AP Calculus AB/BC — Master Problem Set

Unit 1 · Limits & Continuity

L'Hôpital's Rule: If \(\frac{f(x)}{g(x)} \to \frac{0}{0}\) or \(\frac{\infty}{\infty}\), then:

\(\displaystyle\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}\)

Continuity at \(x=a\): (1) \(f(a)\) exists, (2) \(\lim_{x\to a}f(x)\) exists, (3) \(\lim_{x\to a}f(x)=f(a)\).

\(\lim_{x\to 0}\dfrac{\sin x}{x}=1\)

Unit 2 · Derivatives — Definition & Rules

Limit definition:

\(f'(x)=\displaystyle\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\)

Chain Rule: \(\dfrac{d}{dx}[f(g(x))] = f'(g(x))\cdot g'(x)\)

Implicit Differentiation: Differentiate both sides w.r.t. \(x\); treat \(y\) as a function of \(x\), so \(\dfrac{d}{dx}[y^n]=ny^{n-1}\dfrac{dy}{dx}\).

Unit 3 · Applications of Derivatives

Mean Value Theorem: If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), there exists \(c\in(a,b)\) such that:

\(f'(c)=\dfrac{f(b)-f(a)}{b-a}\)

Related Rates: Differentiate an equation relating quantities with respect to time \(t\). Use the chain rule.

Unit 4 · Integrals & FTC

FTC Part 1: \(\dfrac{d}{dx}\!\displaystyle\int_a^x f(t)\,dt = f(x)\)

FTC Part 2: \(\displaystyle\int_a^b f(x)\,dx = F(b)-F(a)\)

\(\int x^n\,dx = \dfrac{x^{n+1}}{n+1}+C,\quad n\neq -1\)

Unit 5 · Applications of Integrals

Area between curves: \(\displaystyle\int_a^b[f(x)-g(x)]\,dx\) where \(f(x)\ge g(x)\).

Disk Method (about x-axis):

\(V=\pi\displaystyle\int_a^b[f(x)]^2\,dx\)

Unit 6 · BC — Advanced Integration

Integration by Parts: \(\displaystyle\int u\,dv = uv - \int v\,du\)

Partial Fractions: Decompose rational functions into simpler fractions before integrating.

\(\int\frac{1}{x^2-1}\,dx = \frac{1}{2}\ln\!\left|\frac{x-1}{x+1}\right|+C\)

Unit 7 · BC — Polar & Parametric

Polar area:

\(A=\dfrac{1}{2}\displaystyle\int_\alpha^\beta [r(\theta)]^2\,d\theta\)

Unit 8 · BC — Sequences & Series

p-Series: \(\displaystyle\sum_{n=1}^\infty\frac{1}{n^p}\) converges if \(p>1\), diverges if \(p\le 1\).

Taylor series for \(\sin x\):

\(\sin x = x - \dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\cdots\)

Taylor series for \(e^x\):

\(e^x = \displaystyle\sum_{n=0}^\infty\frac{x^n}{n!} = 1+x+\frac{x^2}{2!}+\cdots\)

Unit 9 · Differential Equations

Separable DE: Separate variables, integrate both sides.

Exponential model: \(\dfrac{dy}{dt}=ky \;\Rightarrow\; y=Ce^{kt}\)

Logistic growth:

\(\dfrac{dP}{dt}=kP\!\left(1-\dfrac{P}{M}\right)\)

As \(t\to\infty\), \(P\to M\) (carrying capacity).

Core Concepts & Key Formulas