AP Calculus BC Exam (College Board) — Questions and Answers

Question 1: What is d/dx [3^(x^2)]?

2x · ln 3 · 3^(x^2) (Correct answer)
x^2 · 3^(x^2 − 1)
ln 3 · 3^(x^2)
2x · 3^(x^2)

Correct answer: 2x · ln 3 · 3^(x^2)

For a constant base, d/dx a^u = a^u · ln a · u'. With a = 3 and u = x^2, the result is 3^(x^2) · ln 3 · 2x.

Question 2: What is the value of ∫ from 0 to 1 of x·eˣ dx?

0
1 (Correct answer)
e
e − 1

Correct answer: 1

Integration by parts with u = x and dv = eˣ dx gives x·eˣ − eˣ. Evaluating from 0 to 1 gives (e − e) − (0 − 1) = 1.

Question 3: Consider dy/dx = y² with y(0) = 1. On which interval is the particular solution defined, given that the interval must contain x = 0?

All real numbers
−1 < x < 1
x > 1
x < 1 (Correct answer)

Correct answer: x < 1

Separating gives −1/y = x + C, and y(0) = 1 gives C = −1. So y = 1/(1 − x), which has a vertical asymptote at x = 1, and the solution through x = 0 lives on x < 1.

Question 4: A 13-foot ladder leans against a vertical wall. The bottom of the ladder slides away from the wall at 2 ft/s. At the instant the bottom is 5 feet from the wall, what is the rate of change of the height of the top of the ladder on the wall?

-6/5 ft/s
5/6 ft/s
-5/6 ft/s (Correct answer)
-12/5 ft/s

Correct answer: -5/6 ft/s

With x² + y² = 169, differentiating gives x(dx/dt) + y(dy/dt) = 0. At x = 5, y = 12, so 5(2) + 12(dy/dt) = 0 and dy/dt = -5/6 ft/s.

Question 5: What is the value of ∫ from 1 to e of x·ln x dx?

e²/4
(e² − 1)/4
(e² + 1)/2
(e² + 1)/4 (Correct answer)

Correct answer: (e² + 1)/4

By parts with u = ln x and dv = x dx, the antiderivative is (x²/2)ln x − x²/4. Evaluating from 1 to e gives (e²/2 − e²/4) − (−1/4) = (e² + 1)/4.

Question 6: Which of the following series converges?

Σ_{n=1}^∞ n/(2n + 1)
Σ_{n=1}^∞ 1/√n
Σ_{n=2}^∞ 1/(n ln n)
Σ_{n=2}^∞ 1/(n(ln n)²) (Correct answer)

Correct answer: Σ_{n=2}^∞ 1/(n(ln n)²)

By the integral test, ∫₂^∞ dx/(x(ln x)²) = 1/ln 2 is finite, so the series converges. The other three series diverge.

Question 7: What is lim(x→∞) (3x² − 5x + 1)/(2x² + 7)?

3
0
∞
3/2 (Correct answer)

Correct answer: 3/2

The numerator and denominator have the same degree, so the limit is the ratio of the leading coefficients, 3/2.

Question 8: Air is pumped into a spherical balloon at a rate of 100 cm³/s. How fast is the radius increasing at the instant the radius is 5 cm? (V = (4/3)πr³)

π/4 cm/s
1 cm/s
1/π cm/s (Correct answer)
4/π cm/s

Correct answer: 1/π cm/s

Differentiating gives dV/dt = 4πr²(dr/dt). With r = 5, 100 = 100π(dr/dt), so dr/dt = 1/π cm/s.

Question 9: A tank holds 10 gallons of water at time t = 0. Water flows in at a rate of R(t) = 12 − 3t gallons per minute for 0 ≤ t ≤ 6, with t in minutes. How many gallons are in the tank at t = 6?

18
38
28 (Correct answer)
10

Correct answer: 28

The net change is ∫ from 0 to 6 of (12 − 3t) dt = 72 − 54 = 18 gallons. Adding the initial 10 gives 28 gallons.

Question 10: What is the value of the derivative of f(x) = arctan(2x) at x = 1/2?

4/5
2
1/2
1 (Correct answer)

Correct answer: 1

d/dx arctan(2x) = 2/(1 + (2x)^2). At x = 1/2 this is 2/(1 + 1) = 1.

Question 11: What is lim(x→2) |x − 2|/(x − 2)?

−1
Does not exist (Correct answer)
1
0

Correct answer: Does not exist

From the right the expression equals 1, and from the left it equals −1. The one-sided limits differ, so the two-sided limit does not exist.

Question 12: A population satisfies dP/dt = 0.2P(1 − P/500). For what value of P is the population growing most rapidly?

400
100
500
250 (Correct answer)

Correct answer: 250

The growth rate 0.2P(1 − P/500) is a downward-opening parabola in P. Its maximum is halfway between the zeros at P = 0 and P = 500, which is P = 250.

Question 13: Which statement describes the series Σ_{n=1}^∞ (−1)ⁿ/√n?

It diverges because its terms do not approach 0.
It converges conditionally. (Correct answer)
It diverges by the alternating series test.
It converges absolutely.

Correct answer: It converges conditionally.

The terms 1/√n decrease to 0, so the alternating series test shows convergence. The series of absolute values, Σ1/√n, is a p-series with p = 1/2 ≤ 1 and diverges.

Question 14: Euler's method with two steps of size 0.5 is used to approximate y(1) for the differential equation dy/dx = x + y with y(0) = 1. What is the approximation?

2.25
2.5 (Correct answer)
3.0
2.0

Correct answer: 2.5

At (0, 1) the slope is 1, so y(0.5) ≈ 1 + 0.5(1) = 1.5. At (0.5, 1.5) the slope is 2, so y(1) ≈ 1.5 + 0.5(2) = 2.5.

Question 15: A left Riemann sum with 4 subintervals of equal width is used to approximate ∫ from 0 to 4 of x² dx. What is its value?

30
22
21
14 (Correct answer)

Correct answer: 14

The width is 1 and the left endpoints are 0, 1, 2, 3. The sum is 1·(0 + 1 + 4 + 9) = 14.

Question 16: What is the area enclosed by the polar curve r = 1 + cos θ for 0 ≤ θ ≤ 2π?

π/2
3π/2 (Correct answer)
3π
π

Correct answer: 3π/2

Area = ½∫₀^{2π}(1 + cos θ)² dθ = ½(2π + 0 + π) = 3π/2.

Question 17: What is the value of ∫ from 0 to ∞ of e^(−2x) dx?

2
1/2 (Correct answer)
The integral diverges
1

Correct answer: 1/2

The integral is the limit as b→∞ of [−(1/2)e^(−2x)] from 0 to b. That is 0 − (−1/2) = 1/2.

Question 18: A particle moves along a line with position s(t) = t³ − 9t² + 15t for t ≥ 0. At which time is the particle speeding up?

t = 3
t = 4
t = 2 (Correct answer)
t = 0.5

Correct answer: t = 2

v(t) = 3(t − 1)(t − 5) and a(t) = 6t − 18. At t = 2, v = −9 and a = −6 have the same sign, so the speed is increasing.

Question 19: What is the relative maximum value of f(x) = x·e^(−x)?

e
1/e (Correct answer)
1/e²
1

Correct answer: 1/e

f′(x) = e^(−x)(1 − x) = 0 at x = 1, and f″(1) = e^(−1)(1 − 2) < 0, so x = 1 is a maximum. The value is f(1) = 1/e.

Question 20: What is the area enclosed by one petal of the rose curve r = cos 2θ, taken from θ = −π/4 to θ = π/4?

π/8 (Correct answer)
π/16
π/2
π/4

Correct answer: π/8

Area = ½∫_{−π/4}^{π/4} cos²2θ dθ = ¼∫(1 + cos 4θ) dθ = ¼(π/2) = π/8.

Question 21: The tangent line to f(x) = √x at x = 25 is used to approximate √26. What is the approximation, and is it an overestimate or an underestimate of the true value?

5.01, underestimate
5.2, overestimate
5.1, underestimate
5.1, overestimate (Correct answer)

Correct answer: 5.1, overestimate

L(x) = 5 + (1/10)(x − 25), so L(26) = 5.1. Since f″(x) = −¼x^(−3/2) < 0, f is concave down and tangent lines lie above the graph, so the approximation is an overestimate.

Question 22: What is lim(x→2⁻) 1/(x − 2)?

1/2
−∞ (Correct answer)
+∞
0

Correct answer: −∞

For x slightly less than 2, x − 2 is a small negative number, so 1/(x − 2) is a large negative number. The limit is −∞.

Question 23: What is the area of the region enclosed by the curves x = y² and x = y + 2?

11/2
7/2
9
9/2 (Correct answer)

Correct answer: 9/2

The curves intersect where y² = y + 2, at y = −1 and y = 2, and x = y + 2 is the right-hand curve. The area is ∫₋₁² (y + 2 − y²) dy = 9/2.

Question 24: Which expression equals ∫ 1/((x − 1)(x + 2)) dx?

(1/3)·ln|(x − 1)(x + 2)| + C
−(1/3)·ln|(x − 1)/(x + 2)| + C
ln|(x − 1)/(x + 2)| + C
(1/3)·ln|(x − 1)/(x + 2)| + C (Correct answer)

Correct answer: (1/3)·ln|(x − 1)/(x + 2)| + C

Partial fractions give 1/((x−1)(x+2)) = (1/3)/(x−1) − (1/3)/(x+2). Integrating gives (1/3)ln|x−1| − (1/3)ln|x+2| + C.

Question 25: The region bounded by y = x², the x-axis, and the line x = 2 is revolved about the y-axis. What is the volume of the solid generated?

16π
32π/5
8π (Correct answer)
4π

Correct answer: 8π

Using washers in y, the outer radius is 2 and the inner radius is √y, for 0 ≤ y ≤ 4. The volume is π∫₀⁴ (4 − y) dy = π(16 − 8) = 8π.

Question 26: If g(x) = x^2/(x + 1), what is g'(1)?

3/4 (Correct answer)
1/4
5/4
1/2

Correct answer: 3/4

By the quotient rule, g'(x) = [2x(x+1) − x^2(1)]/(x+1)^2. At x = 1 this is (4 − 1)/4 = 3/4.

Question 27: A farmer has 200 feet of fencing to enclose a rectangular pen along a straight river. No fence is needed along the river, so only three sides are fenced. What is the maximum possible area of the pen?

5,000 ft² (Correct answer)
10,000 ft²
4,000 ft²
2,500 ft²

Correct answer: 5,000 ft²

With width x and length y along the river, 2x + y = 200, so A = x(200 − 2x). A′ = 200 − 4x = 0 gives x = 50, y = 100, and A = 5,000 ft².

Question 28: What is the value of ∫ from 0 to 1 of x/(x² + 1) dx?

ln 2
π/4
(1/2)·ln 2 (Correct answer)
1/2

Correct answer: (1/2)·ln 2

Let u = x² + 1, so du = 2x dx. The integral becomes (1/2)∫ from 1 to 2 of du/u = (1/2)ln 2.

Question 29: If dy/dx = xy and y(0) = 2, what is y(2)?

2e² (Correct answer)
e²
4e²
2e⁴

Correct answer: 2e²

Separating gives dy/y = x dx, so ln|y| = x²/2 + C. Since y(0) = 2, y = 2e^(x²/2), and y(2) = 2e².

Question 30: In the Maclaurin series for f(x) = e^(−x²), what is the coefficient of x⁴?

1/24
1/2 (Correct answer)
1/4
−1/2

Correct answer: 1/2

Substitute u = −x² into eᵘ = 1 + u + u²/2 + …. The u²/2 term is (−x²)²/2 = x⁴/2, so the coefficient is 1/2.

Question 31: If y = ln(cos(3x)), what is dy/dx?

tan(3x)
−3 tan(3x) (Correct answer)
−tan(3x)
−3 cot(3x)

Correct answer: −3 tan(3x)

Apply the chain rule twice: d/dx ln(u) = u'/u with u = cos(3x), and u' = −3 sin(3x). This gives −3 sin(3x)/cos(3x) = −3 tan(3x).

Question 32: What is d/dx [arcsin(x^2)]?

2x / (1 + x^4)
2x / √(1 − x^2)
2x / √(1 − x^4) (Correct answer)
1 / √(1 − x^4)

Correct answer: 2x / √(1 − x^4)

d/dx arcsin(u) = u'/√(1 − u^2). With u = x^2 this gives 2x/√(1 − x^4).

Question 33: If F(x) = ∫ from 0 to x² of √(1 + t³) dt, what is F′(x)?

x²·√(1 + x⁶)
2x·√(1 + x³)
2x·√(1 + x⁶) (Correct answer)
√(1 + x⁶)

Correct answer: 2x·√(1 + x⁶)

By the Fundamental Theorem of Calculus with the chain rule, F′(x) = √(1 + (x²)³) · d/dx(x²) = 2x·√(1 + x⁶).

Question 34: What is the value of ∫ from 0 to 3 of 1/(x² + 9) dx?

π/36
1/3
π/12 (Correct answer)
π/4

Correct answer: π/12

∫ 1/(x² + 9) dx = (1/3)arctan(x/3) + C. Evaluating from 0 to 3 gives (1/3)(π/4 − 0) = π/12.

Question 35: The Maclaurin series for arctan x is Σ_{n=0}^∞ (−1)ⁿ x^(2n+1)/(2n+1) for |x| ≤ 1. What is the sum of Σ_{n=0}^∞ (−1)ⁿ / (3ⁿ(2n + 1))?

π√3/3
π√3/6 (Correct answer)
π/6
π/3

Correct answer: π√3/6

Setting x = 1/√3 gives x^(2n+1) = 3^(−n)/√3, so arctan(1/√3) = π/6 equals (1/√3) times the given sum. The sum is therefore √3 · π/6 = π√3/6.

Question 36: A function f has derivative f′(x) = (x − 1)(x + 2)². Which statement about the relative extrema of f is correct?

f has a relative maximum at x = −2 and a relative minimum at x = 1.
f has a relative minimum at x = 1 and no other relative extrema. (Correct answer)
f has relative minima at both x = −2 and x = 1.
f has a relative maximum at x = 1 only.

Correct answer: f has a relative minimum at x = 1 and no other relative extrema.

f′ changes from negative to positive at x = 1, giving a relative minimum. At x = −2 the factor (x + 2)² never changes sign, so f′ stays negative and there is no extremum.

Question 37: The alternating series Σ_{n=1}^∞ (−1)^(n+1)/n² is approximated by the sum of its first 3 terms. By the alternating series error bound, the absolute error is at most:

1/9
1/25
1/4
1/16 (Correct answer)

Correct answer: 1/16

The error is bounded by the first omitted term, which is the 4th term, 1/4² = 1/16.

Question 38: A particle's position is r(t) = ⟨t³ − t, 2t²⟩. What is its acceleration vector at t = 2?

⟨12, 8⟩
⟨11, 8⟩
⟨12, 4⟩ (Correct answer)
⟨6, 4⟩

Correct answer: ⟨12, 4⟩

The velocity is ⟨3t² − 1, 4t⟩ and the acceleration is ⟨6t, 4⟩, which equals ⟨12, 4⟩ at t = 2.

Question 39: What is the average value of f(x) = 3x² on the interval [1, 3]?

13 (Correct answer)
26
14
9

Correct answer: 13

The average value is (1/(3 − 1))·∫ from 1 to 3 of 3x² dx = (1/2)(27 − 1) = 13.

Question 40: A curve is defined parametrically by x = t² and y = t³. What is dy/dx at t = 2?

2
3 (Correct answer)
6
12

Correct answer: 3

dy/dx = (dy/dt)/(dx/dt) = 3t²/(2t) = 3t/2, which equals 3 at t = 2.

Question 41: What type of discontinuity does f(x) = (x² − x − 6)/(x − 3) have at x = 3?

Removable discontinuity (Correct answer)
Oscillating discontinuity
Infinite discontinuity
Jump discontinuity

Correct answer: Removable discontinuity

The numerator factors as (x − 3)(x + 2), so f(x) = x + 2 for x ≠ 3. The limit is 5 even though f(3) is undefined, which makes the discontinuity removable.

Question 42: What is the absolute maximum value of f(x) = x³ − 12x on the closed interval [−3, 5]?

−16
9
65 (Correct answer)
16

Correct answer: 65

f′(x) = 3x² − 12 = 0 gives x = ±2. Comparing f(−3) = 9, f(−2) = 16, f(2) = −16 and f(5) = 65, the largest value is 65.

Question 43: What is the interval of convergence of Σ_{n=1}^∞ (x − 3)ⁿ / (n·2ⁿ)?

(1, 5)
(1, 5]
[1, 5) (Correct answer)
[1, 5]

Correct answer: [1, 5)

The ratio test gives radius 2, so the series converges on (1, 5). At x = 1 it is the alternating harmonic series, which converges. At x = 5 it is the harmonic series, which diverges.

Question 44: What is the slope of the tangent line to the circle x^2 + y^2 = 25 at the point (4, −3)?

3/4
4/3 (Correct answer)
−4/3
−3/4

Correct answer: 4/3

Implicit differentiation gives 2x + 2y(dy/dx) = 0, so dy/dx = −x/y. At (4, −3) this is −4/(−3) = 4/3.

Question 45: Let f(x) = x^2 for x ≤ 1 and f(x) = 2x − 1 for x > 1. Which statement is true at x = 1?

f is differentiable at x = 1 and f'(1) = 2 (Correct answer)
f is continuous but not differentiable at x = 1
f is differentiable at x = 1 and f'(1) = 1
f is not continuous at x = 1

Correct answer: f is differentiable at x = 1 and f'(1) = 2

Both pieces equal 1 at x = 1, so f is continuous. The left derivative is 2x = 2 and the right derivative is 2, so f'(1) = 2.

AP Calculus BC Exam (College Board)

Tests college-level single-variable calculus, including limits, differentiation, integration, differential equations, parametric, polar and vector-valued functions, and infinite series. Scored 1-5 by the College Board. The multiple-choice section is half of the exam score.

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105 min

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60%

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