Assignment 1 -- Expressions and Graphing
620-295 Semester I 2010

Reformatted 7 April 2010

1. Some definitions.   Define the following:

	$\tan x$		$\cosh x$		$\tanh x$		${\sin }^{-1}x$
	$\arctan x$		${\csc }^{-1}x$		$\mathrm{arccosh}x$		${\mathrm{sech}}^{-1}x$

2. Basic properties.   Prove the following basic statements:

	$\ln x$ is the inverse function to $e^x$.		$\ln \left(xy\right)=\ln x+\ln y$.
	${\displaystyle \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots }$.		$\sin \left(x+y\right)=\sin x\cos y+\cos x\sin y$.
	${\displaystyle \sinh x=x+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+\cdots }$.		$\sinh \left(x+y\right)=\sinh x\cosh y+\cosh x\sinh y$.
	${\displaystyle \mathrm{arccosh}x=\log (x+\sqrt{x^2-1})}$.

3. Trig Identities.   Review your trig by proving the following equalities:

	${\sin }^{2}A{\cot }^{2}A=\left(1-\sin A\right)\left(1+\sin A\right)$.		${\displaystyle \frac{\sec A-1}{\sec A+1}+\frac{\cos A-1}{\cos A+1}=0}$.
	${\displaystyle \frac{\sin A}{\csc A}+\frac{\cos A}{\sec A}=1}$.		${\displaystyle \tan 3\alpha =\frac{3\tan \alpha -{\tan }^3\alpha }{1-3{\tan }^2\alpha }}$.
	${\displaystyle \cos \left(\pi /4-x\right)-\cos \left(\pi /4+x\right)=\sqrt{2}\sin x}$.		${\displaystyle \cot \left(x/2\right)=\frac{1+\cos x}{\sin x}}$.
	${\displaystyle \csc A\sec A=2\csc 2A}$.		${\displaystyle \frac{\sin \alpha +\sin 3\alpha }{\cos \alpha +\cos 3\alpha }=\tan 2\alpha }$.
	${\displaystyle {\cos }^2\theta =\frac{{\cot }^2\theta }{1+{\cot }^2\theta }}$.		${\displaystyle \frac{2{\tan }^{2}A}{1+{\tan }^{2}A}=1-\cos 2A}$.

4. Inverse expressions.   Review your inverse trig functions by proving the following equalities:

	${\displaystyle \cos \left({\sin }^{-1}x\right)=\sqrt{1-x^2}}$		${\displaystyle {\sin }^{-1}\left(-x\right)=-{\sin }^{-1}x}$
	${\displaystyle \mathrm{arcsinh}\left(\frac{x}{\sqrt{1-x^2}}\right)=\mathrm{arctanh}x}$

5. Derivatives.   Review the basics of derivatives with the following:

Let $D:\mathbb{Q}\left[x\right]\to \mathbb{Q}\left[x\right]$ be a function such that

(D1) If $f,g\in \mathbb{Q}\left[x\right]$ then $D(f+g)=D\left(f\right)+D\left(g\right)$,

(D2) If $c\in \mathbb{Q}$ and $f\in \mathbb{Q}\left[x\right]$ then $D\left(cf\right)=cD\left(f\right)$,

(D3) If $f,g\in \mathbb{Q}\left[x\right]$ then $D\left(fg\right)=fD\left(g\right)+D\left(f\right)g$ and

(D4) $D\left(x\right)=1$.

	Prove that if $n\in {\mathbb{Z}}_{>0}$ then $D\left(x^n\right)=n{x}^{n-1}$.
	Prove that if $f=c_0+c_{1}x+c_2{x}^2+c_3{x}^3+c_4{x}^4+⃛$ then $c_k=\frac{1}{k!}\left(D^{k}f\right){|}_{x=0}$.

6. Series expansions.   Find a series expansion for each of the following:

	$\cos x$		${\displaystyle \frac{1}{1+x^2}}$		$\cosh x$		$\sinh x$
	${\displaystyle \frac{1}{1+x}}$		$x\sin 3x$		${\displaystyle \int e^{x^3}dx}$		${\displaystyle {\int }_0^{\mathrm{t}}\sin \left(x^2\right)\hspace{0.17em}dx}$

7. Alternate formulations of series expansions.  

	Find the Taylor series for $e^x$ at the point $a=-3$.
	Find an alternate expression for the series ${\displaystyle \sum _{n=2}^{\mathrm{\infty }}n(n-1)x^n}$.
	Find the sum of the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n{2}^{n+1}}}$.

8. Some basic graphs.   Graph the following:

	$f\left(x\right)=x^2$		$f\left(x\right)=x^{100}$		$f\left(x\right)=x^{-100}$		$f\left(x\right)=\cot x$
	$f\left(x\right)=x^{1/4}$		$f\left(x\right)=x^{-1/4}$		$f\left(x\right)=\mathrm{arccot}x$

9. Additional graphing.   Make sure your graphing skills are at a level where they won't hamper performance later in the course by graphing the following:

	$f\left(x\right)=\left\{\begin{array}{cc}2+x\text{,}& \text{if }x>0\text{,}\\ 2-x\text{,}& \text{if }x\le 0\text{.}\end{array}\right.$		$f\left(x\right)=x^3$		$f\left(x\right)=2x^3+x^2+20x$
	$f\left(x\right)=x^3{\left(x-2\right)}^2$		${\displaystyle f\left(x\right)=\sqrt{x+2}}$		${\displaystyle f\left(x\right)=\frac{x^2}{\sqrt{x+1}}}$
	${\displaystyle f\left(x\right)=\left\{\begin{array}{ll}1\text{,}& \text{if }x>0\text{,}\\ 0\text{,}& \text{if }x=0\text{,}\\ -1\text{,}& \text{if }x<0\text{.}\end{array}\right.}$		${\displaystyle f\left(x\right)=x^{2/3}{\left(6-x\right)}^{1/3}}$		${\displaystyle f\left(x\right)=\sin 2x-x}$
	${\displaystyle y=e^{-x}}$

10. General ellipses and hyperbolas.   Recall how to graph ellipses and hyperbolas by graphing the following:

	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.

11. Using graphs to view sequences.   Graph the following sequences:

	$a_n=n\text{!}$.		$a_n={\displaystyle \frac{n}{n(n+1)}}$		$a_n={\displaystyle \frac{n}{2n+1}}$		$a_n={\displaystyle \frac{i^n}{n^2}}$
	${\displaystyle a_n=\frac{{\left(n\text{!}\right)}^2{5}^n}{\left(2n\right)\text{!}}}$.		$a_n=\sqrt{n}$.

12. Sequences in recursive form.   Graph the following sequences:

	Let $a\in ℝ$ with $a>0$. Fix a positive real number $x_1$ and let $x_{n+1}={\displaystyle \frac{1}{2}(x_n+\frac{a}{x_n})}$.
	Graph the sequence given by $a_1=0$,   $a_{2k}=\frac{1}{2}{a}_{2k+1}$,   and $a_{2k+1}=\frac{1}{2}+a_{2k}$.

13. Detecting continuity from a graph.  

	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 $$f\left(x\right)=\left\{\begin{array}{ll}1+x^2\text{,}& \text{if }0\le x\le 1\text{,}\\ 2-x\text{,}& \text{if }x>1\text{.}\end{array}\right.$$ For which values of $x$ is the function continuous?
	Graph $$f\left(x\right)=\left\{\begin{array}{ll}\sin x\text{,}& \text{if }x<0\text{,}\\ x\text{,}& \text{if }x\ge 0\text{.}\end{array}\right.$$ For which values of $x$ is the function continuous?
	Graph $$f\left(x\right)=\left\{\begin{array}{ll}{x}^3-x^2+2x-2\text{,}& \text{if }x\ne 1\text{,}\\ 4\text{,}& \text{if }x=1\text{.}\end{array}\right.$$ For which values of $x$ is the function continuous?

14. Detecting existence of limits from a graph.  

	Graph ${\displaystyle y=\ln \left(x\right)}$ and explain why ${\displaystyle \underset{x\to -1}{\lim }\ln \left(x\right)}$ does not exist.
	Graph ${\displaystyle y=2^{1/\left(1-x\right)}}$ and explain why ${\displaystyle \underset{x\to 1}{\lim }{2}^{1/\left(1-x\right)}}$ does not exist.