spec / calculus.mdon commit differentiation (a225a5d)
   1# Differential calculus
   2
   3## Limits
   4
   5$$\lim_{x \rightarrow a}f(x)$$
   6
   7$L^-$ - limit from below
   8
   9$L^+$ - limit from above
  10
  11$\lim_{x \to a} f(x)$ - limit of a point
  12
  13- Limit exists if $L^-=L^+$
  14- If limit exists, point does not.
  15
  16Limits can be solved using normal techniques (if div 0, factorise)
  17
  18## Limit theorems
  19
  201. For constant function $f(x)=k$, $\lim_{x \rightarrow a} f(x) = k$
  212. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$
  223. $\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G$
  234. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$
  24
  25Corollary: $\lim_{x \rightarrow a} c \times f(x)=cF$ where $c=$ constant
  26
  27## Solving limits for $x\rightarrow\infty$
  28
  29Factorise so that all values of $x$ are in denominators.
  30
  31e.g.
  32
  33$$\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2$$
  34
  35
  36## Continuous functions
  37
  38A function is continuous if $L^-=L^+=f(x)$ for all values of $x$.
  39
  40## Gradients of secants and tangents
  41
  42Secant (chord) - line joining two points on curve
  43
  44Tangent - line that intersects curve at one point
  45
  46given $P(x,y) \quad Q(x+\delta x, y + \delta y)$:
  47gradient of chord joining $P$ and $Q$ is ${m_{PQ}}={\operatorname{rise} \over \operatorname{run}} = {\delta y \over \delta x}$
  48
  49As $Q \rightarrow P, \delta x \rightarrow 0$. Chord becomes tangent (two infinitesimal points are equal).
  50
  51Can also be used with functions, where $h=\delta x$.
  52
  53## First principles derivative
  54
  55$$\lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={dy \over dx} = f^\prime(x)$$
  56
  57$$m_{\operatorname{tangent}}=\lim_{h \rightarrow 0}f^\prime(x)$$
  58
  59
  60
  61$$m_{\operatorname{chord PQ}}=f^\prime(x)$$
  62
  63first principles derivative:
  64$${m_{\operatorname{tangent at P}} =\lim_{h \rigzhtarrow 0}}{{f(x+h)-f(x)}\over h}$$
  65
  66
  67
  68
  69## Euler's number as a limit
  70
  71$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$