User:Humanist Geek/sandbox

$$ s=\int_a^b\sqrt{1+ \left [ f'(x) \right ]^2} \, dx $$

$$ \int \!u\, dv = uv - \int \!v\,du $$

$$ \sin^2 x = \frac{1-\cos 2x}{2} $$

$$ \cos^2 x = \frac{1+\cos 2x}{2} $$

$$ \frac{0}{0}~,~\frac{\infty}{\infty}~,~0 \cdot \infty~,~ 1^{\infty}~,~{\infty}^0~,~0^0~,~\infty-\infty $$

$$ \frac{dy}{dx} = \frac{\tfrac{dy}{dt}}{\tfrac{dx}{dt}}$$

$$ \frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\!\!\left[\frac{dy}{dx}\right]}{\frac{dx}{dt}}$$

$$ s = \int_a^b\!\sqrt{\left(\tfrac{dx}{dt}\right)^2 + \left(\tfrac{dy}{dt}\right)^2}\, dt$$

$$ \begin{array}{rcl} x\!\!\!&=&\!\!\!r\cos \theta \\ y\!\!\!&=&\!\!\!r\sin \theta \\ \end{array} $$

$$ \tan \theta = \frac{y}{x}$$

$$ r^2=x^2+y^2\!$$ $$ \begin{array}{lcl} ~\!\!\!\!\!&~& \tan \theta = \dfrac{y}{x} \\ ~\!\!\!\!\!&~& \\ ~\!\!\!\!\!&~& r^2 = x^2 + y^2 \\ \end{array} $$ $$ \begin{array}{lcl} ~\!\!\!\!\!&~& \tan \theta = \frac{y}{x} \\ ~\!\!\!\!\!&~& r^2 = x^2 + y^2 \\ \end{array} $$

$$ \begin{array}{lcl} ~\!\!\!\!\!&~& \tan \theta = \dfrac{y}{x} \\ ~\!\!\!\!\!&~& r^2 = x^2 + y^2 \\ \end{array} $$ $$ \begin{array}{lcl} x = r\cos \theta \\ y = r\sin \theta \\ \tan \theta = \dfrac{y}{x} \\ r^2 = x^2 + y^2 \\ \end{array} $$

$$ \frac{dy}{dx}=\frac{\tfrac{dy}{d\theta}}{\tfrac{dx}{d\theta}}=\frac{f'(\theta)\sin \theta + f(\theta)\cos \theta}{f'(\theta)\cos \theta - f(\theta)\sin \theta}$$ $$ A = \frac{1}{2}\int_\alpha^\beta \left[f(\theta) \right]^2 \, d\theta $$ $$ s = \int_\alpha^\beta \!\sqrt{\left[ f(\theta) \right]^2 + \left[ f'(\theta) \right]^2} \, d\theta $$

$$ \begin{array}{lcl} \sum a_n \text{ is absolutely convergent if} \\ \sum \left| a_n \right|\text{ converges} \\ \end{array} $$

$$ \begin{array}{lcl} \sum a_n \text{ is conditionally convergent if} \\ \sum a_n \text{ converges but }\sum \left| a_n \right|\text{ diverges} \\ \end{array} $$

$$ \sum a_n \text{ is conditionally convergent if } \sum a_n \text{ converges but } \sum \left| a_n \right|\text{ diverges}$$ $$ f(x)=\sum_{n=0} ^ {\infin } \frac {f^{\left(n\right)}(c)}{n!} \, (x-c)^{n} = f(c) + f'(c)(x-c) + \cdots + \frac {f^{(n)}(c)}{n!} \, (x-c)^{n} + \cdots $$ $$ \begin{array}{lcl} f(x) = \displaystyle \sum_{n=0} ^ {\infin } \dfrac {f^{\left(n\right)}(c)}{n!} \, (x-c)^{n} = f(c)~+\\ f'(c)(x-c) + \tfrac{f(c)}{2!} \, (x-c)^2 + \tfrac{f'(c)}{3!} \, (x-c)^3 \\ +\cdots + \tfrac {f^{(n)}(c)}{n!} \, (x-c)^{n} + \cdots \\ \end{array} $$

$$ \begin{array}{lcl} f(x) = \displaystyle \sum_{n=0} ^ {\infin } \dfrac {f^{\left(n\right)}(c)}{n!} \, (x-c)^{n} \\ = f(c) + f'(c)(x-c) + \cdots + \tfrac {f^{(n)}(c)}{n!} \, (x-c)^{n} + \cdots \\ \end{array} $$

$$ \begin{array}{lcl} f(x) = \displaystyle \sum_{n=0} ^ {\infin } \dfrac {f^{\left(n\right)}(c)}{n!} \, (x-c)^{n} \\ = f(c) + f'(c)(x-c) + \tfrac{f''(c)}{2!} \, (x-c)^2 \\ +\tfrac{f^{(3)}(c)}{3!} \, (x-c)^3 + \cdots + \tfrac {f^{(n)}(c)}{n!} \, (x-c)^{n} + \cdots \\ \end{array} $$