calculus

# taylor expansion

• taylor expansion: $f(x) \approx f(x_0) + \frac{f’(x_0)}{1!}(x-x_0) + \frac{f’‘(x_0)}{2!}(x-x_0)^2 + …$

# Single-variable calculus

Derivatives:

$\frac{d}{dx}x^n = nx^{n-1}$

$\frac{d}{dx}a^x = a^{x}ln(a)$

$\frac{d}{dx}ln(x) = 1/x$

$\frac{d}{dx}tan(x)= sec^2(x)$

$\frac{d}{dx}cot(x)= -csc^2(x)$

$\frac{d}{dx}sec(x)= sec(x)tan(x)$

$\frac{d}{dx}csc(x)= -csc(x)cot(x)$

$\int tan = ln|sec|$

$\int cot = ln|sin|$

$\int sec = ln|sec+tan|$

$\int csc = ln|csc-cot|$

$\int \frac{du}{\sqrt{a^2-u^2}} = sin^{-1}(\frac{u}{a})$

$\int \frac{du}{u\sqrt{u^2-a^2}} = \frac{1}{a}sec^{-1}(\frac{u}{a})$

$\int \frac{du}{a^2+u^2} = \frac{1}{a} tan^{-1}(\frac{u}{a})$

Continuous: left limit = right limit = value

Differentiable: continuous and no sharp points / asymptotes

L’Hospital’s - for indeterminate forms: $(\frac{f(x)}{g(x)})’ = \frac{f’(x)}{g’(x)}$

Integration by parts: $\int{udv}=uv-\int{duv}$, LIATE

Expansions:

$e^x = \sum{\frac{x^n}{n!}}$

$sin(x) = \sum_0^\infty{\frac{(-1)^n x^{2n+1}}{(2n+1)!}}$

$cos(x) = \sum_0^\infty{\frac{(-1)^n x^{2n}}{(2n)!}}$

Geometric Sum: $a_{1st}\frac{1-r^{n+1}}{1-r}$

# Multivariable calculus

• Polar: r,$\theta$,z
• Spherical: $\rho,\theta,\phi$
• Clairut’s Thm: Conservative function $f_{xy}=f_{yx}$
• Lagrangian - solves minimize f subject to g = c
• solution will always be tangent to f
• $\nabla f = \lambda \nabla g$ - gives us n constraints
• remember g = c is a constraint too
• to do this efficiently, define the Lagrangian $L(x, \lambda) = f - \lambda \cdot g$
• taking deriv wrt $\lambda$ and setting = 0 enforces g = c
• taking deriv wrt other variables and setting = 0 enforces other conditions
• therefore final eq just becomes $\nabla L = 0$