AP Calculus BC 7-Day Cram Plan

title: "AP Calculus BC 7-Day Cram Plan" description: "A comprehensive 7-day AP Calculus BC study schedule: daily tables covering limits, derivatives, integrals, parametric, polar, series, with mini-FRQs and a full mock exam on Day 7." date: "2026-01-15" examDate: "May AP Exam" topics:

• Daily Study Schedule
• Integration
• Parametric & Polar
• Series & Convergence
• FRQ Practice

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:

1. Define variables and identify the objective (what are we maximizing/minimizing?).
2. Write constraint equation(s).
3. Express objective in terms of one variable.
4. Take derivative, set = 0, solve.
5. 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:

1. Geometric: $\sum ar^n$ ⟹ converges iff $|r| < 1$.
2. $p$-series: $\sum \frac{1}{n^p}$ ⟹ converges iff $p > 1$.
3. Ratio test: compute $L = \lim \left| \frac{a_{n+1}}{a_n} \right|$. Converges if $L < 1$, diverges if $L > 1$.
4. Alternating: $\sum (-1)^n a_n$ where $a_n$ decreasing and $\lim a_n = 0$ ⟹ converges.
5. 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. 🎯