AP Calculus AB/BC — Master Problem Set

Unit 1 · AB & BC

Limits & Continuity

Limit Laws: If both limits exist, sums/products/quotients of limits obey arithmetic.

$\lim_{x\to a}[f(x)\pm g(x)] = L \pm M$

L'Hôpital's Rule: For $0/0$ or $\infty/\infty$ indeterminate forms only.

$\lim_{x\to a}\dfrac{f(x)}{g(x)}=\lim_{x\to a}\dfrac{f'(x)}{g'(x)}$

Squeeze Theorem: $g(x)\le f(x)\le h(x)$ near $a$ and $g,h\to L$ $\Rightarrow$ $f\to L$.

Continuity requires: $f(a)$ exists, $\lim_{x\to a}f(x)$ exists, and both are equal.

Quick Example

Evaluate $\displaystyle\lim_{x\to 0}\dfrac{\sin x}{x}$.

Answer: 1 (famous standard limit — memorize this!)

Unit 2 · AB & BC

Differentiation: Definition & Rules

Definition: $f'(x)=\displaystyle\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}$

Power Rule:

$\dfrac{d}{dx}[x^n]=nx^{n-1}$

Product Rule: $(uv)'=u'v+uv'$

Quotient Rule: $\left(\dfrac{u}{v}\right)'=\dfrac{u'v - uv'}{v^2}$

Chain Rule: $[f(g(x))]'=f'(g(x))\cdot g'(x)$

Key trig derivatives: $(\sin x)'=\cos x$, $(\cos x)'=-\sin x$, $(\tan x)'=\sec^2 x$

Quick Example

Find $\dfrac{d}{dx}[\sin(x^2)]$.

Chain rule: $\cos(x^2)\cdot 2x = 2x\cos(x^2)$

Unit 3 · AB & BC

Applications of Derivatives

Mean Value Theorem: If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then $\exists c$ such that $f'(c)=\dfrac{f(b)-f(a)}{b-a}$.

Critical Points: $f'(c)=0$ or $f'(c)$ undefined. Use 1st or 2nd derivative test.

Concavity: $f''>0$ → concave up; $f''<0$ → concave down. Inflection point where $f''$ changes sign.

Related Rates: Differentiate an equation with respect to $t$ implicitly.

Quick Example

$f(x)=x^3-3x$. Find local extrema.

$f'(x)=3x^2-3=3(x-1)(x+1)$. Critical pts: $x=\pm1$. Local max at $x=-1$, local min at $x=1$.

Unit 4 · AB & BC

Integration: Definition & Techniques

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

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

$u$-substitution: Replace inner function; $du$ absorbs the chain-rule factor.

Integration by Parts (BC):

$\int u\,dv = uv - \int v\,du$

Key antiderivatives: $\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C$ ($n\ne-1$); $\int\dfrac{1}{x}\,dx=\ln|x|+C$

Quick Example

Find $\displaystyle\int_0^1 2x\,e^{x^2}\,dx$.

Let $u=x^2$, $du=2x\,dx$. Integral $=\left[e^u\right]_0^1 = e^1-e^0 = e-1$.

Unit 5 · AB & BC

Applications of Integration

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

Disk/Washer Method (about $x$-axis):

$V=\pi\displaystyle\int_a^b\left[R(x)^2-r(x)^2\right]dx$

Average value: $f_{\text{avg}}=\dfrac{1}{b-a}\displaystyle\int_a^b f(x)\,dx$

Accumulation: Net change $=\displaystyle\int_a^b f'(x)\,dx = f(b)-f(a)$

Quick Example

Average value of $f(x)=x^2$ on $[0,3]$.

$\dfrac{1}{3}\displaystyle\int_0^3 x^2\,dx = \dfrac{1}{3}\cdot\dfrac{27}{3}=3$

Unit 6 · BC Only

Differential Equations & Parametric/Polar

Separable DE: Separate variables, integrate both sides.

$\dfrac{dy}{dx}=ky \;\Rightarrow\; y=Ce^{kx}$

Euler's Method: $y_{n+1}=y_n + h\cdot f'(x_n,y_n)$

Parametric arc length (BC):

$L=\displaystyle\int_a^b\sqrt{\left(\tfrac{dx}{dt}\right)^2+\left(\tfrac{dy}{dt}\right)^2}\,dt$

Polar area (BC):

$A=\dfrac{1}{2}\displaystyle\int_\alpha^\beta r^2\,d\theta$

Quick Example

Solve $\dfrac{dy}{dx}=2y$, $y(0)=3$.

$y=3e^{2x}$

Unit 7 · BC Only

Infinite Series

Geometric series: $\displaystyle\sum_{n=0}^\infty ar^n = \dfrac{a}{1-r}$, $|r|<1$

Ratio Test: Converges if $\displaystyle\lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|<1$

Taylor Series:

$f(x)=\displaystyle\sum_{n=0}^\infty\dfrac{f^{(n)}(a)}{n!}(x-a)^n$

Key Maclaurin series (must memorize):
$e^x=\displaystyle\sum_{n=0}^\infty\dfrac{x^n}{n!}$, $\sin x=\displaystyle\sum_{n=0}^\infty\dfrac{(-1)^n x^{2n+1}}{(2n+1)!}$, $\cos x=\displaystyle\sum_{n=0}^\infty\dfrac{(-1)^n x^{2n}}{(2n)!}$

Alternating Series Error: $|$error$|\le |a_{n+1}|$ (next unused term)

Quick Example

Sum of $\displaystyle\sum_{n=0}^\infty \left(\dfrac{1}{3}\right)^n$.

Geometric: $a=1$, $r=\tfrac{1}{3}$. Sum $=\dfrac{1}{1-1/3}=\dfrac{3}{2}$.

AP Calculus
AB / BC

Core Concepts & Key Formulas

AP CalculusAB / BC

Core Concepts & Key Formulas

AP Calculus
AB / BC