AP Calculus AB/BC

\(\displaystyle\lim_{x\to a}f(x)=L\) iff both one-sided limits equal \(L\)
L'Hôpital: \(\displaystyle\lim\frac{f}{g}\overset{0/0}{=}\lim\frac{f'}{g'}\)
Squeeze: \(g\le f\le h,\;\lim g=\lim h=L\Rightarrow\lim f=L\)

Power: \((x^n)'=nx^{n-1}\)
Chain: \([f(g(x))]'=f'(g(x))\cdot g'(x)\)
Product: \((uv)'=u'v+uv'\)
Quotient: \(\left(\tfrac{u}{v}\right)'=\tfrac{u'v-uv'}{v^2}\)

Critical pts: \(f'(c)=0\) or undefined
Concave up: \(f''>0\); Inflection: \(f''\) changes sign
Mean Value Thm: \(f'(c)=\dfrac{f(b)-f(a)}{b-a}\)

FTC I: \(\displaystyle\frac{d}{dx}\int_a^x f(t)\,dt=f(x)\)
FTC II: \(\displaystyle\int_a^b f(x)\,dx=F(b)-F(a)\)
\(u\)-sub: \(\displaystyle\int f(g(x))g'(x)\,dx=\int f(u)\,du\)

Ratio Test: \(L=\lim\left|\dfrac{a_{n+1}}{a_n}\right|\); conv. if \(L<1\)
Taylor: \(\displaystyle f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n\)
\(e^x=\sum\tfrac{x^n}{n!},\;\sin x=\sum\tfrac{(-1)^n x^{2n+1}}{(2n+1)!}\)

\(\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}\)
Arc length (param): \(\displaystyle\int_a^b\!\sqrt{\!\left(\tfrac{dx}{dt}\right)^2\!+\!\left(\tfrac{dy}{dt}\right)^2}\,dt\)
Polar area: \(\displaystyle A=\tfrac{1}{2}\int_\alpha^\beta r^2\,d\theta\)