1 · Limits & Continuity
A function $f$ is continuous at $x=a$ if $\lim_{x\to a}f(x)=f(a)$.
L'Hôpital's Rule: if $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then
$\displaystyle\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}$.
2 · Derivatives — Core Rules
Power rule: $\dfrac{d}{dx}[x^n]=nx^{n-1}$ | Product: $(uv)'=u'v+uv'$
Quotient: $\left(\dfrac{u}{v}\right)'=\dfrac{u'v-uv'}{v^2}$
Chain: $\dfrac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)$
3 · Derivatives — Trig & Exponential
$(\sin x)'=\cos x$ | $(\cos x)'=-\sin x$ | $(\tan x)'=\sec^2 x$
$(e^x)'=e^x$ | $(\ln x)'=\dfrac{1}{x}$ | $(a^x)'=a^x\ln a$
4 · Mean Value & Rolle's Theorem
MVT: If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then
$\exists\, c$ s.t. $f'(c)=\dfrac{f(b)-f(a)}{b-a}$.
Rolle's: Same conditions + $f(a)=f(b)$ $\Rightarrow$ $f'(c)=0$.
5 · Definite Integral & FTC
FTC I: $\dfrac{d}{dx}\int_a^x f(t)\,dt = f(x)$
FTC II: $\int_a^b f(x)\,dx = F(b)-F(a)$
Common: $\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C,\quad n\neq -1$
6 · Techniques of Integration
$u$-substitution: $\int f(g(x))g'(x)\,dx=\int f(u)\,du$
Integration by parts: $\int u\,dv = uv - \int v\,du$
Partial fractions for rational functions.
7 · Differential Equations
Separable: $\dfrac{dy}{dx}=f(x)g(y)$ → $\dfrac{dy}{g(y)}=f(x)\,dx$
Logistic: $\dfrac{dP}{dt}=kP\!\left(1-\dfrac{P}{M}\right)$, carrying capacity $M$.
Euler's method: $y_{n+1}=y_n+h\cdot f(x_n,y_n)$.
8 · Parametric & Polar (BC)
$\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}$ | Arc length: $\displaystyle L=\int_a^b\sqrt{\!\left(\tfrac{dx}{dt}\right)^2\!+\!\left(\tfrac{dy}{dt}\right)^2}\,dt$
Polar area: $A=\dfrac{1}{2}\displaystyle\int_\alpha^\beta r^2\,d\theta$
9 · Infinite Series (BC)
Taylor: $f(x)=\displaystyle\sum_{n=0}^{\infty}\dfrac{f^{(n)}(a)}{n!}(x-a)^n$
Maclaurin for $e^x$: $\displaystyle\sum_{n=0}^{\infty}\dfrac{x^n}{n!}$
Ratio test: converges if $\displaystyle\lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|<1$.
10 · Volumes of Revolution
Disk/Washer: $V=\pi\displaystyle\int_a^b[f(x)]^2\,dx$ or $\pi\displaystyle\int_a^b\left([R(x)]^2-[r(x)]^2\right)dx$
Shell: $V=2\pi\displaystyle\int_a^b x\,f(x)\,dx$
Example 1 · Limit with L'Hôpital
Find $\displaystyle\lim_{x\to 0}\dfrac{\sin x}{x}$.
Both numerator and denominator → 0. Apply L'Hôpital: $\dfrac{\cos x}{1}\Big|_{x=0}=\mathbf{1}$.
Example 3 · FTC I
Find $\dfrac{d}{dx}\displaystyle\int_1^x \ln(t^2+1)\,dt$.
By FTC I: $\mathbf{\ln(x^2+1)}$
Example 4 · Taylor Series
Write the first three non-zero terms of the Maclaurin series for $e^{-x}$.
$e^{-x}=1-x+\dfrac{x^2}{2!}-\cdots = \mathbf{1-x+\dfrac{x^2}{2}}$