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\noindent
\parbox{2.25in}{\bf Math 3220}
\hfill {\Large \bf Homework \#6} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Apr. $3^\text{rd}$, 2019}

\vspace{0.2in}

%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
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\begin{description}
\item[\#1 By the Definition] Use the limit definition of the derivative to find the derivative of $f(x)=3x^2-x+5$.

\item[\#2 Piecewise] Let $f(x) = \left\{ \begin{array}{ccc} x+1 & & x < 2 \\ x^2-3x+5 & & x \geq 2 \end{array} \right.$

\begin{enumerate}[(a)]
\item Show that $f$ is differentiable everywhere. 

You can use the usual derivative rules for $x \not=2$. For $x=2$, use the limit definition and approach from the left and right separately.

\item Suppose we change our function a little: $g(x) = \left\{ \begin{array}{ccc} x+100 & & x < 2 \\ x^2-3x+5 & & x \geq 2 \end{array} \right.$

Where is $g$ differentiable? Explain your answer.
\end{enumerate}

\item[\#3 Badly Behaved] Let $f(x) = \left\{ \begin{array}{ccc} 0 & & x \in \mathbb{Q} \\ x^2 & & x \in \mathbb{R}-\mathbb{Q} \end{array} \right.$

This is a lot like Dirichlet's function. 

\begin{enumerate}[(a)]
\item Show that $f$ is discontinuous at $x=a$ for all $a\not=0$.

{\it Suggestion:} Let $a\not=0$. Approach via a sequence of rationals and then via irrationals. 

\item Show that $f$ is differentiable at $x=0$. [You must use the limit definition of the derivative.]

\item Where is $f(x)$ continuous? Where is $f(x)$ differentiable? Explain you answers using your results from parts (a) and (b).
\end{enumerate}

\item[\#4 Easy Rule] Let $f(x)$ be differentiable at $x=a$. Let $k$ be some constant. Show that $(kf)'(a) = k\,f'(a)$ (i.e., we can pull constants out of derivatives).

\item[RESUBMIT] Type up Homework \#5 Problem \#5 and its solution in \LaTeX. 

\end{description}

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