AP Calculus BC FRQ Practice Guide

The AP Exam gives you 6 free-response questions. Each tests predictable patterns the College Board has reused for 20+ years. Master these templates, and you'll know exactly what to write when you see an FRQ on exam day.

The 6 FRQ archetypes

Archetype 1: Rate In / Rate Out (AB-shared)

Pattern: Water flows into/out of a tank or container. You're given a rate function and asked to find accumulated water, average rate, or when the container overflows.

Template:

1. Set up accumulation integral: $\text{amount} = \int_a^b (\text{rate in} - \text{rate out}) \, dt$.
2. Evaluate the integral.
3. Interpret units (gallons, liters, cubic feet).

Worked Example:

Water flows into a tank at a rate of $(20 + t)$ gallons/hour, and drains at $15$ gallons/hour, for $0 \leq t \leq 10$. The tank initially holds $50$ gallons.

(a) Find the amount of water in the tank at $t = 10$.

Solution: $\text{Amount} = 50 + \int_0^{10} [(20 + t) - 15] \, dt = 50 + \int_0^{10} (5 + t) \, dt$ $= 50 + \left[ 5t + \frac{t^2}{2} \right]_0^{10} = 50 + (50 + 50) = 150 \text{ gallons}$

(b) Find the average rate of water flow (net) over $[0, 10]$.

Solution: $\text{Average rate} = \frac{1}{10} \int_0^{10} (5 + t) \, dt = \frac{100}{10} = 10 \text{ gallons/hour}$

Archetype 2: Particle Motion (AB-shared + BC parametric/vector)

Pattern: A particle moves along a curve. Given position, velocity, or parametric equations, find speed, direction, arc length, or displacement.

Template:

• Speed: $\sqrt{(dx/dt)^2 + (dy/dt)^2}$.
• Arc length: $\int_a^b \sqrt{(dx/dt)^2 + (dy/dt)^2} \, dt$.
• Displacement: $\int_a^b v(t) \, dt$ (can be negative).
• Distance traveled: $\int_a^b |v(t)| \, dt$ (always positive).

Worked Example (BC):

A particle moves along a curve with $x(t) = t^2$, $y(t) = t^3$, for $0 \leq t \leq 2$.

(a) Find the speed of the particle at $t = 1$.

Solution: $\frac{dx}{dt} = 2t, \quad \frac{dy}{dt} = 3t^2$ $\text{Speed} = \sqrt{(2 \cdot 1)^2 + (3 \cdot 1^2)^2} = \sqrt{4 + 9} = \sqrt{13}$

(b) Find the arc length of the curve from $t = 0$ to $t = 2$.

Solution: $L = \int_0^2 \sqrt{(2t)^2 + (3t^2)^2} \, dt = \int_0^2 \sqrt{4t^2 + 9t^4} \, dt = \int_0^2 t\sqrt{4 + 9t^2} \, dt$ Using $u = 4 + 9t^2$, $du = 18t \, dt$: $L = \frac{1}{18} \int_4^{40} \sqrt{u} \, du = \frac{1}{18} \cdot \frac{2}{3} u^{3/2} \big|_4^{40} = \frac{1}{27} [40^{3/2} - 8] \approx 9.07$

Archetype 3: Polar Curves (BC-only)

Pattern: A curve is defined in polar form $r = f(\theta)$. Find area enclosed, area between curves, or arc length.

Template:

• Area: $A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta$.
• Area between: $A = \frac{1}{2} \int_{\alpha}^{\beta} (r_{\text{outer}}^2 - r_{\text{inner}}^2) \, d\theta$.

Worked Example:

Find the area enclosed by $r = 2 + 2\cos \theta$.

Solution: The curve is a cardioid. By symmetry, it completes one loop for $0 \leq \theta \leq 2\pi$. $A = \frac{1}{2} \int_0^{2\pi} (2 + 2\cos \theta)^2 \, d\theta = \frac{1}{2} \int_0^{2\pi} (4 + 8\cos \theta + 4\cos^2 \theta) \, d\theta$ Using $\cos^2 \theta = \frac{1 + \cos 2\theta}{2}$: $A = \frac{1}{2} \left[ 4\theta + 8\sin \theta + 2(\theta + \frac{\sin 2\theta}{2}) \right]_0^{2\pi} = \frac{1}{2} \cdot 12\pi = 6\pi$

Archetype 4: Area and Volume (AB-shared)

Pattern: Find area between curves or volume by rotating about an axis.

Template:

• Area between: $\int_a^b (\text{top} - \text{bottom}) \, dx$ or $\int_c^d (\text{right} - \text{left}) \, dy$.
• Volume (disk): $V = \pi \int_a^b [R(x)]^2 \, dx$.
• Volume (washer): $V = \pi \int_a^b ([R(x)]^2 - [r(x)]^2) \, dx$.

💡 Tip: Always identify the axis of rotation. If rotating about $y = k$ (not the $x$-axis), adjust the radius: $R = |f(x) - k|$.

Archetype 5: Series and Convergence (BC-only)

Pattern: Determine if a series converges, find its sum, estimate error, or find the interval of convergence for a power series.

Template for convergence:

1. Geometric: Does it have the form $\sum ar^n$? If $|r| < 1$, it converges to $\frac{a}{1-r}$.
2. $p$-series: Does it have the form $\sum \frac{1}{n^p}$? Converges iff $p > 1$.
3. Ratio test: Compute $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$. If $L < 1$, converges; $L > 1$, diverges; $L = 1$, test inconclusive.
4. Alternating: If $(-1)^n a_n$ where $a_n$ is decreasing and $\lim a_n = 0$, then converges (but may not converge absolutely).

Template for Lagrange error: $|R_n(x)| \leq \frac{M |x - a|^{n+1}}{(n+1)!}$ where $M = \max |f^{(n+1)}(t)|$ on the interval.

Worked Example:

(a) Determine if $\sum_{n=1}^{\infty} \frac{n}{2^n}$ converges. If so, justify your test.

Solution: Use the ratio test. $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \frac{(n+1)/2^{n+1}}{n/2^n} = \lim_{n \to \infty} \frac{n+1}{2n} = \frac{1}{2} < 1$ By the Ratio Test, the series converges.

(b) Find the interval of convergence for $\sum_{n=0}^{\infty} \frac{(x-2)^n}{n+1}$.

Solution: Use the ratio test on $(x - 2)^n$. $\lim_{n \to \infty} \left| \frac{(x-2)^{n+1}}{n+2} \cdot \frac{n+1}{(x-2)^n} \right| = |x - 2| \lim_{n \to \infty} \frac{n+1}{n+2} = |x - 2|$ Converges when $|x - 2| < 1$, so $1 < x < 3$ (open interval).

Test $x = 1$: $\sum \frac{(-1)^n}{n+1}$ — alternating series, decreasing to 0, converges.

Test $x = 3$: $\sum \frac{1}{n+1} = \sum \frac{1}{n}$ (harmonic) — diverges.

IOC: $[1, 3)$.

Archetype 6: Taylor Polynomials and Power Series (BC-only)

Pattern: Write a Taylor polynomial about $x = a$, find Lagrange error bound, or use a known series to approximate a value.

Template: $P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x - a)^k$

Worked Example:

Write the 4th-degree Taylor polynomial for $f(x) = e^x$ centered at $x = 0$. Use it to approximate $e^{0.1}$ and find a bound on the error.

Solution: All derivatives of $e^x$ are $e^x$, so $f^{(k)}(0) = 1$ for all $k$. $P_4(x) = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24}$ $P_4(0.1) = 1 + 0.1 + 0.005 + 0.000167 + 0.0000042 \approx 1.1052$

Error bound: $|R_4(0.1)| \leq \frac{M(0.1)^5}{5!}$ where $M = \max |e^t|$ on $[0, 0.1]$. Since $e^t \leq e^{0.1} < 1.11$ on $[0, 0.1]$: $|R_4(0.1)| < \frac{1.11 \cdot (0.1)^5}{120} \approx 0.000000925$

⚠️ FRQ trap: Always compute the maximum of the derivative explicitly. Use calculus or endpoint analysis; don't just guess a bound.

Scoring rubric checklist

Every FRQ rubric demands:

1. Setup: Clearly state the integral, differential equation, or formula you're using.
2. Execution: Show all algebraic steps; don't jump to the answer.
3. Justification: If asked to justify, cite the theorem by name (MVT, IVT, First Derivative Test, Ratio Test, etc.).
4. Units: Always include them on applied problems.
5. Answer: Box your final answer or clearly state it in a sentence.

Ready to drill? Pick one archetype, solve 2-3 problems, then move to the next. Browse more FRQ examples →. You've got the patterns — now execute. 🎯