## MATH 221 Lecture 15

Last update: 6 August 2012

## Lecture 15

A function $f\left(x\right)$ is continuous at $x=a$ if it doesn't jump at $x=a$,

$\text{i.e.}\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{0.5em}{0ex}}\underset{x\to a}{lim}f\left(x\right)=f\left(a\right)$

Not continuous at $x=a$.

${\frac{df}{dx}|}_{x=a}=\underset{\Delta x\to 0}{lim}\frac{f\left(a+\Delta x\right)-f\left(a\right)}{\Delta x}$

in terms of the graph

$\underset{\Delta x\to 0}{lim}\frac{f\left(a+\Delta x\right)-f\left(a\right)}{\Delta x}=\text{slope of}\phantom{\rule{0.5em}{0ex}}f\phantom{\rule{0.5em}{0ex}}\text{at the point}\phantom{\rule{0.5em}{0ex}}x=a.$

A function $f\left(x\right)$ differentiable at $x=a$ if the derivative $\phantom{\rule{0.4em}{0ex}}{\frac{df}{dx}|}_{x=a}\phantom{\rule{0.4em}{0ex}}$ exists,
$\phantom{\rule{1em}{0ex}}\text{i.e.}\phantom{\rule{1em}{0ex}}\text{if the slope of the graph of}\phantom{\rule{0.5em}{0ex}}f\left(x\right)\phantom{\rule{0.5em}{0ex}}\text{at}\phantom{\rule{0.5em}{0ex}}x=a\phantom{\rule{0.5em}{0ex}}\text{exists.}$

Example:$\phantom{\rule{1em}{0ex}}$ $\text{Graph}\phantom{\rule{0.5em}{0ex}}f\left(x\right)=|x|=\left\{\begin{array}{ccc}x,& & \text{if}\phantom{\rule{0.5em}{0ex}}x\ge 0,\\ -x,& & \text{if}\phantom{\rule{0.5em}{0ex}}x\le 0,\end{array}\right\$

Then

${\frac{df}{dx}|}_{x=a}=\left\{\begin{array}{ccc}1,& & \text{if}\phantom{\rule{0.5em}{0ex}}a>0,\\ -1,& & \text{if}\phantom{\rule{0.5em}{0ex}}a<0,\\ \text{does not exist},& & \text{if}\phantom{\rule{0.5em}{0ex}}a=0,\end{array}$

So $f$ is not differentiable at $x=0$.

Example:$\phantom{\rule{1em}{0ex}}$ $\text{Graph}\phantom{\rule{0.5em}{0ex}}y={x}^{\frac{1}{3}}$

 Notes: (a) $y={x}^{\frac{1}{3}}\phantom{\rule{0.5em}{0ex}}\text{is the same as}\phantom{\rule{0.5em}{0ex}}{y}^{3}=x.$ is a basic circle of radius 1

$\frac{dy}{dx}=\frac{1}{3}{x}^{-\frac{2}{3}}=\frac{1}{3{x}^{\frac{2}{3}}}$

.

A function $f\left(x\right)$ is increasing at $x=a$ if it is going up at $x=a,$
$\phantom{\rule{1em}{0ex}}\text{i.e.}\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{0.5em}{0ex}}f\left(a+\Delta x\right)>f\left(x\right)\phantom{\rule{0.5em}{0ex}}\text{for all small}\phantom{\rule{0.5em}{0ex}}\Delta x>0,$
$\phantom{\rule{1em}{0ex}}\text{i.e.}\phantom{\rule{1em}{0ex}}\text{if the slope of}\phantom{\rule{0.5em}{0ex}}f\left(x\right)\phantom{\rule{0.5em}{0ex}}\text{at}\phantom{\rule{0.5em}{0ex}}x=a\phantom{\rule{0.5em}{0ex}}\text{is positive,}$
$\phantom{\rule{1em}{0ex}}\text{i.e.}\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{0.5em}{0ex}}{\frac{df}{dx}|}_{x=a}>0$.

A function $f\left(x\right)$ is decreasing at $x=a$ if it is going down at $x=a,$
$\phantom{\rule{1em}{0ex}}\text{i.e.}\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{0.5em}{0ex}}f\left(a+\Delta x\right)0,$
$\phantom{\rule{1em}{0ex}}\text{i.e.}\phantom{\rule{1em}{0ex}}\text{if the slope of}\phantom{\rule{0.5em}{0ex}}f\left(x\right)\phantom{\rule{0.5em}{0ex}}\text{at}\phantom{\rule{0.5em}{0ex}}x=a\phantom{\rule{0.5em}{0ex}}\text{is negative,}$
$\phantom{\rule{1em}{0ex}}\text{i.e.}\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{0.5em}{0ex}}{\frac{df}{dx}|}_{x=a}<0$.

$f$ is concave up at $x=a$ if it is right side up bowl shaped $x=a,$
$\phantom{\rule{1em}{0ex}}\text{i.e.}\phantom{\rule{1em}{0ex}}\text{if the slope of}\phantom{\rule{0.5em}{0ex}}f\phantom{\rule{0.5em}{0ex}}\text{is getting larger at}\phantom{\rule{0.5em}{0ex}}x=a,$
$\phantom{\rule{1em}{0ex}}\text{i.e.}\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{0.5em}{0ex}}\frac{df}{dx}\phantom{\rule{0.5em}{0ex}}\text{is increasing at}\phantom{\rule{0.5em}{0ex}}x=a,$
$\phantom{\rule{1em}{0ex}}\text{i.e.}\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{0.5em}{0ex}}{\frac{{d}^{2}f}{d{x}^{2}}|}_{x=a}>0$.

$f$ is concave down at $x=a$ if it is upside down bowl shaped $x=a,$
$\phantom{\rule{1em}{0ex}}\text{i.e.}\phantom{\rule{1em}{0ex}}\text{if the slope of}\phantom{\rule{0.5em}{0ex}}f\phantom{\rule{0.5em}{0ex}}\text{is getting smaller at}\phantom{\rule{0.5em}{0ex}}x=a,$
$\phantom{\rule{1em}{0ex}}\text{i.e.}\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{0.5em}{0ex}}\frac{df}{dx}\phantom{\rule{0.5em}{0ex}}\text{is decreasing at}\phantom{\rule{0.5em}{0ex}}x=a,$
$\phantom{\rule{1em}{0ex}}\text{i.e.}\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{0.5em}{0ex}}{\frac{{d}^{2}f}{d{x}^{2}}|}_{x=a}<0$.

A point of inflection is a point where $f$ changes from concave up to concave down, or from concave down to concave up.

A local maximum is a point $x=a$ where $f\left(a\right)$ is bigger then the $f\left(x\right)$ around it.

A local minimum is a point $x=a$ where $f\left(a\right)$ is smaller then the $f\left(x\right)$ around it.
$\phantom{\rule{1em}{0ex}}\text{i.e.}\phantom{\rule{1em}{0ex}}f\left(a\right).

A critical point is a point where a maximum or minimum might occur.

 Note: (1) If $f\left(x\right)$ is continuous and differentiable and $x=a$ is a maximum then ${\frac{df}{dx}|}_{x=a}=0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\frac{{d}^{2}f}{d{x}^{2}}|}_{x=a}<0$ (2) If $f\left(x\right)$ is continuous at $x=a,$ $f\left(x\right)$ is differentiable at $x=a,$ ${\frac{df}{dx}|}_{x=a}=0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\frac{{d}^{2}f}{d{x}^{2}}|}_{x=a}>0\phantom{\rule{1em}{0ex}}\text{then}$ $x=a$ is a minimum.

 Where can a maximum or minimum occur? (a) A point $x=a$ where $f\left(x\right)$ is differentiable and ${\frac{df}{dx}|}_{x=a}=0$. x y a (b) A point $x=a$ where $f\left(x\right)$ is not continuous. x y 1 2 3 1 2 $f\left(x\right)=\left\{\begin{array}{ccc}{x}^{2}+1,& & \text{if}\phantom{\rule{0.5em}{0ex}}0\le x\le 1,\\ 2-x,& & \text{if}\phantom{\rule{0.5em}{0ex}}x>1,\end{array}$ $x=1$ is a maximum (c) A point $x=a$ on the boundary of where $f\left(x\right)$ is defined. x y 1 2 3 1 2 $f\left(x\right)=\left\{\begin{array}{ccc}{x}^{2}+1,& & \text{if}\phantom{\rule{0.5em}{0ex}}0\le x\le 1,\\ 2-x,& & \text{if}\phantom{\rule{0.5em}{0ex}}x>1,\end{array}$ $x=0$ is a minimum

## Notes and References

These are a typed copy of lecture notes given by Arun Ram on October 11, 2000.