differential calculus

A function is said to be differentiable if it has a derivative at every point in its domain. More formally, a function $f:\mathbb{R}\to \mathbb{R}$ is differentiable at a point $a$ if the following limit exists:

$$f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}.$$

Differentiable => continuous. The converse does not hold.

mean value theorem §

If $f$ is a continuous function on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, then there exists a point $c$ in $(a,b)$ such that the tangent at $c$ is parallel to the secant line through the endpoints $(a,f(a))$ and $(b,f(b))$, that is,

$$f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}.$$

differential equations §

• ordinary
• partial
• stochastic