$$\lim_{x \to c} f(x)=L\iff \forall\epsilon>0,\exists \delta>0~s.t.~0<|x-c|<\delta\longrightarrow|f(x)-L|<\epsilon$$

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

$$\text{Constant Rule:}~~\frac{d}{dx}c=0$$ $$\text{Constant Multiple Rule:}~~\frac{d}{dx}[cf(x)]=cf^{'}(x)$$ $$\text{Power Rule:}~~\frac{d}{dx}x^n=nx^{n-1}$$ $$\text{Sum & Difference Rule:}~~\frac{d}{dx}[f(x)\pm g(x)]=f^{'}(x)\pm g^{'}(x)$$ $$\text{Product Rule:}~~\frac{d}{dx}[f(x)g(x)]=f^{'}(x)g(x)+f(x)g^{'}(x)$$ $$\text{Quotient Rule:}~~\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{f^{'}(x)g(x)-f(x)g^{'}(x)}{[g(x)]^2}$$ $$\text{Chain Rule:}~~\frac{d}{dx}[f(g(x))]=f^{'}(g(x))g^{'}(x)$$

$$\sum_{i=1}^{n}1=n$$ $$\sum_{i=1}^{n}i=\frac{n(n+1)}{2}$$ $$\sum_{i=1}^{n}i^2=\frac{n(n+1)(2n+1)}{6}$$ $$\sum_{i=1}^{n}i^3=\left [ \frac{n(n+1)}{2} \right ]^2$$

$$\int_{a}^{b}f(x)dx=\lim_{n\to\infty}\sum_{i=1}^{n}f(x_i^*)\Delta x$$ $$\text{where:}~~\Delta x=\frac{b-a}{n}$$

$$\text{Part 1:}~~F(x)=\int_{a}^{x}f(t)dt$$ $$\text{Part 2:}~\int_{a}^{b}f(x)dx=F(b)-F(a)$$

$$\frac{d }{dx}\int_{g(x)}^{h(x)}f(t)dt=f(h(x))h'(x)-f(g(x))g'(x)$$

$$\frac{\mathrm{d} }{\mathrm{d} x}\sin x=\cos x$$ $$\frac{\mathrm{d} }{\mathrm{d} x}\cos x=-\sin x$$ $$\frac{\mathrm{d} }{\mathrm{d} x}\tan x=\sec^2x$$ $$\frac{\mathrm{d} }{\mathrm{d} x}\cot x=-\csc^2x$$ $$\frac{\mathrm{d} }{\mathrm{d} x}\sec x=\sec x\tan x$$ $$\frac{\mathrm{d} }{\mathrm{d} x}\csc x=-\csc x\cot x$$

$$\frac{\mathrm{d} }{\mathrm{d} x}\sin^{-1}x=\frac{1}{\sqrt{1-x^2}}$$ $$\frac{\mathrm{d} }{\mathrm{d} x}\cos^{-1}x=-\frac{1}{\sqrt{1-x^2}}$$ $$\frac{\mathrm{d} }{\mathrm{d} x}\tan^{-1}x=\frac{1}{{1+x^2}}$$ $$\frac{\mathrm{d} }{\mathrm{d} x}\cot^{-1}x=-\frac{1}{{1+x^2}}$$ $$\frac{\mathrm{d} }{\mathrm{d} x}\sec^{-1}x=\frac{1}{x\sqrt{x^2-1}}$$ $$\frac{\mathrm{d} }{\mathrm{d} x}\csc^{-1}x=-\frac{1}{x\sqrt{x^2-1}}$$

$$\int_{}^{}\sin{x}~dx=-\cos{x}+C$$ $$\int_{}^{}\cos{x}~dx=\sin{x}+C$$ $$\int_{}^{}\tan{x}~dx=\ln|\sec{x}|+C$$ $$\int_{}^{}\cot{x}~dx=\ln|\sin{x}|+C$$ $$\int_{}^{}\sec{x}~dx=\ln|\sec{x}+\tan{x}|+C$$ $$\int_{}^{}\csc{x}~dx=-\ln|\csc{x}+\cot{x}|+C$$

$$\int_{}^{}udv=uv-\int_{}^{}vdu$$

$$L=\int_{a}^{b}|r^{'}(t)|dt$$

$$\text{Single Variable Parametric:}~~\frac{\partial z}{\partial t}=\frac{\partial z}{\partial x}\cdot\frac{dx}{dt}+\frac{\partial z}{\partial y}\cdot\frac{dy}{dt}$$ $$\text{Multi-Variable Parametric:}~~\frac{\partial z}{\partial t}=\frac{\partial z}{\partial x}\cdot\frac{\partial x}{\partial t}+\frac{\partial z}{\partial y}\cdot\frac{\partial y}{\partial t}$$

$$\nabla f=\left\langle\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right\rangle$$

$$D_{\vec u}=\nabla f\cdot \hat u$$

$$\vec n\cdot\left\langle x-x_0,y-y_0,z-z_0\right\rangle=0$$ $$z-z_0=f_x (x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$$ $$z=f_x (x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)+f(x_0,y_0)$$

$$S=\int_{}^{}\int_{}^{}\sqrt{1+\left(\frac{\partial z}{\partial x}\right)^2+\left(\frac{\partial z}{\partial y}\right)^2}dA$$

$$V=\int_{}^{}\int_{}^{}f(x,y)dA$$

$$dV=dxdydz=rdrd\theta=\rho^2\sin{\phi}d\rho d\theta d\phi$$

$$\oint_{}^{}f(x,y)ds=\int_{a}^{b}f(x(t),y(t))|r'(t)|dt$$ $$\int_{}^{}F\cdot dr=\int_{a}^{b}F(r(t))\cdot r'(t)dt$$

$$\oint_{}^{}F\cdot dr=\oint_{}^{}fdx+gdy=\int_{}^{}\int_{}^{}\left(\frac{\partial g}{\partial x}-\frac{\partial f}{\partial y}\right)dA$$