Integrals

Calculus in the ancient times: Method of exhaustion (see wikipedia) User-friendly definition of the integral: $latex lim_{N\to\infty}\frac{b-a}{N}\sum_{i=1}^N f(x_i)$ It would be better if the integral would not depend on the definition of the bins. In the limit $latex N\to\infty$ it doesn't matter if they are evenly spaced or not: $latex lim_{N\to\infty}\sum_{i=1}^N f(x_i)dx_i$ Fundamental theorem of …

Even and Odd Functions

Even functions: $latex f(-x) = f(x)$ e.g. $latex \cos\theta $ Odd functions: $latex f(-x) = -f(x)$ e.g. $latex \sin\theta $ See also wikipedia This explains why in the previous post: $latex D\sin (x^2-1)=2x\cos(1-x^2)=2x\cos(x^2-1)$ Find out what this function looks like using Wolfram|Alpha (click this link)!

Derivatives

$latex Df(x)=f'(x)=\frac{df}{dx}=\frac{d}{dx}f=\dot{f}(x) = \lim_{h \to 0}\frac{f(x+h)-f(x)}{h} $ Introduced in order to fix the idea of velocity: $latex v = \lim_{\Delta t \to 0}\frac{\Delta s}{\Delta t}=\frac{ds}{dt} $ See: Zeno's paradoxes Working example: $latex s(t) = At^3 + Bt + C$ $latex s + \Delta s = A ( t + \Delta t )^3 + B ( …