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\title{Applied Calculus}
\author{Homework 3}
\date{Due in class, October 6, 2015}
\maketitle

\emph{Note: Homework 2, Problems 5 and 6 are also due October 6.}

\begin{enumerate}

\item Compute the derivative of $f(x)=x^2$ in three different ways.
\begin{enumerate}
\item Use the explicit definition of the derivative as the difference
quotient
\[
  f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}h \,.
\]

\item Use the power rule.
\item Use the product rule.
\end{enumerate}

\item Compute the derivative of the following functions.
\begin{enumerate}
\item $\displaystyle f(x) = 3 \, x^4 + \frac12 \, x^2 + 32 $
\item $\displaystyle f(x) = x^{3/2}+x^{-3/2}$
\item $\displaystyle f(x) = x \, \e^x$
\item $\displaystyle f(x) = \frac{e^x}x$
\item $\displaystyle f(x) = \sqrt{1+x^2}$
\item $\displaystyle f(x) = \frac{1+x}{1-x}$
\item $\displaystyle f(x) = x \, \ln x$
\item $\displaystyle f(x) = \ln (rx + x^2)$ where $r$ is a constant
\item $\displaystyle f(x) = \e^{x/k}$ where $k$ is a constant
\end{enumerate}
\hfill(2 Points each)


\item Determine whether the following functions are differentiable.
If not, state the point(s) of non-differentiability.  Explain your
answer.
\begin{enumerate}
\item $\displaystyle f(x) = \lvert x \rvert$
\item $\displaystyle f(x) = x^{1/3}$
\item $\displaystyle f(x) =
\begin{cases}
  0 & \text{for } x < 0 \\
  1 & \text{for } x \geq 0 
\end{cases}
$
\item $\displaystyle f(x) =
\begin{cases}
  0 & \text{for } x < 0 \\
  x^2 & \text{for } x \geq 0 
\end{cases}
$
\item $\displaystyle f(x) = \frac{1+x}{1-x}$

\end{enumerate}
\hfill(2 Points each)


\end{enumerate}




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