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author: Andrew Lorimer
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# Methods - Calculus

## Average rate of change

$$m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}$$

On CAS: Action $\rightarrow$ Calculation $\rightarrow$ Diff $\rightarrow$ ($f(x)$ | $y$) $=\dots$

## Instantaneous rate of change

**Secant** - line passing through two points on a curve  
**Chord** - line segment joining two points on a curve

## 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

$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$

Not differentiable at:

- discontinuous points
- sharp point/cusp
- vertical tangents ($\infty$ gradient)

## Tangents & gradients

**Tangent line** - defined by $y=mx+c$ where $m={dy \over dx}$  
**Normal line** - $\perp$ tangent ($m_{{tan}} \cdot m_{\operatorname{norm}} = -1$)  
**Secant** $={{f(x+h)-f(x)} \over h}$

## Strictly increasing

- **strictly increasing** where $f(x_2) > f(x_1)$ and $x_2 > x_1$
- **strictly decreasing** where $f(x_2) < f(x_1)$ and $x_2 > x_1$
- If $f^\prime (x) > 0$ for all $x$ in interval, then $f$ is **strictly increasing**
- If $f^\prime(x) < 0$ for all $x$ in interval, then $f$ is **strictly decreasing**
- Endpoints are included, even where gradient $=0$

### Solving on CAS

**In main**: type function. Interactive $\rightarrow$ Calculation $\rightarrow$ Line $\rightarrow$ (Normal | Tan line)  
**In graph**: define function. Analysis $\rightarrow$ Sketch $\rightarrow$ (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 ${dy \over dx} = 0$

\begin{center}
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**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

\begin{tabularx}{\columnwidth}{rl}
  
  \hline \(f(x)\) & \(f^\prime(x)\) \\ \hline

  \(kx^n\) & \(knx^{n-1}\)\tabularnewline
  \(g(x) \pm h(x)\) & \(g^\prime (x) \pm h^\prime (x)\)\tabularnewline
  \(c\) & \(0\)\tabularnewline
  \({u \over v}\) &
  \({{(v{du \over dx} - u{dv \over dx}}) \div v^2}\)\tabularnewline
  \(uv\) & \(u{dv \over dx} + v{du \over dx}\)\tabularnewline
  \(f \circ g\) & \({dy \over du} \cdot {du \over dx}\)\tabularnewline
  \(\sin ax\) & \(a\cos ax\)\tabularnewline
  \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\tabularnewline
  \(\cos ax\) & \(-a \sin ax\)\tabularnewline
  \(\cos(f(x))\) & \(f^\prime(x)(-\sin(f(x)))\) \\
  \(e^{ax}\) & \(ae^{ax}\)\tabularnewline
  \(\log_e {ax}\) & \(1 \over x\)\tabularnewline
  \(\log_e f(x)\) & \(f^\prime (x) \over f(x)\)\tabularnewline
  
  \hline

\end{tabularx}