Real Analysis

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

Last update: 8 July 2014

Lecture 5

Graphing Techniques

1. Basic graphs
2. Shifting
3. Scaling
4. Flipping
5. Limits
6. Asymptotes
7. Slopes: Increasing/Decreasing
8. Concave up/Concave down points of inflection.

Basic Graphs

Shifting

Graph ${(x-3)}^2+{(y-2)}^2=1\text{.}$ $$\begin{array}{c} 37.790960 x 1 2 3 y 1 2 (3,2) 1 \end{array}$$ Notes:

(a)	$x^2+y^2=1$ is the basic graph, a circle of radius 1.
(b)	Center is shifted by 3 to the right in the $x\text{-direction}$ and 2 upwards in the $y\text{-direction.}$

Scaling

Graph $2y=\text{sin}\hspace{0.17em}3x\text{.}$ $$\begin{array}{c} y 1 -1 x -π -5π6 -3π3 -π2 -π3 -π6 π6 π3 π2 2π3 5π6 π \end{array}$$ Notes:

(a)	$y=\text{sin}\hspace{0.17em}x$ is the basic graph.
(b)	The $x$ axis is scaled (squished) by 3.
(c)	The $y$ axis is scaled (squished) by 2.

Flipping

Graph $y=-e^{-x}\text{.}$ $$\begin{array}{c} x y -1 \end{array}$$ Notes:

(a)	$y=e^x$ is the basic graph.
(b)	$y=-e^{-x}$ is the same as $-y=e^{-x}\text{.}$
(c)	The $x\text{-axis}$ is flipped. The $y\text{-axis}$ is flipped.

Graph $y=\text{sin}\left(\frac{1}{x}\right)\text{.}$ $$\begin{array}{c} x y 1 -1 -1π -12π 12π 1π 2π \end{array}$$ Notes:

(a)	$y=\text{sin}\hspace{0.17em}x$ is the basic graph.
(b)	Positive $x$ axis is flipped in $x=1\text{.}$ Negative $x$ axis is flipped in $x=1\text{.}$
(c)	As $x\to \infty ,$ $\text{sin}\left(\frac{1}{x}\right)\to 0^{+}\text{.}$ As $x\to -\infty ,$ $\text{sin}\left(\frac{1}{x}\right)\to 0^{-}\text{.}$
(d)	As $x\to 0^{+},$ $\text{sin}\left(\frac{1}{x}\right)$ goes between $+1$ and $-1\text{.}$

Graph $y={\text{sin}}^{-1}x\text{.}$ $$\begin{array}{c} x -1 1 y 3π2 π π2 -π2 -π -3π2 \end{array}$$ Notes:

(a)	$y=\text{sin}\hspace{0.17em}x$ is the basic graph.
(b)	$y={\text{sin}}^{-1}x$ is $\text{sin}\hspace{0.17em}y=x$ so $x$ and $y$ are switched from the $y=\text{sin}\hspace{0.17em}x$ graph.

Example $$f\left(x\right)=\{\begin{array}{cc}{\displaystyle \frac{1-\text{cos}\hspace{0.17em}x}{x^2}},& \text{if}\hspace{0.17em}x\ne 0,\\ 1,& \text{if}\hspace{0.17em}x=0\text{.}\end{array}$$ $$\begin{array}{c} y 1 -1 x -3π -5π2 -2π -3π2 -π -π2 π2 π 3π2 2π 5π2 3π y=cos⁡x y 1 -1 x -3π -5π2 -2π -3π2 -π -π2 π2 π 3π2 2π 5π2 3π y=-cos⁡x \\ x y 2 1 -3π -5π2 -2π -3π2 -π -π2 π2 π 3π2 2π 5π2 3π y=1-cos⁡x x y -1 1 1 y=1x2 \\ x y 12 1 y=2x2 y = f⁡(x) = { 1-cos⁡x x2 , x≠0 , 1 , x=0 , \end{array}$$ Notes:

(a)	As $x\to 0$ $$\begin{array}{ccc}{\displaystyle \underset{x\to 0}{\text{lim}}\frac{1-\text{cos}\hspace{0.17em}x}{x^2}}& =& {\displaystyle \underset{x\to 0}{\text{lim}}\frac{1-(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots )}{x^2}}\\ & =& {\displaystyle \underset{x\to 0}{\text{lim}}\frac{\frac{x^2}{2!}-\frac{x^4}{4!}+\frac{x^6}{6!}-\cdots }{x^2}}\\ & =& {\displaystyle \underset{x\to 0}{\text{lim}}\frac{1}{2!}-\frac{x^2}{4!}+\frac{x^6}{6!}-\cdots =\frac{1}{2}\text{.}}\end{array}$$ (if $x=.0001$ then $x^2=.0000001$ and $x^4=.0000000000001\text{).}$
(b)	At $x=0,$ $f\left(x\right)=1\text{.}$
(c)	At the peaks of $1-\text{cos}\hspace{0.17em}x,$ $\frac{1-\text{cos}\hspace{0.17em}x}{x^2}=\frac{2}{x^2}\text{.}$

A function is continuous at $x=a$ if it doesn't jump at $x=a\text{.}$

A function is continuous at $x=a$ if $$\underset{x\to a}{\text{lim}}f\left(x\right)=f\left(a\right)\text{.}$$

Notes and References

These are notes from a 2010 course on Real Analysis 620-295. This page comes from 100310Lect5.pdf and was given on 10 March 2010.

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