AP Calculus BC practice questions and exam guide

If Σ aₙ converges absolutely with sum S, then:

1. Σ aₙ may diverge
2. Σ |aₙ| < S
3. Rearrangements can change sum to anything (Riemann)
4. Σ |aₙ| ≥ |S| with equality possible — rearrangements preserve sum

Answer: D — Σ |aₙ| ≥ |S| with equality possible — rearrangements preserve sum

D) For absolutely convergent series: Σ |aₙ| ≥ |Σ aₙ| (triangle inequality), with equality if all aₙ same sign. Rearrangements preserve sum. C) Riemann rearrangement applies only to CONDITIONALLY convergent series. A/B) Wrong.

The series Σ_{n=1}^∞ (−1)^n / n^(1/3) is:

1. Absolutely convergent
2. Divergent — fails alternating series test
3. Conditionally convergent — alternating series test passes; Σ 1/n^(1/3) diverges (p = 1/3 < 1)
4. Inconclusive

Answer: C — Conditionally convergent — alternating series test passes; Σ 1/n^(1/3) diverges (p = 1/3 < 1)

C) 1/n^(1/3) decreases to 0 → alternating series converges. Absolute series Σ 1/n^(1/3) is p-series with p = 1/3 < 1 → diverges. Conditionally convergent. A) No, absolute series diverges. B) Test passes. D) Determinable.

Implicit differentiation: if x² + y² = 25, then dy/dx =

1. x/y
2. −x/y (differentiate: 2x + 2y·y' = 0 → y' = −x/y)
3. −y/x
4. 2x

Answer: B — −x/y (differentiate: 2x + 2y·y' = 0 → y' = −x/y)

Differentiate both sides w.r.t. x: 2x + 2y·(dy/dx) = 0 → dy/dx = −x/y. Circle of radius 5: makes sense geometrically (perpendicular to radius). Foundational AB topic.

Apply the ratio test to Σ n!/n^n. The limit L equals:

1. 1 — inconclusive
2. 0 — converges absolutely
3. ∞ — diverges
4. 1/e — converges since 1/e < 1

Answer: D — 1/e — converges since 1/e < 1

D) L = lim |((n+1)!/(n+1)^(n+1)) · (n^n/n!)| = lim (n^n/(n+1)^n) = lim 1/((n+1)/n)^n = 1/e ≈ 0.368 < 1. Series converges. A/B/C) Wrong limit.

A particle moves so that its position is given by parametric equations x(t) = t² and y(t) = 2t³. The speed at t = 1 is:

1. 2
2. √(4 + 36) = √40 ≈ 6.32
3. 6
4. 4

Answer: B — √(4 + 36) = √40 ≈ 6.32

For parametric motion, speed = √[(dx/dt)² + (dy/dt)²]. dx/dt = 2t = 2; dy/dt = 6t² = 6. Speed = √(4 + 36) = √40 = 2√10 ≈ 6.32.

∫ x² dx =

1. 2x + C
2. x²/2 + C
3. x³/3 + C
4. 3x² + C

Answer: C — x³/3 + C

C) Power rule for integration. A) Differentiation. B/D) Wrong exponent or coefficient.

For x(t) = t² − 1, y(t) = t³ − 3t, find dy/dx at t = 2.

1. 9/4 — (3t² − 3)/(2t) = 9/4 at t = 2
2. 6/4 = 3/2
3. 12/4 = 3
4. 3/2 — direct ratio of derivatives 3/2

Answer: A — 9/4 — (3t² − 3)/(2t) = 9/4 at t = 2

A) dy/dx = (dy/dt)/(dx/dt) = (3t² − 3)/(2t) = (12 − 3)/4 = 9/4. B/D) Used wrong derivative for numerator. C) Forgot the −3.

The derivative of ln(x²+1) is:

1. 1/(x²+1)
2. 2x/(x²+1) (chain rule: d/dx[ln u] = u'/u with u = x²+1)
3. ln(2x)
4. x²+1

Answer: B — 2x/(x²+1) (chain rule: d/dx[ln u] = u'/u with u = x²+1)

d/dx[ln u] = u'/u. u = x²+1, u' = 2x. Result: 2x/(x²+1). (A) Forgot the chain rule. (C,D) Misapplied.

L'Hôpital's rule applies when a limit takes the indeterminate form:

1. 0/0 or ∞/∞ — then take the limit of f'/g'
2. Any nonzero ratio
3. Only 0/0
4. Only ∞/∞

Answer: A — 0/0 or ∞/∞ — then take the limit of f'/g'

A) Both classic indeterminate cases. B) Not indeterminate. C/D) Too narrow.

The polar equation r = 2 cos θ represents:

1. A line
2. A parabola
3. An ellipse
4. A circle of radius 1 centered at (1, 0) in Cartesian coordinates

Answer: D — A circle of radius 1 centered at (1, 0) in Cartesian coordinates

D) Standard polar-to-Cartesian conversion. A/B/C) Each is incorrect.

∫₀¹ x · ln(x) dx (taking the limit at 0):

1. Always diverges
2. Equals 1
3. Diverges to infinity
4. Converges to −1/4 — computable via integration by parts

Answer: D — Converges to −1/4 — computable via integration by parts

D) Standard improper integral. A/B/C) Each is incorrect.

If position s(t) = t³ − 6t² + 9t, when is the particle at rest?

1. t = 0 only
2. Never
3. t = 2 only
4. t = 1 and t = 3 (v = s' = 3t² − 12t + 9 = 3(t−1)(t−3) = 0)

Answer: D — t = 1 and t = 3 (v = s' = 3t² − 12t + 9 = 3(t−1)(t−3) = 0)

Velocity v(t) = s'(t) = 3t² − 12t + 9. Factor: 3(t² − 4t + 3) = 3(t−1)(t−3). Zero at t = 1 and t = 3. Common kinematics question. AP commonly asks about direction changes too (sign of v).

Common error: forgetting to test endpoints of interval of convergence?

Power series may converge or diverge at endpoints — must check separately to determine interval (closed, half-open, open).

Geometric series sum (|r| < 1)?

S = a₁/(1 − r). Diverges if |r| ≥ 1.

FTC Part 2?

If g(x) = ∫_a^x f(t) dt, then g'(x) = f(x).

Find dy/dx if y = ln(sin x).

(1/sin x)·cos x = cot x.

Solve dy/dx = y² with y(0) = 1.

Separable: −1/y = x + C; y(0) = 1 → C = −1; y = 1/(1−x).

Common BC FRQ topic: 'particle moves on parametric curve…'?

Asks for speed, distance traveled (∫|v(t)|dt), positions, slope of path (dy/dx), etc.