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\title{Math 120 Homework 5}
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\begin{itemize}
\item Due Friday October 20 at start of class.
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\item If your homework is longer than one page, {\bf staple} the pages together, and put your name on each sheet of paper. {\bf Homework lacking a staple will lose 2 points}.
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\item Each homework problem should be correct as stated. Occasionally, however, I might screw something up and give you an impossible homework problem. If you believe a problem is incorrect, please email me. If you are right, the first person to point out an error will get +1 on that homework, and I will post an updated version. 
\end{itemize}

\subsection*{Differentiation}

\noindent {\bf 1.} We will use the following fact: if $x\neq 1$, then  
\begin{equation}\label{sumFormular}
1+x+x^2+\ldots+x^n= \frac{x^{n+1}-1}{x-1}.
\end{equation}

Using Equation \eqref{sumFormular}, write down a formula for $1+2x+3x^2+4x^3+\ldots+nx^{n-1}$ which is valid for all values $x\neq 1$. 


\medskip

\noindent In lecture, we will prove that if $f(x)=\sin(x)$, then $f^\prime(x) = \cos(x)$, and if $f(x) = \cos(x)$, then $f^\prime(x) = -\sin(x)$. For the next problem you may use these facts freely.

\medskip

\noindent {\bf 2 a.} Let 
$$
f(x) = \left\{\begin{array}{ll} x^2\sin(1/x),&\ x\neq 0, \\ 0,&\ x=0.\end{array}\right.
$$
Prove that $f$ is differentiable at every point $x\in\RR$ (and thus $D(f^\prime)=\RR$). 

\medskip

\noindent {\bf b.} Prove that $f^\prime$ is continuous at every point $x\neq 0$, and is discontinuous at $x=0$.

\medskip

\noindent {\bf c.} Give an example of a function $f$ that is differentiable at every point $x\in\RR$, but whose derivative $f^\prime$ is discontinuous on the set $\{1,2,3\}\subset\RR$.

\subsection*{Newton iteration}

\noindent {\bf 3.} Use Newton's method with an initial guess $x_1=1$ to approximate the (positive) fourth root of 2, and show the computations. It's enough to iterate 3 times (i.e.~compute $x_2,x_2,x_4$), and feel free to use a calculator.

\medskip

\noindent {\bf 4.} Let $f(x) = x^3 - 2x + 2$.
\medskip 

\noindent {\bf a.} Prove that there is at least one point $x\in\RR$ for which $f(x)=0$ (i.e. $f$ has at least one root).
\medskip

\noindent {\bf b.} Let $x_1=0$. Use Newton's method to compute the first few iterates $x_2,\ x_3,\ x_4$. 
\medskip

\noindent {\bf c.}  Prove that for each $n=1,2,3,\ldots$, $|f(x_n)|\geq 1$, where $x_0,x_1,$ etc are the points from part b. (you could do a proof by induction, but you don't have to). 

\medskip
\noindent Remark: problem 4c shows that if you choose a bad starting guess $x_1$, then Newton's method might not find a root of $f$.

\subsection*{Intermediate Value Theorem \& Rolle's theorem}

\medskip
\noindent {\bf 5.} Let $f$ be a function that is continuous on the interval $[0,1],$ and suppose $0\leq f(x)\leq 1$ for all $x\in[0,1].$ Prove that there is at least one point $c\in [0,1]$ with $f(c)=c$.

\medskip

\noindent {\bf 6.} Use Rolle's theorem to prove that for each value of $b$, if we define $f(x) = x^3-3x+b$ then there is at most one point $x\in [-1,1]$ with $f(x)=0$.
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