3.7. calculus¶

3.7.1. taylor expansion¶

  • taylor expansion: \(f(x) \approx f(x_0) + \frac{f'(x_0)}{1!}(x-x_0) + \frac{f''(x_0)}{2!}(x-x_0)^2 + ...\)

3.7.2. 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: \(a_{1st}\frac{1-r^{n+1}}{1-r}\)

3.7.3. 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 \(\nabla L = 0\)