From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Wed, 29 Aug 2018 09:30:34 +0000 (+1000)
Subject: calculus ref for methods
X-Git-Tag: yr11~54
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/2d7ed436d425e98f540b71df27a12f5fd80fc609?ds=inline;hp=5fbda7af797976c32f344854e49ecd298c9769c2

calculus ref for methods
---

diff --git a/methods/calculus-ref.md b/methods/calculus-ref.md
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+---
+geometry: margin=2cm
+<!-- columns: 2 -->
+graphics: yes
+tables: yes
+author: Andrew Lorimer
+---
+
+
+\pagenumbering{gobble}
+
+
+# Methods - Calculus
+
+## Average rate of change
+
+$$m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}$$
+
+Average 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 $\overleftrightarrow{PQ}$
+
+On CAS: (Action|Interactive) -> Calculation -> Diff -> $f(x)$ or $y=\dots$
+
+## Instantaneous rate of change
+
+Secant - line passing through two points on a curve  
+Chord - line segment joining two points on a curve
+
+Estimated by using two given points on each side of the concerned point. Evaluate as in average rate of change.
+
+## Limits & continuity
+
+### Limit theorems
+
+1. For constant function $f(x)=k$, $\lim_{x \rightarrow a} f(x) = k$
+2. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$
+3. $\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G$
+4. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$
+
+A function is continuous if $L^-=L^+=f(x)$ for all values of $x$.
+
+## First principles derivative
+
+$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
+
+## Tangents & gradients
+
+**Tangent line** - defined by $y=mx+c$ where $m={dy \over dx}$  
+**Normal line** - $\perp$ tangent ($m_{\operatorname{tan}} \cdot m_{\operatorname{norm}} = -1$)  
+**Secant** $={{f(x+h)-f(x)} \over h}$
+
+### Solving on CAS
+
+**In main**: type function. Interactive -> Calculation -> Line -> (Normal | Tan line)  
+**In graph**: define function. Analysis -> Sketch -> (Normal | Tan line). Type $x$ value to solve for a point. Return to show equation for line.
+
+## Stationary points
+
+Stationary where $m=0$.  
+Find derivative, solve for ${dy \over dx} = 0$
+
+![](https://cdn.edjin.com/upload/RESOURCE/IMAGE/78444.png){#id .class width=20%}
+
+**Local maximum at point $A$**  
+- $f^\prime (x) > 0$ left of $A$
+- $f^\prime (x) < 0$ right of $A$
+
+**Local minimum at point $B$**  
+- $f^\prime (x) < 0$ left of $B$
+- $f^\prime (x) > 0$ right of $B$
+
+**Stationary** point of inflection at $C$
+
+## Function derivatives
+
+
+| $f(x)$ | $f^\prime(x)$ |
+| ------ | ------------- |
+| $x^n$  | $nx^{n-1}$ |
+| $kx^n$ | $knx^{n-1}$ |
+| $g(x) + h(x)$ | $g^\prime (x) + h^\prime (x)$ |
+| $c$    | $0$ |
+| ${u \over v}$ | ${{v{du \over dx} - u{dv \over dx}} \over v^2}$ |
+| $uv$ | $u{dv \over dx} + v{du \over dx}$ |
+| $f \circ g$ | ${dy \over du} \cdot {du \over dx}$ |
+
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@@ -53,6 +53,8 @@ $$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
 
 **Tangent line** of function $f$ at point $M(a, f(a))$ is the line through $M$ with gradient $f^\prime(a)$.
 
+For $f(x)=x^n, \hspace{0.5em} f^\prime (x) = nx^{n-1}$
+
 ## Tangents and gradients