% TEMPLATE FOR STANDARD EXAM
% (with some sample problem types)
% by laura


\documentclass[11pt]{article}
\usepackage{tikz}
\usetikzlibrary{decorations.pathreplacing,decorations,decorations.pathmorphing}
\usetikzlibrary{patterns}
\usepackage{amsmath}
\usepackage{xypic}
\usepackage{wrapfig}
\usepackage{graphicx}
\usepackage{enumerate}
\usepackage{changepage}

\usepackage{fancyhdr,lastpage}
\pagestyle{fancy}
\cfoot{\thepage\ of \pageref{LastPage}}
\rhead{\hspace{2in}{Name: } $\underline{\hspace{1.7in}}_{\mbox{}}$}

\oddsidemargin=0.5in
\evensidemargin=0.5in
\textwidth=6.3in
\headwidth=6.3in
\topmargin=-.5in
\textheight=9in
\marginparwidth=0pt

\parindent=0in
%\pagestyle{empty}

% Create the problem number counter.  Initialize to zero.
\newcounter{problemnum}

% Specify that problems should be labeled with arabic numerals.
\renewcommand{\theproblemnum}{\arabic{problemnum}}

% Create the part-within-a-problem counter, "within" the problem counter.
% This counter resets to zero automatically every time the PROBLEMNUM counter
% is incremented.
\newcounter{partnum}[problemnum]

% Specify that parts should be labeled with lowercase letters.
\renewcommand{\thepartnum}{\alph{partnum}}

\newenvironment{problem}{
% STATEMENTS TO BE EXECUTED WHEN A NEW PROBLEM IS BEGUN:
%
% Increment the problem number counter, and set the current \ref value to that
% number.
\refstepcounter{problemnum}
%
% Add some vspace to separate from the last problem.
\vspace{0.15in} \par
%
%
% Now put in the "announcement" on the page.
{\Large \bf \theproblemnum. \normalsize %({\it \arabic{curprobpts}
%{point\null\ifnum \value{curprobpts} = 1\else s\fi}\/)
}
}{
% STATEMENTS TO BE EXECUTED WHEN AN OLD PROBLEM IS ENDED:
%
% If no parts to problem, then increment TOTALPOINTS and PAGEPOINTS for the
% entire problem at once.

}


\newenvironment{mproblem}{
% STATEMENTS TO BE EXECUTED WHEN A NEW PROBLEM IS BEGUN:
%
% Increment the problem number counter, and set the current \ref value to that
% number.
\refstepcounter{problemnum}
%
% Add some vspace to separate from the last problem.
\vspace{0.15in} \par
%
%
% Now put in the "announcement" on the page.
\begin{adjustwidth}{1.05in}{0pt}
{\hspace{-1.05in}$\underline{\hspace{0.5in}}_{\mbox{}}$\hspace{0.2in}\Large \bf \theproblemnum. }
}{
% STATEMENTS TO BE EXECUTED WHEN AN OLD PROBLEM IS ENDED:
%
% If no parts to problem, then increment TOTALPOINTS and PAGEPOINTS for the
% entire problem at once.
\end{adjustwidth}
}

\begin{document}

%\Large
% ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
%                             COVER
% ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 
%\Large

% \begin{enumerate}
% \item Ability to calculate complex derivatives (double chain rule, mixed products and chains, quotients, etc.)
% \item Ability to calculate derivatives with unknowns (like $[g(\ln x)]'$)
% \item Knowing basic derivatives: exp, ln, trig, inverse trig
% \item Some sentence parsing in terms of blah is the derivative of blah, or translating between tangent lines and functions
% \item Coping with various notions
% \item Reading derivative values from a chart
% \item Using def of a derivative
% \item Fill in a value to make a function differentiable
% \item Implicit derivatives
% \item Multiple derivatives
% \item Using derivatives to approximate/tangent lines
% \item Derivatives of inverses
% \item Identifying differentiability
% \item Logarithmic differentiation
% \item Related rates
% \end{enumerate}

% \clearpage

\begin{mproblem}
  If $f(x)=x^2\cos(4x+\pi/2)$, what is $f'(\pi)$?\\
  \begin{tabular}{p{2in}p{2in}}
    a.  $2\pi+4\pi^2$ & e. $2\pi$\\
    b. $2\pi-4\pi^2$  & f. $-2\pi$\\
    c. $4\pi^2-2\pi$ & g. $4\pi^2$\\
    d. $-(2\pi+4\pi^2)$ & h. $-4\pi^2$\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
  Find the linearization of $f(x)=x^4$ at $x=2$.\\
  \begin{tabular}{p{2in}p{2in}}
    a. $16+2(x-32)$  & e.   $32+16(x-2)$\\
    b. $2+16(x-32)$ & f.  $16+32(x-2)$\\
    c. $2+x^4(x-32)$ & g. $16+x^4(x-2)$\\
    d. $2+4x^3(x-32)$ & h. $16+4x^3(x-2)$\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
If $f(x)=\frac{(x+1)^3e^x}{(x^3+2)(x^4+2)}$, what is $f'(0)$?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $-1$  & e. $-1/2$ \\
    b. $0$ & f. $e$ \\
    c. $1$ & g. $1/e$ \\
    d.  $1/2$ & h. $e/2$ \\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
Suppose that $g(x)$ is a function with $g'(x)=e^{x^2}$.  If $h(x)=g(\ln x)$, what is $h'(x)$?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $e^{x^2}$  & e. $e^{(\ln x)^2}/x$ \\
    b. $e^{x^2}/x$ & f. $2xe^{x^2}\ln x$\\
    c. $e^{x^2}/\ln x$ & g. $2e^{x^2}$\\
    d. $e^{(\ln x)^2}$  & h. $2x^3$\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
The following chart gives some values of $g,h,g',h'$.\\
\begin{tabular}{|l|l|l|l|l|}
\hline
  x&g(x)&g'(x)&h(x)&h'(x)\\
\hline
2&5&2&5&4\\
3&3&4&1&3\\
4&2&1&3&2\\
5&4&5&2&5\\
\hline
\end{tabular}\\
If $f(x)=h(g(x)\cdot h(x))$, what is $f'(3)$?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $2$  & e. $25$ \\
    b. $5$ & f. $39$ \\
    c. $10$ & g. $60$\\
    d. $24$  & h. $65$\\
  \end{tabular}
\end{mproblem}


\begin{mproblem}
If $f(x)=\frac{x}{|x|+1}$, what is $f'(0)$?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $-2$  & e. $1/2$ \\
    b. $-1$ & f. $1$\\
    c. $-1/2$ & g. $2$\\
    d. $0$  & h. Does not exist\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
If the function 
\[f(x)=\left\{\begin{array}{ll}
4bx^2+2x+2&\text{if }x\leq 1\\
5x^4-x^2+4bx&\text{if }x>1
\end{array}\right.\]
 is differentiable for all $x$ then what is the value of the constant $b$?\\
 \begin{tabular}{p{2in}p{2in}}
    a. $-4$  & e. $1$ \\
    b. $-2$ & f. $2$\\
    c. $-1$ & g. $4$\\
    d. $0$  & h. This function cannot be differentiable for all $x$.\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
What is $\frac{dy}{dx}$ at $(1,1)$ if $y^3-x\ln y+x=2$?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $-2$  & e. $1/2$ \\
    b. $-1$  & f. $1$\\
    c. $-1/2$ & g. $2$\\
    d. $0$  & h. $e$\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
Consider the function $f(x)=2x^3-x^2+2x+1$.  What value should $a$ have so that the tangent lines at $a$ and $a+1$ are parallel?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $-1$  & e. $1/4$ \\
    b. $-1/2$ & f. $1/3$\\
    c. $-1/3$ & g. $1/2$\\
    d. $0$ & h. $1$\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
If $f(x)=\sin^{-1}\sqrt{x}$, what is $f'(1/2)$?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $-\sqrt{2}$  & e. $\sqrt{2}$ \\
    b. $-1$ & f. $1$\\
    c. $-\sqrt{2}/2$ & g. $\sqrt{2}/2$\\
    d. $-1/2$  & h. $1/2$\\
  \end{tabular}
\end{mproblem}

\begin{problem}
What is $\frac{d^{37}}{dx^{37}}xe^x$?
\end{problem}

% \begin{problem}
% A spotlight stands on top of a $10$ meter pole.  A person runs away from the pole at a rate of $20$ meters per second.  The spotlight turns so that it is always pointing directly at the person's feet.  When the person is $50$ meters \emph{from the spotlight}, how quickly is the angle $\theta$ between the spotlight and the pole changing?  (Do not leave unevaluated trig functions in your final answer.)

% \usetikzlibrary{shapes}
% \usetikzlibrary{decorations.text}
% \usetikzlibrary{shapes.callouts}
% \begin{tikzpicture}
%   \draw (0,0)--(0,2);
% \node at (-0.6,1) {$10$ m};
% \draw[<->,dashed] (-0.1,0) -- (-0.1,2);
% \draw (-0.1,2.1) -- (0.1,1.9) -- (0.2,2)--(0,2.2) -- cycle;
% \draw[loosely dashed] (0.15,1.95) -- (2,0);
% \draw [<->,dashed] (0,1.4)  arc (-90:-45:.6);
% \node at (0.3,1.3) {$\theta$};
% \draw (1.8,0) --(2,0.2);
% \draw (2.2,0) --(2,0.2);
% \draw (2,0.2) -- (2,0.6);
% \draw (2,0.6) -- (1.8,0.52);
% \draw (2,0.6) -- (2.2,0.52);
% \draw (2,0.61) arc (-90:270:0.1);
% \node[draw,ellipse callout,text width=2.2cm,font=\tiny,callout absolute pointer={(2,0.64)}] at (3.2,1.5) {Eek, the spotlight is chasing me!};
% \end{tikzpicture}
% \end{problem}


\begin{mproblem}
  What is the derivative of $xe^{|x|}$ at $0$?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $-1$ & e. $e$\\
    b. $0$ & f. $-e$\\
    c. $e^{-1}$ & g. $-e^{-1}$\\
    d. $1$ & h. Not differentiable at $0$\\
  \end{tabular}
\end{mproblem}


\begin{mproblem}
  Let $f(x)=\left\{\begin{array}{ll}x^2\sin\frac{1}{x}&\text{if }x\neq 0\\0&\text{if }x=0\end{array}\right.$.
  What is the derivative of $f(x)$ at $0$?  (Remember that $\lim_{x\rightarrow 0}x\sin\frac{1}{x}=0$.)\\
  \begin{tabular}{p{2in}p{2in}}
    a. $-\pi$ & e. $1$\\
    b. $-\pi/2$ & f. $\pi/2$\\
    c. $-1$ & g. $\pi$\\
    d. $0$ & h. Not differentiable at $0$\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
If $g,h$ are differentiable functions and $f(x)=g(e^{h(x)})$, what is $f'(x)$?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $g'(e^{h(x)})$& e. $g'(e^{h(x)})e^{h(x)}$\\
    b. $g'(h(x)e^{h(x)})$ & f. $g'(e^{h(x)})e^{h'(x)}$\\
    c. $g'(h'(x)e^{h(x)})$ & g. $g'(e^{h(x)})e^{h(x)}h'(x)$\\
    d. $g'(e^{h'(x)})$ & h. $g'(e^{h(x)})h'(x)$\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
Approximate the value of $\cos(\pi/2+0.01)$.\\
  \begin{tabular}{p{2in}p{2in}}
    a. $0.01$  & e. $\pi/2$ \\
    b. $0.01\sqrt{2}/2$ & f. $0$ \\
    c. $-0.01$ & g. $\sqrt{2}/2$\\
    d. $-0.01\sqrt{2}/2$  & h. $-\sqrt{2}/2$\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
The derivative (with respect to $x$) of $4bx^2-4x+2$ at $-1$ is $4$.  What is $b$?
  \begin{tabular}{p{2in}p{2in}}
    a. $1$  & e. $-1/2$ \\
    b. $-1$ & f. $2$\\
    c. $0$ & g. $-2$\\
    d. $1/2$  & h. $4$\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
What is $\frac{dy}{dx}$ at $(3,3)$ on the curve $xe^y=ye^x$?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $0$  & e. $-3$ \\
    b. $1$ & f. $e^3$\\
    c. $-1$ & g. $-e^3$\\
    d.  $e$ & h. $-e^{-3}$\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
The following chart gives some values of $f$ and $f'$.\\
\begin{tabular}{|l|l|l|}
\hline
  x&f(x)&f'(x)\\
\hline
2&4&3\\
3&3&4\\
4&2&1\\
5&1&5\\
\hline
\end{tabular}\\
What is $\frac{d}{dx}f^{-1}(4)$?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $1/5$  & e. $1$ \\
    b. $1/4$ & f. $2$\\
    c. $1/3$ & g. $3$\\
    d. $1/2$  & h. $4$\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
If $f(x)=(\sin^{-1}x)(\cos^{-1}x)$, what is $f'(\sqrt{2}/2)$?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $-\pi$  & e. $1$ \\
    b. $-\pi/2$ & f. $\pi/4$\\
    c.  $-\pi/4$ & g. $\pi/2$\\
    d.  $0$ & h. $\pi$\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
Let $f(x)$ be a function such that $f(0)=0$ and $\lim_{h\rightarrow 0}\frac{f(h)}{h}=2$.  What is $f'(0)$?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $-2$  & e. $1/2$ \\
    b. $-1$ & f. $1$\\
    c. $-1/2$ & g. $2$\\
    d. $0$  & h. $f$ is not differentiable at $0$.\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
If $f(x)=2^{2^{x}}$, what is $f'(0)$?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $(\ln 2)2^{\ln 2}$  & e. $2^{2(\ln 2)^2}$ \\
    b. $2^{(\ln 2)2^{\ln 2}}$  & f. $2^{\ln 2}$ \\
    c. $2(\ln 2)^2$ & g. $2$\\
    d. $(\ln 2)2^{2\ln 2}$  & h. $\ln 2$\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
  Consider the function
\[f(x)=\left\{\begin{array}{ll}
2/x&\text{if }x<1\\
-x^2+3&\text{if }1\leq x<2\\
x^4/16&\text{if }2<x
\end{array}\right..\]
At which points does this function fail to be differentiable?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $\{0,1,2\}$ & e. $\{0\}$ \\
    b. $\{0,1\}$ & f. $\{1\}$\\
    c. $\{0,2\}$ & g. $\{2\}$\\
    d. $\{1,2\}$ & h. This function is differentiable everywhere\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
Find the tangent line to the curve $4x^3y-2x^2y^2=4x^3$ at the point $(1,2)$.\\
  \begin{tabular}{p{2in}p{2in}}
    a. $y-1=x(x-2)$  & e. $y-1=-(x-2)$ \\
    b. $y-1=4x^3(x-2)$ & f. $y-1=2(x+1)$\\
    c. $y+1=x-2$ & g. $y-2=x+1$\\
    d. $y+1=2(x-1)$  & h. $y-2=-(x-1)$\\
  \end{tabular}
\end{mproblem}

\begin{problem}
  Find the derivative of
\[\frac{x^2\cos x}{(x^2+4)(x-1)\ln x}.\]
\end{problem}


\begin{problem}
A strange swimming pool has been built in the shape of an upside-down pyramid with a square base.   At ground level, the base of the pyramid is a square with each side 6 feet long.  The pyramid is 9 feet deep and is filled part way up with water.

\usetikzlibrary{snakes}
\begin{tikzpicture}
\draw (0,0) -- (-2,2) -- (2,2) -- cycle;

\node at (0,2.3) {$6$ ft};
\draw[<->,dashed] (-2,2.1) -- (2,2.1);
\draw[<->,dashed] (2.2,2) -- (2.2,0);
\node at (2.6, 1) {$9$ ft};
\draw[<->,dashed] (-1.2,0) -- (-1.2,1);
\node at (-1.4,0.5) {$h$};
\node at (0,-0.5) {Side View};

\draw[snake=coil,segment aspect=0,segment length=5pt,segment amplitude=1pt] (-1,1) -- (1,1);
\draw[<->,dashed] (-1,1.1) -- (1,1.1);
\node at (0,1.2) {$s$};


\draw (4,0) -- (6,0) -- (6,2) -- (4,2) -- cycle;
\draw[snake=coil,segment aspect=0,segment length=5pt,segment amplitude=1pt] (4.5,0.5) -- (5.5,0.5) -- (5.5,1.5) -- (4.5,1.5) -- cycle;
\draw[<->,dashed] (4.5,1.6) -- (5.5,1.6);
\node at (5,1.7) {$s$};
\draw[<->,dashed] (5.6,0.5) -- (5.6,1.5);
\node at (5.7,1) {$s$};
\node at (5,-0.5) {Top View};

\draw[<->,dashed] (4,2.1) -- (6,2.1);
\node at (5,2.3) {$6$ ft};
\draw[<->,dashed] (6.1,0) -- (6.1,2);
\node at (6.5,1) {$6$ ft};

\end{tikzpicture}

\bigskip

The owner begins filling it with 1.5 ft$^3$/min of water.  How quickly is the water level rising when the water is 3 feet deep?  (The volume of a pyramid---any pyramid---is $\frac{1}{3}hb$ where $h$ is the height and $b$ is the area of the base.)
\end{problem}


\begin{mproblem}
Estimate the value of $\sqrt[3]{8.1}$.\\
  \begin{tabular}{p{2in}p{2in}}
    a. $2+1/12$  & e.   $2+1/80$\\
    b. $2+1/16$ & f.  $2+1/90$\\
    c. $2+1/30$ & g. $2+1/120$\\
    d. $2+1/40$ & h. $2+1/160$\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
If $f(x)=\frac{(x+2)^2(x+3)^3}{e^x(x^2+4)}$, what is $f'(0)$?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $-72$  & e. $72$ \\
    b. $-54$ & f. $54$ \\
    c. $-27$ & g. $27$ \\
    d.  $-9$ & h. $9$ \\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
Suppose that $g(x)$ is a function with $g'(x)=\frac{\sin x}{x}$.  If $h(x)=g(\sin^{-1} x)$, what is $h'(x)$?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $\frac{1}{\sqrt{1-(\sin^{-1}x)^2}}$  & e. $\frac{\sin^{-1}x}{\sqrt{1-x^2}}$ \\
    b. $\frac{1}{(\sin^{-1}x)\sqrt{1-x^2}}$ & f. $\frac{x}{\sqrt{1-x^2}}$\\
    c. $\frac{x}{\sin^{-1}x+\sqrt{1-x^2}}$ & g. $\frac{x}{(\sin^{-1}x)\sqrt{1-x^2}}$\\
    d. $\frac{x}{\sin^{-1}x}$  & h. $2x^3$\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
The following chart gives some values of a function $g$ and $g'$:\\
\begin{tabular}{|l|l|l|}
\hline
  x&g(x)&g'(x)\\
\hline
2&8&2\\
3&3&6\\
4&2&1\\
5&4&5\\
\hline
\end{tabular}\\
If $f(x)=\ln g(x)$, what is $f'(3)$?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $2$  & e. $\ln 2$ \\
    b. $4$ & f. $e^3$ \\
    c. $5$ & g. $\ln 4$\\
    d. $8$  & h. $e^5$\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
If the function 
\[f(x)=\left\{\begin{array}{ll}
bx^2+b^2x&\text{if }x\leq b\\
2x^3+bx-b^2&\text{if }x>b
\end{array}\right.\]
 is differentiable for all $x$ and $b<0$ then what is the value of the constant $b$?\\
 \begin{tabular}{p{2in}p{2in}}
    a. $-4$  & e. $-1/2$ \\
    b. $-3$ & f. $-1/3$\\
    c. $-2$ & g. $-1/4$\\
    d. $-1$  & h. This function cannot be differentiable for all $x$.\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
What is $\frac{dy}{dx}$ at $(1,1)$ if $xy^2+\ln x-\ln  y+y=2$?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $-2$  & e. $1/2$ \\
    b. $-1$  & f. $1$\\
    c. $-1/2$ & g. $2$\\
    d. $0$  & h. $e$\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
Consider the function $f(x)=2x^2-3x+1$.  What value should $a$ have so that the tangent lines at $a$ is parallel to the line $y=x$?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $-1$  & e. $1/4$ \\
    b. $-1/2$ & f. $1/3$\\
    c. $-1/3$ & g. $1/2$\\
    d. $0$ & h. $1$\\
  \end{tabular}
\end{mproblem}

\begin{mproblem}
If $f(x)=\tan^{-1}e^x$, what is $f'(0)$?\\
  \begin{tabular}{p{2in}p{2in}}
    a. $-\sqrt{2}$  & e. $\sqrt{2}$ \\
    b. $-1$ & f. $1$\\
    c. $-\sqrt{2}/2$ & g. $\sqrt{2}/2$\\
    d. $-1/2$  & h. $1/2$\\
  \end{tabular}
\end{mproblem}

\begin{problem}
What is $\frac{d^{36}}{dx^{36}}x\sin x$?
\end{problem}

\begin{problem}
  A fish (on the surface of the water) is being reeled in from a point $12m$ above the surface of the water.  The fishing line being shortened at a rate of $4m/
s$.

\xyoption{curve}
\diagram
\ar@{--}[4,0]_{12m}\xline[4,4]\\
&\\
&\\
&&&&&&\\
\ar@{~}[0,4]&&\ar @{-} @/^/ [-1,1]_\theta&&\propto\\
\enddiagram

How quickly is $\theta$ changing when the fishing line is $25m$ long?


\end{problem}

\begin{problem}
  You have fallen off a boat into a river.  Your car, with a change of clothes, is parked on the side of the river, and you would like to get there as quickly as possible so you can change into dry clothes.

The river bank is $8$ miles away from you at the nearest spot, and your car is $4$ miles away from that down the river bank.

You can swim at a rate of $1$ mile/hour, and you can walk on land at a rate of $\sqrt{17}$ miles/hour, so your fastest route is to swim to some point on the edge of the river bank and then run the rest of the way.

How far away from the car (the value $x$ in the picture below) should you reach the river bank so that you get to the car as quickly as possible?  (Remember that the time it takes to travel a distance is $\frac{\mathrm{distance}}{\mathrm{speed}}$.)

\begin{tikzpicture}
  \draw (0,0) -- (6,0);
\draw[decoration={brace,amplitude=6pt},decorate] (1,-2.5) -- (1,0) node[midway, left=3pt]{8 miles};
\draw[fill=black] (1,-2.5) circle (0.05);
\node at (1,-2.9) {You};
\draw[fill=black] (5,0) circle (0.05);
\node at (5,0.4) {Car};
\draw[decoration={brace,amplitude=6pt},decorate] (1,0) -- (5,0) node[midway, above=3pt]{4 miles};
\draw[decoration={brace,mirror, amplitude=6pt},decorate] (3,0) -- (5,0) node[midway, below=3pt]{x};
\draw[dashed] (1,-2.5) -- (3,0);
\node at (7,0) {River Bank};
\node at (4,-1) {River};
\end{tikzpicture}

\end{problem}

\begin{problem}
  Some values of the functions $f$ and $g$ are given in the table below.

\begin{tabular}{|l|c|c|c|c|}
\hline
$x$&$f(x)$&$f'(x)$&$g(x)$&$g'(x)$\\
\hline
 $0$&$1$&$2$&$-1$&$3$\\
\hline
$2$&$-3$&$-1$&$2$&$-2$\\
\hline
\end{tabular}

If $h(x)=f(1-x)g(1+x)$, what is $h'(1)$?
\end{problem}

\begin{problem}
  The curves (i), (ii), and (iii) in the graph below are the graphs of
  a function $f$ and its first and second derivatives.  Which curve is
  $f$, which is $f'$, and which is $f''$?  Explain.

  \includegraphics[scale=0.5]{CurveSketch}
\end{problem}

\begin{problem}
  Let $f(x)=\sin^{-1}\left(2e^{x}\right)$.  What is $f'(\ln (3/10))$?
\end{problem}

\begin{problem}
  A curve is given implicitly by the equation $x^2-y^2=x^2y$.  What is the tangent line to this curve at the point $(\sqrt{2}/2,1/2)$?
\end{problem}

\begin{problem}
  A rectangular field adjacent to a river is to be enclosed.  Fencing along the river costs \$5 per food but fencing along the other three sides costs only \$3 per foot.  The area of the field is to be 4800 square feet.  What are the dimensions of the cheapest possible rectangle field with the right size?  (Be sure to check that you have found a minimum and not a maximum!)
\end{problem}

\begin{problem}
  Find all critical points of $\frac{x^2+4}{x}$.  Identify which are local minima and which are local maxima.
\end{problem}

\begin{problem}
  What are the global minimum and maximum values of $f(x)=\frac{x}{1+x^2}$.  (This should be the largest and smallest values $f(x)$ can ever have, not the the $x$-values you plug in to get them.)
  \begin{tabular}{p{2in}p{2in}}
    a. $2$ and $-2$  & d. $2$ and $0$ \\
    b. $1$ and $-1$ & e. $4$ and $-4$\\
    c. $1/2$ and $-1/2$ & f. $\infty$ and $-\infty$\\
  \end{tabular}
\end{problem}

\begin{problem}
  Suppose you have two functions $f$ and $g$, both of which are differentiable everywhere. Suppose that $f'(x)=x^2$ and $g'(3)=17$.  What additional piece of information do you need in order to compute $(f\circ g)'(3)$?
\end{problem}

\begin{problem}
  Let $f(x)=x^3-3x+2$.

  \begin{enumerate}
  \item Where are the critical points of $f(x)$?
  \item On which intervals is $f(x)$ increasing?
  \item On which intervals is $f(x)$ concave up?
  \end{enumerate}
\end{problem}
\end{document}

Problems to include:
Related rates
xEasy linearization
xFinding derivative using chart for chain rule
xFind a value making a function differentiable
xA multistep derivative
xAn implicit derivative
xA problem where log. diff is appropriate
xA derivative with just names
An inverse
Interval of differentiability


\begin{mproblem}

  \begin{tabular}{p{2in}p{2in}}
    a.   & e.  \\
    b.  & f. \\
    c.  & g. \\
    d.   & h. \\
  \end{tabular}
\end{mproblem}
\vfill