AP Calculus BC 7-Day Cram Plan

You have seven days to prepare. This plan spreads topics across the week while building momentum toward a full mock exam on Day 7. Aim for 5-6 hours per day, splitting between review and practice.

The 7-day schedule

Day	Focus	Topics	Practice	Time
Mon	Limits & Derivatives (AB foundation)	Limits at ∞, chain rule, implicit diff	20 MCQ + 2 mini-FRQ	5 hrs
Tue	Applications of Derivatives	Optimization, related rates, curve sketching	15 MCQ + 1 full FRQ on optimization	5 hrs
Wed	Integration (AB + techniques)	$u$ -sub, parts, partial fractions, improper integrals	20 MCQ + 3 integration-by-parts drills	5.5 hrs
Thu	Parametric, Polar, Vectors	Motion, $dy/dx$ , speed, arc length, polar area	15 MCQ + 1 parametric FRQ + 1 polar FRQ	5.5 hrs
Fri	Series & Convergence Tests	Geometric, $p$ -series, ratio test, alternating test	18 MCQ + convergence analysis drills	5 hrs
Sat	Taylor Series & Power Series	Maclaurin, Lagrange error, interval of convergence	12 MCQ + 2 series FRQ templates	5 hrs
Sun	Full Mock Exam (45 MCQ + 6 FRQ, timed)	All topics, mixed difficulty	Simulate exam conditions, 3-hour exam	4 hrs

Monday: Limits & Derivatives

Concepts to drill: limits from graphs, algebraic limit manipulation, L'Hôpital's rule, continuity, chain rule, implicit differentiation.

Key formulas:

$\lim_{x \to \infty} \frac{ax^m + \ldots}{bx^n + \ldots} = \frac{a}{b}$ if $m = n$ ; 0 if $m < n$ ; ∞ if $m > n$ .
$(f(g(x)))' = f'(g(x)) \cdot g'(x)$ .
Implicit: differentiate both sides, collect $\frac{dy}{dx}$ terms, solve.

💡 Tip: Implicit differentiation appears on ~2-3 MCQs every year. Practice it until it's muscle memory.

Tuesday: Applications of Derivatives

Concepts to drill: critical points, optimization setup, related rates, concavity, second derivative test.

Optimization template:

Define variables and identify the objective (what are we maximizing/minimizing?).
Write constraint equation(s).
Express objective in terms of one variable.
Take derivative, set = 0, solve.
Justify it's a max/min using 1st or 2nd Derivative Test.

⚠️ FRQ trap: "It's a max because the graph shows it" earns zero justification points. Write the sign change or second derivative test.

Wednesday: Integration Techniques

Technique	Form	Formula	Example
$u$ -substitution	$\int f(g(x)) g'(x) \, dx$	Let $u = g(x)$	$\int 2x e^{x^2} dx = e^{x^2} + C$
By parts	$\int u \, dv$	$uv - \int v \, du$	$\int x \cos x \, dx = x \sin x + \cos x + C$
Partial fractions	$\frac{P}{(x-a)(x-b)}$	$\frac{A}{x-a} + \frac{B}{x-b}$	$\frac{1}{(x-1)(x+2)} = \frac{1/3}{x-1} - \frac{1/3}{x+2}$
Improper integrals	$\int_a^{\infty} f(x) \, dx$	$\lim_{t \to \infty} \int_a^t f(x) \, dx$	$\int_1^{\infty} \frac{1}{x^2} dx = 1$ (converges)

Wednesday evening: Complete 3 integration-by-parts problems with products of polynomials and trig functions.

Thursday: Parametric, Polar, Vectors

Parametric motion:

Velocity: $\vec{v}(t) = \langle \frac{dx}{dt}, \frac{dy}{dt} \rangle$ .
Speed: $|\vec{v}(t)| = \sqrt{(dx/dt)^2 + (dy/dt)^2}$ .
Arc length: $\int_a^b \sqrt{(dx/dt)^2 + (dy/dt)^2} \, dt$ .
$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ .

Polar curves $r = f(\theta)$ :

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

⚠️ FRQ trap: When writing parametric arc length, always show the integrand $\sqrt{(dx/dt)^2 + (dy/dt)^2}$ before evaluating. Partial credit requires the setup.

Friday: Series & Convergence Tests

Test decision tree:

Geometric: $\sum ar^n$ ⟹ converges iff $|r| < 1$ .
$p$ -series: $\sum \frac{1}{n^p}$ ⟹ converges iff $p > 1$ .
Ratio test: compute $L = \lim \left| \frac{a_{n+1}}{a_n} \right|$ . Converges if $L < 1$ , diverges if $L > 1$ .
Alternating: $\sum (-1)^n a_n$ where $a_n$ decreasing and $\lim a_n = 0$ ⟹ converges.
Integral test: $\sum a_n$ and $\int_1^{\infty} a_x \, dx$ behave the same.

Friday task: Given 6 random series, determine convergence and state which test you used.

Saturday: Taylor & Maclaurin Series

Series	Formula
$e^x$	$\sum_{n=0}^{\infty} \frac{x^n}{n!}$
$\sin x$	$\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$
$\cos x$	$\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$
$\frac{1}{1-x}$	$\sum_{n=0}^{\infty} x^n$ (for $\\|x\\| < 1$ )
$\ln(1+x)$	$\sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n}$ (for $-1 < x \leq 1$ )

Lagrange error bound: $\|R_n(x)\| \leq \frac{M \|x-a\|^{n+1}}{(n+1)!}$ where $M = \max \|f^{(n+1)}\|$ .

Saturday task:

Write a Taylor polynomial about $x = a$ from the derivative formula.
Find the interval of convergence, testing endpoints.
Estimate Lagrange error for a specific value.

Sunday: Full Mock Exam

Simulate real exam conditions: 3 hours, 45 MCQ + 6 FRQ, no interruptions.

First 90 min: 28 non-calculator MCQs (aim for 85%+).
Next 50 min: 17 calculator MCQs.
Last 90 min: 6 FRQs (Part A 30 min with no calc, Part B 60 min with calc).

Scoring: 40 MCQ points + 60 FRQ points = 108 total. Aim for 70+.

Resources: Review the AP Calculus BC topic library → or revisit FRQ patterns → before each daily section. Good luck. 🎯