Problem Set: Graphing
620-205 Semester I 2010

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

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 28 February 2010

(1) Graphs of the Basic Functions
(2) Graphing Polynomials
(3) Graphing Rational Functions
(4) Graphing Sequences
(5) Graphing Other Functions
(6) Where is a Function Continuous?
(7) Existence of Limits

Graphs of the Basic Functions

	Graph $f\left(x\right)=\left|x\right|$.
	Graph $f\left(x\right)=⌊x⌋$.
	Graph $f\left(x\right)=2$.
	Graph $f\left(x\right)=x$.
	Graph $f\left(x\right)=x^2$.
	Graph $f\left(x\right)=x^3$.
	Graph $f\left(x\right)=x^4$.
	Graph $f\left(x\right)=x^5$.
	Graph $f\left(x\right)=x^6$.
	Graph $f\left(x\right)=x^{100}$.
	Graph $f\left(x\right)=x^{-1}$.
	Graph $f\left(x\right)=x^{-2}$.
	Graph $f\left(x\right)=x^{-3}$.
	Graph $f\left(x\right)=x^{-4}$.
	Graph $f\left(x\right)=x^{-100}$.
	Graph $f\left(x\right)=e^x$.
	Graph $f\left(x\right)=\sin x$.
	Graph $f\left(x\right)=\cos x$.
	Graph $f\left(x\right)=\tan x$.
	Graph $f\left(x\right)=\cot x$.
	Graph $f\left(x\right)=\sec x$.
	Graph $f\left(x\right)=\csc x$.
	Graph $f\left(x\right)=\sqrt{x}$.
	Graph $f\left(x\right)=x^{1/3}$.
	Graph $f\left(x\right)=x^{1/4}$.
	Graph $f\left(x\right)=x^{1/5}$.
	Graph $f\left(x\right)=x^{1/6}$.
	Graph ${\displaystyle f\left(x\right)=\frac{1}{\sqrt{x}}}$.
	Graph $f\left(x\right)=x^{-1/3}$.
	Graph $f\left(x\right)=x^{-1/4}$.
	Graph $f\left(x\right)=\ln x$.
	Graph $f\left(x\right)=\arcsin x$.
	Graph $f\left(x\right)=\arccos x$.
	Graph $f\left(x\right)=\arctan x$.
	Graph $f\left(x\right)=\mathrm{arccot}x$.
	Graph $f\left(x\right)=\mathrm{arcsec}x$.
	Graph $f\left(x\right)=\mathrm{arccsc}x$.

Graphing Polynomials

	Graph $f\left(x\right)=a$, where $a$ is a constant.
	Graph $f\left(x\right)=ax+b$, where $a$ and $b$ are constants.
	Graph $f\left(x\right)=a\left(x-c\right)+b$, where $a,,,b$ and $c$ are constants.
	Graph $f\left(x\right)=\left\{\begin{array}{cc}2-x\text{,}& \text{if }x\ge 1\text{,}\\ x\text{,}& \text{if }0\le x\le 1\text{.}\end{array}\right.$
	Graph $f\left(x\right)=\left\{\begin{array}{cc}2+x\text{,}& \text{if }x>0\text{,}\\ 2-x\text{,}& \text{if }x\le 0\text{.}\end{array}\right.$
	Graph $f\left(x\right)=\left\{\begin{array}{cc}1-x\text{,}& \text{if }x<1\text{,}\\ x^2-1\text{,}& \text{if }x\ge 1\text{.}\end{array}\right.$
	Graph $f\left(x\right)=2x-x^2$.
	Graph $f\left(x\right)=x-x^2-27$.
	Graph $f\left(x\right)=3x^2-2x-1$.
	Graph $f\left(x\right)=x^3$.
	Graph $f\left(x\right)=x^3-x+1$.
	Graph $f\left(x\right)=x^3-x-1$.
	Graph $f\left(x\right)={\left(x-2\right)}^2\left(x-1\right)$.
	Graph $f\left(x\right)=2x^3-21x^2+36x-20$.
	Graph $f\left(x\right)=2x^3+x^2+20x$.
	Graph $f\left(x\right)=1-x^4$.
	Graph $f\left(x\right)=3x^4-4x^3-12x^2+5$.
	Graph $f\left(x\right)==3x^4-16x^3+18x^2$.
	Graph $f\left(x\right)=x^5-4x^4+4x^3$.
	Graph $f\left(x\right)=x^3{\left(x-2\right)}^2$.
	Graph $f\left(x\right)={\left(x-2\right)}^4{\left(x+1\right)}^3\left(x-1\right)$.

Graphing Rational Functions

	Graph $f\left(x\right)=y$, where $x^2+y^2=1$.
	Graph $f\left(x\right)=\sqrt{1-x^2}$.
	Graph $f\left(x\right)=\sqrt{a^2-x^2}$, where $a$ is a constant.
	Graph ${\displaystyle f\left(x\right)=y}$, where ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$, and $h,k$ and $r$ are constants.
	Graph ${\displaystyle f\left(x\right)=y}$, where $x^2+y^2-2hx-2ky+h^2+k^2=r^2$, and $h,k$ and $r$ are constants.
	Graph ${\displaystyle f\left(x\right)=y}$ where ${\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1}$, and $a$ and $b$ are constants.
	Graph ${\displaystyle f\left(x\right)=y}$, where ${\displaystyle x=a\cos \theta }$, $y=b\sin \theta$, and $a$ and $b$ are constants.
	Graph ${\displaystyle f\left(x\right)=\left(b/a\right)\sqrt{a^2-x^2}}$, where $a$ and $b$ are constants.
	Graph ${\displaystyle f\left(x\right)=y}$, where $x^2-y^2=1$.
	Graph ${\displaystyle f\left(x\right)=y}$, where ${\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1}$, and $a$ and $b$ are constants.
	Graph ${\displaystyle f\left(x\right)=a{x}^2-b}$, where $a$ and $b$ are constants.
	Graph ${\displaystyle f\left(x\right)=y}$, where $x=2y^2-1$.
	Graph ${\displaystyle f\left(x\right)=y}$, where $x=\cos 2\theta$ and $y=\cos \theta$.
	Graph ${\displaystyle f\left(x\right)=b\sqrt{x-a}}$, where $a$ and $b$ are constants.
	Graph ${\displaystyle f\left(x\right)=\sqrt{x+2}}$.
	Graph ${\displaystyle f\left(x\right)=-\sqrt{x+2}}$.
	Graph ${\displaystyle f\left(x\right)=y}$, where $y^2\left(x^2-x\right)=x^2-1$.
	Graph ${\displaystyle f\left(x\right)=y}$, where ${\displaystyle x=\frac{y^2-1}{y^2+1}}$.
	Graph ${\displaystyle f\left(x\right)=\frac{\sqrt{1+x}}{\sqrt{1-x}}}$.
	Graph ${\displaystyle f\left(x\right)=\frac{x^2}{\sqrt{x+1}}}$.
	Graph ${\displaystyle f\left(x\right)=x\sqrt{32-x^2}}$.
	Graph ${\displaystyle f\left(x\right)=x\sqrt{1-x^2}}$.

Graphing sequences

	Graph the sequence $a_n=n$.
	Graph the sequence $a_n={(-1)}^{n}n$.
	Graph the sequence $a_n=n^2$.
	Graph the sequence $a_n=12n-n^3$.
	Graph the sequence $a_n=n\text{!}$.
	Graph the sequence ${\displaystyle a_n=\frac{1}{n}}$.
	Graph the sequence ${\displaystyle a_n=3-\frac{1}{n}}$.
	Graph the sequence ${\displaystyle a_n=\frac{1}{n^p}}$.
	Graph the sequence ${\displaystyle a_n=\frac{1}{n\text{!}}}$.
	Graph the sequence $a_n={\displaystyle \frac{n}{n(n+1)}}$.
	Graph the sequence $a_n={\displaystyle \frac{1}{n}-\frac{1}{n+1}}$.
	Graph the sequence $a_n={\displaystyle \frac{(-1)}{n+1}}$.
	Graph the sequence $a_n={\displaystyle \frac{{(-1)}^{n+1}}{n}}$.
	Graph the sequence $a_n={\displaystyle {(-1)}^n(1+\frac{1}{n})}$.
	Graph the sequence $a_n={\displaystyle \frac{n}{2n+1}}$.
	Graph the sequence $a_n={\displaystyle \frac{2n}{n+1}}$.
	Graph the sequence $a_n={\displaystyle \frac{n}{n^2+1}}$.
	Graph the sequence $a_n={\displaystyle \frac{3n+1}{2n+5}}$.
	Graph the sequence $a_n={\displaystyle \frac{n^2-1}{2n^2+3}}$.
	Graph the sequence $a_n={\displaystyle \frac{i^n}{n^2}}$.
	Graph the sequence $a_n={\displaystyle \frac{n+2i}{n}}$.
	Graph the sequence $a_n={\displaystyle \frac{4n+3}{4n^2+3n+1}}$.
	Graph the sequence $a_k={\displaystyle \frac{1}{{(3k^4-7k^2+5)}^{\frac{1}{3}}}}$.
	Graph the sequence ${\displaystyle a_n=\frac{{\left(n\text{!}\right)}^2}{\left(2n\right)\text{!}}}$.
	Graph the sequence ${\displaystyle a_n=\frac{{\left(n\text{!}\right)}^2{5}^n}{\left(2n\right)\text{!}}}$.
	Graph the sequence $a_n={(-1)}^n$.
	Graph the sequence $a_n=n^{1/n}$.
	Graph the sequence $a_n={\displaystyle {(1+\frac{1}{n})}^n}$.
	Graph the sequence $a_n=e^{in\pi /7}$.
	Graph the sequence $a_n=\sqrt{n}$.
	Graph the sequence $a_n={\displaystyle \frac{1}{\sqrt{n}}}$.
	Graph the sequence $a_n=\sqrt{n+1}-\sqrt{n}$.
	Graph the sequence $a_n=\sqrt{n}(\sqrt{n+1}-\sqrt{n})$.
	Let $x\in ℝ$ with $\left|x\right|<1$. Graph the sequence $a_n=x^n$.
	Let $x\in ℝ$ with $x>0$. Graph the sequence $a_n=x^{1/n}$.
	Let $x\in ℝ$. Graph the sequence $a_n={\displaystyle {(1+\frac{x}{n})}^n}$.
	Let $x\in ℝ$. Graph the sequence $a_n={\displaystyle \frac{1-x^{n+1}}{1-x}}$.
	Let $x\in ℝ$. Graph the sequence $a_n={\displaystyle 1+x+\cdots +x^n}$.
	Graph the sequence given by $a_1=3$ and $a_n={\displaystyle \frac{1}{2}(a_{n-1}+\frac{5}{a_{n-1}})}$.
	Let $a\in ℝ$ with $a>0$. Fix a positive real number $x_1$. Graph the sequence given by $x_{n+1}={\displaystyle \frac{1}{2}(x_n+\frac{a}{x_n})}$.
	Let $\alpha ,\beta \in {ℝ}_{>0}$. Graph the sequence given by $a_1=\alpha$ and $a_{n+1}=\sqrt{\beta +a_n}$.
	Let $\alpha ,\beta \in {ℝ}_{>0}$. Graph the sequence given by $a_1=\alpha$ and $a_{n+1}=\beta +\sqrt{a_n}$.
	Graph the sequence given by $x_1=1$ and $x_{n+1}={\displaystyle \frac{1}{2+x_n}}$.
	Fix a real number $x_1$ between 0 and 1. Graph the sequence given by $x_{n+1}={\displaystyle \frac{1}{7}(x_n^3+2)}$. Estimate the solution to $x^3-7x+2=0$ to three decimal places and verify that the limit is a solution to the equation $x^3-7x+2=0$.
	Graph the sequence given by $a_1=0$,   $a_{2k}=\frac{1}{2}{a}_{2k+1}$,   and $a_{2k+1}=\frac{1}{2}+a_{2k}$.

Graphing Other Functions

	Graph $f\left(x\right)=⌊x⌋$.
	Graph $f\left(x\right)=\left|x\right|$.
	Graph $f\left(x\right)=\left|x-5\right|$.
	Graph ${\displaystyle f\left(x\right)=\left|x^2-1\right|}$.
	Graph ${\displaystyle f\left(x\right)=\left\{\begin{array}{ll}1\text{,}& \text{if }x>0\text{,}\\ 0\text{,}& \text{if }x=0\text{,}\\ -1\text{,}& \text{if }x<0\text{.}\end{array}\right.}$
	Graph ${\displaystyle f\left(x\right)={\left(x-1\right)}^{1/3}}$.
	Graph ${\displaystyle f\left(x\right)=x^{2/3}}$.
	Graph ${\displaystyle f\left(x\right)=\frac{1}{{\left(x-1\right)}^{2/3}}}$.
	Graph ${\displaystyle f\left(x\right)=x{\left(1-x\right)}^{2/5}}$.
	Graph ${\displaystyle f\left(x\right)=x^{2/3}{\left(6-x\right)}^{1/3}}$.
	Graph ${\displaystyle f\left(x\right)=y}$, where $\sqrt{x}+\sqrt{y}=1$.
	Graph ${\displaystyle f\left(x\right)=y}$, where $x^{2/3}+y^{2/3}=a^{2/3}$, where $a$ is a constant.
	Graph ${\displaystyle f\left(x\right)=y}$, where $x=a{\cos }^3\theta$ and $y=a{\sin }^3\theta$.
	Graph ${\displaystyle f\left(x\right)=\sin x}$.
	Graph ${\displaystyle f\left(x\right)=\sin 2x-x}$.
	Graph ${\displaystyle y=\sin x-\cos x}$, for $-\pi /3<x<0$.
	Graph ${\displaystyle y\left(x\right)=2\cos x-\sin 2x}$.
	Graph ${\displaystyle y=\frac{\sin x}{x}}$.
	Graph ${\displaystyle y=\sin \left(1/x\right)}$.
	Graph ${\displaystyle y=e^{-x}}$.
	Graph ${\displaystyle y=e^{1/x}}$.
	Graph ${\displaystyle y=e^{-x^2}}$.
	Graph ${\displaystyle y=\ln \left(4-x^2\right)}$.

Where is a Function Continuous?

	Graph $f\left(x\right)=x^2+3x+4$. For which values of $x$ is the function continuous?
	Graph $$f\left(x\right)=\left\{\begin{array}{ll}\frac{x^2-x-6}{x-3}\text{,}& \text{if }x\ne 3\text{,}\\ 5\text{,}& \text{if }x=3\text{.}\end{array}\right.$$ For which values of $x$ is the function continuous?
	Graph $$f\left(x\right)=\left\{\begin{array}{ll}\frac{\sin 3x}{x}\text{,}& \text{if }x\ne 0\text{,}\\ 1\text{,}& \text{if }x=0\text{.}\end{array}\right.$$ For which values of $x$ is the function continuous?
	Graph $$f\left(x\right)=\left\{\begin{array}{ll}\frac{1-\cos x}{x^2}\text{,}& \text{if }x\ne 0\text{,}\\ 1\text{,}& \text{if }x=0\text{.}\end{array}\right.$$ For which values of $x$ is the function continuous?
	Let $k\in ℝ$. Graph $$f\left(x\right)=\left\{\begin{array}{ll}\frac{\sin 2x}{5x}\text{,}& \text{if }x\ne 0\text{,}\\ k\text{,}& \text{if }x=0\text{.}\end{array}\right.$$ For which values of $k$ is the function continuous?
	Graph $$f\left(x\right)=\left\{\begin{array}{ll}x-1\text{,}& \text{if }1\le x<2\text{,}\\ 2x-3\text{,}& \text{if }2\le x\le 3\text{.}\end{array}\right.$$ For which values of $x$ is the function continuous?
	Graph $$f\left(x\right)=\left\{\begin{array}{ll}\cos x\text{,}& \text{if }x\ge 0\text{,}\\ -\cos x\text{,}& \text{if }x<0\text{.}\end{array}\right.$$ For which values of $x$ is the function continuous?
	Graph $$f\left(x\right)=\left\{\begin{array}{ll}\sin \left(1/x\right)\text{,}& \text{if }x\ne 0\text{,}\\ 0\text{,}& \text{if }x=0\text{.}\end{array}\right.$$ For which values of $x$ is the function continuous?
	Let $a\in ℝ$. Graph $$f\left(x\right)=\left\{\begin{array}{ll}ax+5\text{,}& \text{if }x\le 2\text{,}\\ x-1\text{,}& \text{if }x>2\text{.}\end{array}\right.$$ For which values of $x$ is the function continuous at $x=2$?
	Graph $$f\left(x\right)=\left\{\begin{array}{ll}1+x^2\text{,}& \text{if }0\le x\le 1\text{,}\\ 2-x\text{,}& \text{if }x>1\text{.}\end{array}\right.$$ For which values of $x$ is the function continuous?
	Graph $f\left(x\right)=2x-\left|x\right|$. For which values of $x$ is the function continuous?
	Let $a\in ℝ$. Graph $$f\left(x\right)=\left\{\begin{array}{ll}2x-1\text{,}& \text{if }x<2\text{,}\\ a\text{,}& \text{if }x=2\text{,}\\ x+1\text{,}& \text{if }x>2\text{.}\end{array}\right.$$ For which values of $a$ is the function continuous?
	Graph $$f\left(x\right)=\left\{\begin{array}{ll}\frac{\left|x-a\right|}{x-a}\text{,}& \text{if }x\ne a\text{,}\\ 1\text{,}& \text{if }x=a\text{.}\end{array}\right.$$ For which values of $x$ is the function continuous?
	Graph $$f\left(x\right)=\left\{\begin{array}{ll}\frac{x-\left|x\right|}{2}\text{,}& \text{if }x\ne 0\text{,}\\ 2\text{,}& \text{if }x=0\text{.}\end{array}\right.$$ For which values of $x$ is the function continuous?
	Graph $$f\left(x\right)=\left\{\begin{array}{ll}\sin x\text{,}& \text{if }x<0\text{,}\\ x\text{,}& \text{if }x\ge 0\text{.}\end{array}\right.$$ For which values of $x$ is the function continuous?
	Graph $$f\left(x\right)=\left\{\begin{array}{ll}\frac{x^n-1}{x-1}\text{,}& \text{if }x\ne 1\text{,}\\ n\text{,}& \text{if }x=1\text{.}\end{array}\right.$$ For which values of $x$ is the function continuous?
	Graph $f\left(x\right)=\cos x$. For which values of $x$ is the function continuous?
	Graph $f\left(x\right)=\cos \hspace{0.17em}\left|x\right|$. For which values of $x$ is the function continuous?
	Graph $f\left(x\right)=⌊x⌋$. For which values of $x$ is the function continuous?
	Graph $$f\left(x\right)=\left\{\begin{array}{ll}{x}^3-x^2+2x-2\text{,}& \text{if }x\ne 1\text{,}\\ 4\text{,}& \text{if }x=1\text{.}\end{array}\right.$$ For which values of $x$ is the function continuous?
	Graph $f\left(x\right)=\left|x\right|+\left|x-1\right|\text{, for}-1\le x\le 2$. For which values of $x$ is the function continuous?

Existence of Limits

	Graph ${\displaystyle y=\left(\frac{1}{x}\right)}$ and explain why ${\displaystyle \underset{x\to 0}{\lim }\left(\frac{1}{x}\right)}$ does not exist.
	Graph ${\displaystyle y=\tan \left(x\right)}$ and explain why ${\displaystyle \underset{x\to \pi /2}{\lim }\tan \left(x\right)}$ does not exist.
	Graph ${\displaystyle y=\sec \left(x\right)}$ and explain why ${\displaystyle \underset{x\to \pi /2}{\lim }\sec \left(x\right)}$ does not exist.
	Graph ${\displaystyle y=\csc \left(x\right)}$ and explain why ${\displaystyle \underset{x\to 0}{\lim }\csc \left(x\right)}$ does not exist.
	Graph ${\displaystyle y=\ln \left(x\right)}$ and explain why ${\displaystyle \underset{x\to -1}{\lim }\ln \left(x\right)}$ does not exist.
	Graph ${\displaystyle y=\sin \left(\frac{1}{x}\right)}$ and explain why ${\displaystyle \underset{x\to 0}{\lim }\sin \left(\frac{1}{x}\right)}$ does not exist.
	Graph ${\displaystyle y=\cos \left(x\right)}$ and explain why ${\displaystyle \underset{x\to \mathrm{\infty }}{\lim }\cos \left(x\right)}$ does not exist.
	Graph ${\displaystyle y=\mathrm{sgn}\left(x\right)}$ , where $\mathrm{sgn}\left(x\right)=\left\{\begin{array}{ll}1\text{,}& \text{if }x>0\\ 0\text{,}& \text{if }x=0\\ -1\text{,}& \text{if }x<0\end{array}\right.$. Explain why ${\displaystyle \underset{x\to 0}{\lim }\mathrm{sgn}\left(x\right)}$ does not exist.
	Graph ${\displaystyle y=2^{1/x}}$ and explain why ${\displaystyle \underset{x\to 0}{\lim }{2}^{1/x}}$ does not exist.
	Graph ${\displaystyle y=2^{1/\left(1-x\right)}}$ and explain why ${\displaystyle \underset{x\to 1}{\lim }{2}^{1/\left(1-x\right)}}$ does not exist.

References

[Ca] S. Carnie, 620-143 Applied Mathematics, Course materials, 2006 and 2007.

[Ho] C. Hodgson, 620-194 Mathematics B and 620-211 Mathematics 2 Notes, Semester 1, 2005.

[Wi] P. Wightwick, UMEP notes, 2010.