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\title{Math 3110: Homework 9}
%\author{Zach Norwood}
\date{\textcolor{navy}{Due \textcolor{red}{Monday (!)}, 15 April 2019}}

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\begin{document}
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\noindent
You will be graded on both the accuracy and the clarity of your solutions.
One purpose of the homework is to give you an opportunity to practice your proof-writing skills.

You are welcome\,---\,encouraged, even!\,---\,to collaborate on homework, but you should not 
copy solutions from any source, nor should you submit anything that you don't understand.

\addtocounter{problem}{-1}

\begin {problem} [don't submit any of this for a grade]
\hfill
 \begin {enumerate} [(a)]
  \item Review old homework problems and make sure you know how to do them.
  (But seriously.)
  
  \item Make sure you know how to deduce the standard corollaries to the 
  Mean Value Theorem: a differentiable function is constant iff its derivative is $0$,
  is increasing iff its derivative is never negative, etc.
  
  \item Prove that if $\displaystyle\lim_{x\to c}f(x) > 0$, then there 
  is a neighborhood of $c$ on which $f$ is positive.
 \end {enumerate}

And here are some good practice problems from the textbook:
 \begin{enumerate} [(a), resume]
  \item 4.4.7, 4.4.13.
  \item 4.2.7--4.2.11.
  \item 5.3.2--5.3.5 and 5.3.7--5.3.8.
 \end{enumerate}
\end {problem}

\begin {problem}
 Prove that the function $x\mapsto \sqrt x$ is uniformly continuous on $[0,\infty)$.
\end {problem}

\begin {problem} [Constructing familiar functions II]
Review Homework 8, Problem 3.
Fix a real number $b>1$.
 \begin{enumerate} [(a)]
 \item[($\star$)] (Don't turn anything in for this one.)
 What do you think $2^\pi$ should mean? 
  \item Prove that if $r$ is a rational number then
  \[
   b^r = \sup\set{ b^t}{ t\in\Q, \, t<r }.
  \]
 It is therefore consistent with our previous definition to define, for any $x\in\R$,
 \[
  b^x = \sup\set{ b^t }{ t\in\Q,\, t< x}.
 \]
 \item
 Prove that for all $x,y\in\R$ we have $b^{x+y} = b^x\cdot b^y$.
 \item Prove that the function $\operatorname{exp}_b\colon\R\to\R$ defined by
 $\operatorname{exp}_b(x) = b^x$ is continuous and strictly increasing.
 \item Prove that the range of $\operatorname{exp}_b$ is the set $\set{y\in\R}{y>0}$.
 \end{enumerate}
\end {problem}

\begin {problem} [Compare with Homework 7, Problem 7]
 Suppose that $f\colon [a,b]\to\R$ is a one-to-one differentiable function whose derivative
 is never $0$: for all $x\in[a,b]$, $f'(x)\ne0$. Prove that its inverse $f\inv$ is differentiable with
 derivative given by
 \[ 
  (f\inv)'(y) = \frac{1}{f'(f\inv(x))}.
 \]
 Use this to produce the familiar formula for the derivative of the square-root function.
\end {problem}

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\begin {problem} [essentially 4.3.11 from the textbook]
 Suppose that $f\colon\R\to\R$ is a function, and assume that there is a constant $c$ such that,
 for all $x,y\in\R$,
 \begin{equation}	\label{eq:cmp}	\tag{$\star$}
  \abs{ f(x) - f(y) } \le c\abs{x - y}.
 \end{equation}
 \begin{enumerate} [(a)]
  \item
  Explain (in words) what our assumption~\eqref{eq:cmp} means geometrically.
  \item
  Prove that $f$ is continuous (on $\R$).
  \item
  Show that if $y_1$ is any real number, then the sequence $(f^n(y_1))_{n\in\N}$ is Cauchy
  and therefore converges.
  (``$f^n$'' means $n$ applications of $f$, so $f^0(y_1) = y_1$ and $f^{n+1}(y_1) = f(f^{n}(y_1))$.)
  \item
  Let $y_\infty$ be the limit of the sequence from the previous part.
  Prove that $y_\infty$ is a fixed point of $f$ (meaning $f(y_\infty) = y_\infty$)
  and that it is the \emph{unique} fixed point of $f$.
  \item
  Conclude that if $z_1$ is any real number, then the sequence $(f^n(z_1))_{n\in\N}$ 
  converges to the same $y_\infty$ from the previous parts.
  \item
  Assuming standard facts about the sine and cosine functions (which we haven't yet proved!),
  what does this problem tell you about sine and cosine?  
 \end {enumerate}
\end {problem}

\begin {problem}	
 \hfill
 \begin{enumerate} [(a)]
  \item Suppose that $g\colon [0,5] \to \R$ is differentiable, $g(0) = 0$, and $g$ has bounded derivative
  on $[0,5]$: say $\abs{g'(x)}\le M$. Show for all $x\in [0,5]$ that $\abs{g(x)}\le Mx$.
  
  \item Suppose that $h\colon [0,5]\to \R$ is twice-differentiable (meaning that it's differentiable,
  and its derivative is differentiable), that $h'(0) = h(0) = 0$ and $\abs{h''(x)}\le M$.
  Show for all $x\in[0,5]$ that $\abs{h(x)} \le Mx^2/2$.
  
  \item
  Can you give a geometric interpretation of the previous two parts?
 \end{enumerate}
\end {problem}

\begin {problem}
For this problem, assume that $\fcos \colon \R\to\R$ is function with the following properties:
 \begin {enumerate} [(i)]
  \item $\fcos$ is differentiable;
  \item for all $x\in\R$, $\fcos(x) \in[-1,1]$;
  \item for all integers $n\in\Z$, $\fcos(2n) = 1$ and $\fcos(2n+1) = -1$ and $\fcos(n + \frac12) = 0$.%
 \end{enumerate}
(We might later prove that $x\mapsto \cos(\pi x)$ has these properties, but 
we haven't yet defined sine or cosine or $\pi$!!!)
 \begin {enumerate} [(a)]
  \item
  Prove that the function $f\colon \R\to\R$ given by
  \[
   f(x) = \begin{cases}
   		\fcos(1/x)	&\text{ if } x\ne 0 \\
		0		&\text{ if } x = 0
	\end{cases}
  \]
  has the intermediate value property but is not continuous at $x = 0$.
  \item
  Prove that the function given by
  \[
   g(x) = \begin{cases}
   		x \cdot\fcos(1/x)	&\text{ if } x\ne 0 \\
		0			&\text{ if } x = 0
	\end{cases}
  \]
  is continuous but not differentiable at $x = 0$.
  
  (\emph{Hint}: Look at the animation on the course webpage.)
  \item
    Prove that the function given by
  \[
   h(x) = \begin{cases}
   		x^2 \fcos(1/x)	&\text{ if } x\ne 0 \\
		0			&\text{ if } x = 0
	\end{cases}
  \]
  is differentiable everywhere (in particular at $x = 0$), but its derivative is discontinuous
  at $x = 0$.
 \end {enumerate}
\end {problem}


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