Andrew's git - notes.git/blob - methods/calculus-ref.md

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   6author: Andrew Lorimer
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  10\pagenumbering{gobble}
  11
  12
  13# Methods - Calculus
  14
  15## Average rate of change
  16
  17$$m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}$$
  18
  19Average rate of change between $x=[a,b]$ given two points $P(a, f(a))$ and $Q(b, f(b))$ is the gradient $m$ of line $\overleftrightarrow{PQ}$
  20
  21On CAS: (Action|Interactive) -> Calculation -> Diff -> $f(x)$ or $y=\dots$
  22
  23## Instantaneous rate of change
  24
  25Secant - line passing through two points on a curve  
  26Chord - line segment joining two points on a curve
  27
  28Estimated by using two given points on each side of the concerned point. Evaluate as in average rate of change.
  29
  30## Limits & continuity
  31
  32### Limit theorems
  33
  341. For constant function $f(x)=k$, $\lim_{x \rightarrow a} f(x) = k$
  352. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$
  363. $\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G$
  374. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$
  38
  39A function is continuous if $L^-=L^+=f(x)$ for all values of $x$.
  40
  41## First principles derivative
  42
  43$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
  44
  45Not differentiable at:
  46
  47- discontinuous points
  48- sharp point/cusp
  49- vertical tangents ($\infty$ gradient)
  50
  51## Tangents & gradients
  52
  53**Tangent line** - defined by $y=mx+c$ where $m={dy \over dx}$  
  54**Normal line** - $\perp$ tangent ($m_{\operatorname{tan}} \cdot m_{\operatorname{norm}} = -1$)  
  55**Secant** $={{f(x+h)-f(x)} \over h}$
  56
  57### Solving on CAS
  58
  59**In main**: type function. Interactive -> Calculation -> Line -> (Normal | Tan line)  
  60**In graph**: define function. Analysis -> Sketch -> (Normal | Tan line). Type $x$ value to solve for a point. Return to show equation for line.
  61
  62## Stationary points
  63
  64Stationary where $m=0$.  
  65Find derivative, solve for ${dy \over dx} = 0$
  66
  67![](https://cdn.edjin.com/upload/RESOURCE/IMAGE/78444.png){#id .class width=20%}
  68
  69**Local maximum at point $A$**  
  70- $f^\prime (x) > 0$ left of $A$
  71- $f^\prime (x) < 0$ right of $A$
  72
  73**Local minimum at point $B$**  
  74- $f^\prime (x) < 0$ left of $B$
  75- $f^\prime (x) > 0$ right of $B$
  76
  77**Stationary** point of inflection at $C$
  78
  79## Function derivatives
  80
  81
  82| $f(x)$ | $f^\prime(x)$ |
  83| ------ | ------------- |
  84| $x^n$  | $nx^{n-1}$ |
  85| $kx^n$ | $knx^{n-1}$ |
  86| $g(x) + h(x)$ | $g^\prime (x) + h^\prime (x)$ |
  87| $c$    | $0$ |
  88| ${u \over v}$ | ${{v{du \over dx} - u{dv \over dx}} \over v^2}$ |
  89| $uv$ | $u{dv \over dx} + v{du \over dx}$ |
  90| $f \circ g$ | ${dy \over du} \cdot {du \over dx}$ |
  91