MATH 221 Lecture 15

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 6 August 2012

Lecture 15

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

$\text{i.e.}\text{if}\underset{x\to a}{lim}f\left(x\right)=f\left(a\right)$

Not continuous at $x=a$.

Think about

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

in terms of the graph

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

A function $f\left(x\right)$ differentiable at $x=a$ if the derivative ${\displaystyle {\left.\frac{df}{dx}\right|}_{x=a}}$ exists,
$\text{i.e.}\text{if the slope of the graph of}f\left(x\right)\text{at}x=a\text{exists.}$

Example:$$ ${\displaystyle \text{Graph}f\left(x\right)=\left|x\right|=\left\{\begin{array}{ccc}x,& & \text{if}x\ge 0,\\ -x,& & \text{if}x\le 0,\end{array}\right.}$

Then

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

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

Example:$$ $\text{Graph}y=x^{\frac{1}{3}}$

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

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

${\displaystyle \text{So}{\frac{dy}{dx}|}_{x=0}=\infty .\text{So}f\left(x\right)\text{is }\text{not}\text{ differentiable at}x=0}$.

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

A function $f\left(x\right)$ is decreasing at $x=a$ if it is going down at $x=a,$
$\text{i.e.}\text{if}f(a+\Delta x)<f\left(x\right)\text{for all small}\Delta x>0,$
$\text{i.e.}\text{if the slope of}f\left(x\right)\text{at}x=a\text{is negative,}$
${\displaystyle \text{i.e.}\text{if}{\frac{df}{dx}|}_{x=a}<0}$.

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

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

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.
$\text{i.e.}f\left(a\right)<f(a+\Delta x)\text{for small}\Delta x$.

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 ${\displaystyle {\frac{df}{dx}|}_{x=a}=0\text{and}{\frac{d^{2}f}{d{x}^2}|}_{x=a}<0}$
(2)	If $f\left(x\right)$ is continuous at $x=a,$ $f\left(x\right)$ is differentiable at $x=a,$ ${\displaystyle {\frac{df}{dx}|}_{x=a}=0\text{and}{\frac{d^{2}f}{d{x}^2}|}_{x=a}>0\text{then}}$ $x=a$ is a minimum.

Where can a maximum or minimum occur?
(a)	A point $x=a$ where $f\left(x\right)$ is differentiable and ${\displaystyle {\frac{df}{dx}|}_{x=a}=0}$.
(b)	A point $x=a$ where $f\left(x\right)$ is not continuous. $x=1$ is a maximum
(c)	A point $x=a$ on the boundary of where $f\left(x\right)$ is defined. $x=0$ is a minimum

Notes and References

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

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