AP Calculus BC 3-Day Cram Plan

You have three days until the AP Calculus BC exam. This is your moment to drill the highest-frequency topics and lock in the BC-specific patterns that appear in nearly every exam: series convergence, parametric motion, and polar integration.

This plan assumes ~4 focused hours per day. Do not skip topics; shorten the practice sets instead. BC has 45 MCQs (60% AB, 40% BC-only) and 6 FRQs (at least 2 are BC-heavy). You need both foundations and BC mastery.

Day 1: AB Review + Integration Techniques (4 hrs)

Start with the fundamentals that underpin everything in BC.

What to review (90 min)

• Limits & derivatives: limits at infinity, product/quotient/chain rules, implicit differentiation, derivatives of $e^x$, $\ln x$, inverse trig.
• Critical points & concavity: sign charts, first/second derivative tests, MVT hypotheses.
• Riemann sums & FTC: $\frac{d}{dx} \int_a^{g(x)} f(t) \, dt = f(g(x)) \cdot g'(x)$ (don't forget the chain rule).
• Integration by parts: $\int u \, dv = uv - \int v \, du$. Memorize the LIATE priority.
• Partial fractions: decompose $\frac{P(x)}{(x-a)(x-b)}$ and integrate term-by-term.
• Improper integrals: $\int_a^{\infty} f(x) \, dx = \lim_{t \to \infty} \int_a^t f(x) \, dx$. Know when they converge.

What to practice (2.5 hrs)

• 15 mixed MCQs on derivatives and AB-level integrals.
• 5 integration-by-parts problems (at least 2 with trigonometric products).
• 3 partial fractions decompositions.

💡 Highest leverage: Integration by parts is tested almost every year in at least one FRQ. Drill $\int x e^x \, dx$ and $\int x \sin x \, dx$ until your hands do it automatically.

Day 2: Parametric, Polar, and Vector Functions (4 hrs)

These two areas account for ~30% of all BC MCQs and guarantee at least one FRQ.

What to review (90 min)

• Parametric functions: $x = f(t)$, $y = g(t)$. Key formulas:
  • $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$
  • Speed: $\sqrt{(dx/dt)^2 + (dy/dt)^2}$
  • Arc length: $\int_a^b \sqrt{(dx/dt)^2 + (dy/dt)^2} \, dt$
• Polar curves $r = f(\theta)$:
  • Area in polar: $A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta$
  • Arc length: $L = \int_{\alpha}^{\beta} \sqrt{r^2 + (dr/d\theta)^2} \, d\theta$
• Vector functions: $\vec{r}(t) = \langle x(t), y(t) \rangle$. Speed, acceleration, velocity analysis.
• Particle motion in 2D: position from velocity, velocity from acceleration, displacement vs distance.

What to practice (2.5 hrs)

• 1 parametric FRQ (find $dy/dx$ at a point, then speed or arc length).
• 1 polar FRQ (find area enclosed by a curve or between two curves).
• 20 mixed MCQs on parametric/polar/vector topics (calculator allowed).

⚠️ FRQ trap: When asked for speed, never write $\sqrt{v_x + v_y}$. Write $\sqrt{(dx/dt)^2 + (dy/dt)^2}$ explicitly. Show the components.

Day 3: Series, Convergence Tests, and Full FRQ Drills (4 hrs)

Series convergence is the most heavily tested BC-only topic on both MCQ and FRQ.

What to review (90 min)

• Convergence tests (know the names and when to use):
  • Geometric series: $\sum ar^n$ converges iff $|r| < 1$ to $\frac{a}{1-r}$.
  • p-series: $\sum \frac{1}{n^p}$ converges iff $p > 1$.
  • Integral test: $\sum a_n$ and $\int a_x \, dx$ converge/diverge together.
  • Comparison & limit comparison tests: compare to known series.
  • Ratio test: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$. Converges if $L < 1$, diverges if $L > 1$.
  • Alternating series test: if $a_n$ decreasing and $\lim a_n = 0$, then $\sum (-1)^n a_n$ converges.
• Taylor/Maclaurin series: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n$.
  • Memorize: $e^x = \sum \frac{x^n}{n!}$, $\sin x = \sum \frac{(-1)^n x^{2n+1}}{(2n+1)!}$, $\cos x = \sum \frac{(-1)^n x^{2n}}{(2n)!}$, $\frac{1}{1-x} = \sum x^n$.
• Lagrange error bound: $|R_n(x)| \leq \frac{M |x-a|^{n+1}}{(n+1)!}$ where $M = \max |f^{(n+1)}|$ on the interval.
• Interval of convergence: find where series converges; test endpoints separately.

What to practice (2.5 hrs — full timed set)

• 1 full FRQ on Taylor polynomial and Lagrange error.
• 1 full FRQ on interval of convergence with endpoint testing.
• 25 mixed MCQs (calculator + non-calculator), strictly timed.

The night before

Skim our last-minute review checklist. Get 8 hours of sleep — your brain consolidates short-term memory overnight, and fatigue kills accuracy on series tests.

Calculator must-knows

On the calculator section, these functions earn easy points:

1. nDeriv(f(x), x, a) — numerical derivative (useful for rate of change in parametric problems).
2. fnInt(f(x), x, a, b) — definite integral (for arc length, polar area, series approximations).
3. Graph + intersect — find where curves meet (polar/parametric intersection problems).
4. Solve $f(x) = 0$ — find bounds for area integrals.

Common point-leaks

• Forgetting the "$\frac{1}{2}$" in polar area formula: $A = \frac{1}{2} \int r^2 \, d\theta$, not $\int r^2 \, d\theta$.
• Writing $\sqrt{v_x + v_y}$ instead of $\sqrt{v_x^2 + v_y^2}$ for speed.
• Missing "test endpoints" when finding interval of convergence (IOC must include endpoint analysis).
• Confusing the ratio test: $L < 1$ ⟹ converge; $L > 1$ ⟹ diverge; $L = 1$ ⟹ inconclusive.
• Forgetting the $\frac{1}{n!}$ in Taylor series or dropping the alternating sign.
• Not citing which convergence test you used (College Board demands it on FRQs).

What this 3-day plan deliberately skips

Euler's method, logistic differential equations, and some advanced integration techniques (like reduction formulas) will not receive full coverage. If tested: accept 2-3 lost points and focus on series + parametric instead.

Ready to start?

Open the AP Calculus BC topic library → and attack Day 1 topic by topic. Mix in 3-5 worked examples per topic. You've prepared for this — now lock it in. 🎯