AP Calculus AB Exam (College Board Advanced Placement) — Questions and Answers

Question 1: The region bounded by y = √x, the x-axis, and the line x = 4 is revolved about the x-axis. What is the volume of the resulting solid?

32π/5
8π (Correct answer)
4π
16π

Correct answer: 8π

By the disk method, V = π∫₀⁴ (√x)² dx = π∫₀⁴ x dx = π(16/2) = 8π.

Question 2: Let f(x) = x³ − 3x on the interval [0, 2]. At which value of c in (0, 2) does the Mean Value Theorem guarantee that f'(c) equals the average rate of change of f on [0, 2]?

1
4/3
√2
2/√3 (Correct answer)

Correct answer: 2/√3

The average rate of change is (f(2) − f(0))/2 = (2 − 0)/2 = 1. Setting f'(c) = 3c² − 3 = 1 gives c² = 4/3, so c = 2/√3, which lies in (0, 2).

Question 3: A left Riemann sum with 4 subintervals of equal width is used to approximate ∫ from 0 to 4 of x² dx. What is the value of the sum?

14 (Correct answer)
21
30
20

Correct answer: 14

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

Question 4: If f(x) = x² / (x + 1), what is f′(1)?

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

Correct answer: 3/4

By the quotient rule, f′(x) = [2x(x + 1) − x²] / (x + 1)² = (x² + 2x) / (x + 1)². At x = 1 this is 3/4.

Question 5: What is d/dx [arctan(3x)]?

3 / (1 + 9x²) (Correct answer)
9 / (1 + 9x²)
3 / (1 + 3x²)
1 / (1 + 9x²)

Correct answer: 3 / (1 + 9x²)

The derivative of arctan u is u′ / (1 + u²). With u = 3x, this gives 3 / (1 + 9x²).

Question 6: Suppose f(2) = 1 and f′(2) = 4. If h(x) = [f(x)]³, what is h′(2)?

4
3
48
12 (Correct answer)

Correct answer: 12

By the chain rule, h′(x) = 3[f(x)]² · f′(x). At x = 2, h′(2) = 3(1)²(4) = 12.

Question 7: What is lim(x→2) (x² − 4)/(x − 2)?

4 (Correct answer)
The limit does not exist
0
2

Correct answer: 4

Factor the numerator as (x − 2)(x + 2) and cancel the common factor. That leaves lim(x→2) (x + 2) = 4.

Question 8: Let F(x) = ∫ from 0 to x of (t² − 4t + 3) dt. At which value of x does F have a local maximum?

x = 0
x = 1 (Correct answer)
x = 3
x = 2

Correct answer: x = 1

F′(x) = x² − 4x + 3 = (x − 1)(x − 3). F′ changes from positive to negative at x = 1, so F has a local maximum there.

Question 9: What is the particular solution to dy/dx = e^(x − y) with y(0) = ln 2?

y = x + ln 2
y = ln(e^x + 1) (Correct answer)
y = ln(e^x) + 2
y = e^x + ln 2 − 1

Correct answer: y = ln(e^x + 1)

Writing e^(x−y) = e^x/e^y gives e^y dy = e^x dx, so e^y = e^x + C. Using y(0) = ln 2 gives 2 = 1 + C, so C = 1 and y = ln(e^x + 1).

Question 10: What is the average value of f(x) = x² on the interval [0, 3]?

9
3 (Correct answer)
9/2
3/2

Correct answer: 3

The average value is (1/(3−0))·∫ from 0 to 3 of x² dx = (1/3)(9) = 3.

Question 11: Which function is the solution to dy/dx = y − 4 with y(0) = 1?

y = 1 + 4e^x
y = −3e^x + 1
y = 4 − 3e^x (Correct answer)
y = 4 + 3e^x

Correct answer: y = 4 − 3e^x

Separating gives dy/(y − 4) = dx, so y − 4 = Ce^x. Using y(0) = 1 gives C = −3, so y = 4 − 3e^x.

Question 12: The base of a solid is the region in the first quadrant bounded by y = 1 − x² and the coordinate axes. Cross-sections perpendicular to the x-axis are squares. What is the volume of the solid?

8/15 (Correct answer)
4/15
1/5
2/3

Correct answer: 8/15

Each square has side 1 − x², so V = ∫₀¹ (1 − x²)² dx = ∫₀¹ (1 − 2x² + x⁴) dx = 1 − 2/3 + 1/5 = 8/15.

Question 13: Let f(x) = x⁴ − 4x³. Which statement about f at its critical points is correct?

f has a local minimum at x = 3 and no local extremum at x = 0. (Correct answer)
f has a local minimum at both x = 0 and x = 3.
f has a local maximum at x = 0 and a local minimum at x = 3.
f has a local minimum at x = 0 and a local maximum at x = 3.

Correct answer: f has a local minimum at x = 3 and no local extremum at x = 0.

f'(x) = 4x²(x − 3), so the critical points are x = 0 and x = 3. f' is negative on both sides of 0, so there is no extremum there. f' changes from negative to positive at 3, so that is a local minimum.

Question 14: What is the value of ∫ from 0 to 1 of x/(x² + 1) dx?

1/2
(1/2)·ln 2 (Correct answer)
ln 2
2·ln 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 (1/u) du = (1/2)·ln 2.

Question 15: Water is in a tank at the rate of change V′(t) = 6 − 2t gallons per minute for 0 ≤ t ≤ 6, where V(t) is the volume of water in gallons. If the tank holds 20 gallons at t = 0, what is the maximum volume of water in the tank on this interval?

29 (Correct answer)
18
20
38

Correct answer: 29

V′ changes from positive to negative at t = 3, so the maximum is at t = 3. V(3) = 20 + ∫₀³ (6 − 2t) dt = 20 + 9 = 29.

Question 16: A function f is twice differentiable with f''(x) = x(x − 2)². At which x-values does the graph of f have a point of inflection?

x = 0 and x = 2
x = 0 only (Correct answer)
f has no inflection points
x = 2 only

Correct answer: x = 0 only

An inflection point requires f'' to change sign. f'' is zero at x = 0 and x = 2, but (x − 2)² is never negative, so the sign of f'' changes only at x = 0.

Question 17: Let f(x) = x² for x ≠ 1 and f(1) = 3. Which statement is true?

f is continuous at x = 1 because f(1) is defined
lim(x→1) f(x) = 3
lim(x→1) f(x) = 1, but f is not continuous at x = 1 (Correct answer)
lim(x→1) f(x) does not exist

Correct answer: lim(x→1) f(x) = 1, but f is not continuous at x = 1

Limits ignore the value at the point, so lim(x→1) f(x) = lim(x→1) x² = 1. Because this differs from f(1) = 3, f is not continuous at x = 1.

Question 18: A 13-foot ladder leans against a vertical wall. The bottom of the ladder slides away from the wall at a constant 2 ft/s. How fast is the top of the ladder sliding down the wall at the instant the bottom is 5 feet from the wall?

5/6 ft/s (Correct answer)
12/5 ft/s
2/5 ft/s
6/5 ft/s

Correct answer: 5/6 ft/s

From x² + y² = 169, x(dx/dt) + y(dy/dt) = 0. When x = 5, y = 12, so 5(2) + 12(dy/dt) = 0 and dy/dt = -5/6 ft/s, so the top slides down at 5/6 ft/s.

Question 19: What is d/dx [5^x]?

5^x / ln 5
5^x
x · 5^(x−1)
5^x ln 5 (Correct answer)

Correct answer: 5^x ln 5

For a constant base a > 0, d/dx [a^x] = a^x ln a. With a = 5, the derivative is 5^x ln 5.

Question 20: Consider dy/dx = y(2 − y) with initial condition y(0) = 1. What is lim as x → ∞ of y?

1
does not exist (y is unbounded)
0
2 (Correct answer)

Correct answer: 2

For 0 < y < 2 the slope y(2 − y) is positive, so y increases. It cannot cross the equilibrium solution y = 2, so it approaches 2.

Question 21: Water flows into a tank at a rate of r(t) = 2t + 1 gallons per minute. If the tank holds 5 gallons at t = 0, how many gallons are in the tank at t = 3 minutes?

20
17 (Correct answer)
14
12

Correct answer: 17

The amount is 5 + ∫ from 0 to 3 of (2t + 1) dt = 5 + [t² + t] from 0 to 3 = 5 + 12 = 17 gallons.

Question 22: If f(x) = x² sin x, what is f′(π)?

2π
0
π²
−π² (Correct answer)

Correct answer: −π²

By the product rule, f′(x) = 2x sin x + x² cos x. At x = π this is 2π(0) + π²(−1) = −π².

Question 23: What is the absolute maximum value of f(x) = x³ − 6x² + 9x + 1 on the closed interval [0, 5]?

5
17
21 (Correct answer)
1

Correct answer: 21

f'(x) = 3(x − 1)(x − 3), so the critical points are x = 1 and x = 3. Evaluating gives f(0) = 1, f(1) = 5, f(3) = 1, f(5) = 21, so the maximum is 21.

Question 24: Let f(x) = x³. What is the value of lim_{h→0} [(2 + h)³ − 8] / h?

12 (Correct answer)
8
6
0

Correct answer: 12

The limit is the definition of f′(2) for f(x) = x³. Since f′(x) = 3x², f′(2) = 12.

Question 25: Which of the following functions is continuous at x = 0 but not differentiable at x = 0?

f(x) = sin x
f(x) = x^(4/3)
f(x) = x^(2/3) (Correct answer)
f(x) = x³

Correct answer: f(x) = x^(2/3)

For f(x) = x^(2/3), f′(x) = (2/3)x^(−1/3), which is unbounded near 0, so there is a cusp at x = 0. The other three functions are differentiable at 0.

Question 26: The derivative of a function f is f'(x) = (x + 2)(x − 1)². Which statement describes f at x = 1?

f has neither a local maximum nor a local minimum at x = 1. (Correct answer)
f has a vertical tangent at x = 1.
f has a local maximum at x = 1.
f has a local minimum at x = 1.

Correct answer: f has neither a local maximum nor a local minimum at x = 1.

f'(1) = 0, so the tangent is horizontal, but (x − 1)² is nonnegative, so f' stays positive on both sides of x = 1. Because f' does not change sign, f has no local extremum there.

Question 27: If ∫ from 2 to 6 of f(x) dx = 10 and ∫ from 2 to 4 of f(x) dx = 3, what is ∫ from 6 to 4 of f(x) dx?

13
−13
7
−7 (Correct answer)

Correct answer: −7

By additivity, ∫ from 4 to 6 of f = 10 − 3 = 7. Reversing the limits changes the sign, so ∫ from 6 to 4 of f = −7.

Question 28: The point (1, 2) lies on the curve x² + xy + y² = 7. What is dy/dx at that point?

4/5
−2/5
−5/4
−4/5 (Correct answer)

Correct answer: −4/5

Differentiating implicitly gives 2x + y + x(dy/dx) + 2y(dy/dx) = 0. At (1, 2) this is 4 + 5(dy/dx) = 0, so dy/dx = −4/5.

Question 29: What is the area of the region bounded by the curves x = y² and x = y + 2?

9
3/2
9/2 (Correct answer)
27/2

Correct answer: 9/2

The curves meet at y = −1 and y = 2, and x = y + 2 is to the right of x = y². The area is ∫₋₁² (y + 2 − y²) dy = 9/2.

Question 30: What is the value of ∫ from 0 to 1 of x·e^(x²) dx?

(e − 1)/2 (Correct answer)
e − 1
e/2
(e + 1)/2

Correct answer: (e − 1)/2

Let u = x², so du = 2x dx and x dx = du/2. The integral becomes (1/2)∫ from 0 to 1 of e^u du = (e − 1)/2.

Question 31: Which function is the solution to dy/dx = −x/y with initial condition y(0) = 2?

y = √(4 − x²) (Correct answer)
y = 2e^(−x²/2)
y = −√(4 − x²)
y = √(4 + x²)

Correct answer: y = √(4 − x²)

Separating gives y dy = −x dx, so y²/2 = −x²/2 + C and y² + x² = 4. Since y(0) = 2 is positive, y = √(4 − x²).

Question 32: Let f(x) = x³ + x + 1, which is one-to-one. If g is the inverse function of f, what is g′(3)?

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

Correct answer: 1/4

Since f(1) = 3, g(3) = 1. Then g′(3) = 1 / f′(1) = 1/(3·1² + 1) = 1/4.

Question 33: A spherical balloon is inflated so that its volume increases at 36π cm³/s. At the instant its radius is 3 cm, how fast is its surface area increasing? (V = (4/3)πr³, S = 4πr²)

12π cm²/s
36π cm²/s
8π cm²/s
24π cm²/s (Correct answer)

Correct answer: 24π cm²/s

From dV/dt = 4πr²(dr/dt), 36π = 4π(9)(dr/dt) gives dr/dt = 1 cm/s. Then dS/dt = 8πr(dr/dt) = 8π(3)(1) = 24π cm²/s.

Question 34: If dy/dx = y² and y(0) = 1, what is y(1/2)?

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

Correct answer: 2

Separating gives dy/y² = dx, so −1/y = x + C. With y(0) = 1, C = −1, so y = 1/(1 − x) and y(1/2) = 2.

Question 35: Which of the following is equal to ∫ 2x·cos(x²) dx?

sin(x²) + C (Correct answer)
−sin(x²) + C
2·sin(x²) + C
x²·sin(x²) + C

Correct answer: sin(x²) + C

Let u = x², so du = 2x dx. The integral becomes ∫ cos u du = sin u + C = sin(x²) + C.

Question 36: Evaluate lim (x→0) (e^(2x) − 1 − 2x) / x².

4
0
1
2 (Correct answer)

Correct answer: 2

Direct substitution gives 0/0, so apply L'Hospital's Rule twice. The first pass gives (2e^(2x) − 2)/(2x), still 0/0. The second gives 4e^(2x)/2, which tends to 2.

Question 37: What is the average value of f(x) = sin x on the interval [0, π]?

π/2
2
1/π
2/π (Correct answer)

Correct answer: 2/π

The average value is (1/(π−0))∫₀^π sin x dx = (1/π)(2) = 2/π.

Question 38: A farmer has 200 meters of fencing and will enclose a rectangular pen along a straight river. No fence is needed along the river side. What is the maximum possible area of the pen?

5000 m² (Correct answer)
10000 m²
2500 m²
4000 m²

Correct answer: 5000 m²

With two sides of length x perpendicular to the river and one side of length 200 − 2x parallel to it, A(x) = x(200 − 2x). A'(x) = 200 − 4x = 0 gives x = 50, so A = 50 · 100 = 5000 m².

Question 39: What is d/dx of √(3x² + 1)?

6x / √(3x² + 1)
3x / √(3x² + 1) (Correct answer)
1 / (2√(3x² + 1))
3 / (2√(3x² + 1))

Correct answer: 3x / √(3x² + 1)

By the chain rule, the derivative is (1/2)(3x² + 1)^(−1/2) · 6x = 3x / √(3x² + 1).

Question 40: What is lim(x→−∞) (2x + 1)/√(x² + 4)?

−2 (Correct answer)
0
2
−∞

Correct answer: −2

For large negative x, √(x² + 4) behaves like |x| = −x. The ratio then approaches 2x/(−x) = −2.

Question 41: What is lim(x→3⁻) 2/(x − 3)?

2/3
+∞
−∞ (Correct answer)
0

Correct answer: −∞

As x approaches 3 from the left, x − 3 is a small negative number. Dividing 2 by it gives a large negative value, so the function decreases without bound.

Question 42: Let f(x) = x² on the interval [0, 3]. At what value of c in (0, 3) does f(c) equal the average value of f on [0, 3]?

√3 (Correct answer)
3
3/2
3√3

Correct answer: √3

The average value is (1/3)∫₀³ x² dx = (1/3)(9) = 3. Setting c² = 3 gives c = √3, which lies in (0, 3).

Question 43: What is the value of ∫ from 0 to 2 of (3x² − 2x) dx?

4 (Correct answer)
2
8
12

Correct answer: 4

An antiderivative is x³ − x². Evaluating from 0 to 2 gives (8 − 4) − 0 = 4.

Question 44: A particle moves along a line with position s(t) = t³ − 6t² + 9t for t ≥ 0. At which time is the particle speeding up?

t = 2
t = 0.5
t = 2.5
t = 4 (Correct answer)

Correct answer: t = 4

v(t) = 3(t − 1)(t − 3) and a(t) = 6t − 12. At t = 4, v = 9 and a = 12 have the same sign, so the speed is increasing. At t = 0.5 and t = 2.5 the signs differ, and at t = 2 the acceleration is 0.

Question 45: The function f is continuous on the closed interval [1, 4], with f(1) = −3 and f(4) = 6. Which statement must be true?

f′(c) = 0 for some c in (1, 4)
f has a maximum value of 6 on [1, 4]
f(c) = 0 for some c in (1, 4) (Correct answer)
f(x) > 0 for every x in (1, 4)

Correct answer: f(c) = 0 for some c in (1, 4)

By the Intermediate Value Theorem, a continuous f takes every value between −3 and 6 on (1, 4). Since 0 lies between them, f(c) = 0 for some c in (1, 4).

Question 46: What is lim(x→0) (√(x + 9) − 3)/x?

3
0
1/6 (Correct answer)
1/3

Correct answer: 1/6

Multiply the numerator and denominator by the conjugate √(x + 9) + 3. The expression becomes 1/(√(x + 9) + 3), which approaches 1/6.

AP Calculus AB Exam (College Board Advanced Placement)

Tests college-level first-semester calculus: limits and continuity, differentiation and its applications, integration and accumulation of change, differential equations, and applications of integration. The multiple-choice section is 45 questions (Part A without a calculator, Part B with a graphing calculator), followed by a 6-question free-response section that is not modeled here.

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105 min

Time Limit

60%

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