methods / antidiff.mdon commit [spec] start vector calculus (42fe40b)
   1---
   2geometry: margin=2cm
   3<!-- columns: 2 -->
   4graphics: yes
   5tables: yes
   6author: Andrew Lorimer
   7---
   8
   9# Antidifferentiation
  10
  11If $F'(x)=f(x)$, then $\int f(x) \cdot dx = F(x) + c$
  12
  13$$\int x^n \cdot dx = {x^{n+1} \over {n+1}} + c, \quad n \in \mathbb{N} \cup \{0\}$$
  14
  15Rules:
  16
  17$\int [f(x) \pm g(x)] \cdot dx = \int f(x) \cdot dx \pm \int g(x) \cdot dx$  
  18$\int kf(x) \cdot dx = k \int f(x) \cdot dx$, where $k \in \mathbb{R}$
  19
  20## Applications of differentiation to kinematics
  21
  22Kinematics - straight line motion of a particle
  23
  24Instantaneous velocity - dx/dt
  25
  26## Newton's method
  27
  28$$x_{n+1}=x_n - {f(x_n) \over f^\prime(x_n)}$$
  29
  30or
  31
  32$$x_1=x_0 - {f(x_0) \over f^\prime(x_0)}$$