (x) and log base 10 (102) ≈ 2.0086, which is closest to f'(100)?

(A) 0.0043

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

The area inside the circle r = 3sin(θ) and outside the cardioid r = 1 + sin(θ) is given by

(A) The integral from π/6 to π/2 of 9sin^2(θ) - (1 + sin(θ))^2 dθ

The equation of the tangent to the curve 2x^2 - y^4 = 1 at the point (-1,1) is

(A) y = -x

The definite integral from 1 to 10 of root(1 + (9/x^2)) dx 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 = 3ln(x) + 2

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

(B) $8902

When a series is used to approximate the integral from 0 to 0.3 of e^-x^2 dx, the value of the integral, to two decimal places, is

(B) 0.29

n=0 ∑ to ∞ (3^n / 4^n+1)

(B) 1

If root(x-2) is replaced by u, then the integral from 3 to 6 of root(x-2)/x dx is equivalent to

(B) 2 * the integral from 1 to 2 of u^2du/(u^2 + 2)

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

(B) π(root3)/2

The base of a solid is the first-quadrant region bounded by y = the 4th root(4 - 2x), 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

(B) π/8 * the integral from 0 to 2 of root(4 - 2x)dx

lim(h->0) ((sin(π/2 + h) - 1)/h) is

(C) 0

A cup of coffee placed on a table cools at a rate of dH/dt = -0.05(H - 70)°F 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?

(C) 100°F

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?

(C) 2.217

The average value of f(x) = 3 + |x| on the interval [-2, 4] is

(C) 4 and 2/3

If G(2) = 5 and G'(x) = 10x/(9-x^2), then an estimate of G(2.2) using a tangent-line approximation is

(C) 5.8

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

(C) a cubic function

The integral from 0 to 1 of x^2 * e^2 dx

(C) e - 2

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

(C) θ = 0.47

The nth term of the Taylor series expansion about x=0 of the function f(x) = 1/(1+2x) is

(D) (-1)^n-1 * (2x)^n-1

The set off all x for which the power series n=0 ∑ to ∞ (x^n/((n + 1)*3^n)) converges is

(D) -3≤x<3

A local maximum value of the function y= ln(x)/x is

(D) 1/e

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

(D) 2π

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 N(x) = 2x/(1 + x^3/2) fragments per square meter. How many fragments will be found within 20 meters of the point where the bomb exploded?

(D) 712

The sketch shows the graph of f(x) = x^2 - 4x - 5 and the line x = k. The regions labeled A and B have equal areas if k =

(D) 8

The graph of function f shown above consists of three quarter-circles. Which of the following is (are) equivalent to the integral of 0 to 2 of f(x)dx?

(D) I and II only I. 1/2 * the integral of -2 to 2 of f(x)dx II. The integral of 4 to 2 of f(x)dx

Let f(x) = x^2 - 36/(x-6) if x ≠ 6, 12 if x= = 6. Which of the following statements is (are) true?

(D) I, II, and III I. f is defined at x = 6. II. lim(x->6) f(x) exists. III. f is continuous at x = 6.

Which statement is true?

(D) If f is differentiable at x = c, then f is continuous at x = c.

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

(D) L≤M≤T≤R

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

(D) f(x) = x + 1/x on [-1, 1]

The integral of e^u/(1+e^2u) du is equal to

(D) tan^-1(e^2u) + C

The area bounded by the parabola y = x^2 and the lines y = 1 and y = 9 equals

(E) 104/3

Two objects in motion from t = 0 to t = 3 seconds have positions x1(t) = cos(t^2 + 1) and x2(t) = e^t/2t, respectively. How many times during the 3 seconds do the objects have the same velocity?

(E) 4

Suppose f(3) = 2, f'(3) = 5, and f''(3) = -2. Then d^2/dx^2 (f^2(X)) at x = 3 is equal to

(E) 42

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 dh/dt increasing?

(E) 4<t<12

The graph shows the velocity of an object during the interval 0≤t≤9. The object attains its greatest speed at t =

(E) 8 sec

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

(E) E

Let F(x)= the integral from 5 to x of dt/(1-t^2). Which of the following statements is (are) true?

(E) II and III only II. F(2)>0. III. The graph of F is concave upward.

Which of the following statements about the graph of y = x^2/(x-2) is (are) true?

(E) all three 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.

The graph of f' is shown above. Which statements about f must be true for a<x<b?

(E) all three I. f is increasing. II. f is continuous. III. f is differentiable.

The graph shows the velocity of an object during the interval 0≤t≤9. The object was at the origin at t = 3. It returned to the origin

(E) during 7<t<8 sec

The first-quadrant region bounded by y = 1/(rootx), y = 0, x = q (0<q<1), and x = 1 is rotated about the x-axis. The volume obtained as q approaches 0 from the right equals

(E) none of these

lim(x->2) [x] (where [x] is the greatest integer in x) is

(E) nonexistent