\documentclass[addpoints,12pt]{exam} % can add 'answers' option
\setlength{\parindent}{0mm}
\usepackage{amsmath, amsfonts}
\usepackage{enumerate}
\usepackage{microtype}
\usepackage{tikz}

\newcommand{\pvec}[1]{\vec{#1}\mkern2mu\vphantom{#1}}
\newcommand{\blankunderline}[2]{\makebox[#1]{#2\enspace\hrulefill}}

\newcommand{\dueday}{Friday}
\newcommand{\duewhen}{12/4/20 at 11:59pm}
\newcommand{\hwnum}{8}
\newcommand{\classnumber}{Math 20C -- Swanson }
\newcommand{\classiteration}{Fall 2020}

\pagestyle{headandfoot}
\runningheadrule
\firstpageheader{\classnumber -- \classiteration}{}{Homework \hwnum}
\runningheader{\classnumber}
{Homework, Page \thepage\ of \numpages}
{Due \duewhen}
\firstpagefooter{}{}{}
\runningfooter{}{}{}

\begin{document}
\ 

\begin{center}
  \fbox{\parbox{3in}{\centering\large
  \classnumber -- \classiteration \\
  Homework \hwnum \\
  Due \dueday, \duewhen}}
\end{center}
\vspace{1cm}
%\blankunderline{0.8\textwidth}{Name:}
%\blankunderline{0.15\textwidth}{Section:}

\vspace{1cm}
\begin{itemize}
  \item The graded part of the homework is on WebAssign.
  \item The problems below are also assigned and you are responsible for doing them,
    but they will not be collected or graded.
\end{itemize}

\newpage

\begin{questions}
  \question Let $\vec{r}(t)$ be a differentiable path in $\mathbb{R}^3$.
    Use the differentiation rules on page 218 to evaluate the following
    in terms of $\vec{r}(t)$ and its derivatives.
    \begin{enumerate}[(a)]
      \item Evaluate...
        \begin{enumerate}[(i)]
          \item $\frac{d}{dt} (t^2 \hat\imath \times \vec{r}(t))$
          \item $\frac{d}{dt} (t^2 \hat\imath \cdot \vec{r}(t))$
          \item $\frac{d}{dt} (t^2 \vec{r}(t))$
          \item $\frac{d}{dt} (5t^2 \hat\imath + 3 \vec{r}(t))$
          \item $\frac{d}{dt} \vec{r}(t^2)$
        \end{enumerate}
      \item Evaluate $\frac{d}{dt} ||\vec{r}(t)||$.
        (Hint: recall that $||\vec{r}(t)||^2 = \vec{r}(t) \cdot \vec{r}(t)$.)
      \item Evaluate and simplify $\frac{d}{dt} (\vec{r}(t) \times \pvec{r}'(t))$.
      \item Prove that if the speed is constant, i.e.~$||\pvec{r}'(t)|| = k$ for
        some constant $k$, then the velocity and the acceleration vectors
        are orthogonal. (Hint: what is the value of $\pvec{r}'(t) \cdot \pvec{r}'(t)$ and
        how is it related to the speed? Differentiate this expression.)
    \end{enumerate}
    
    \begin{solution}
      
    \end{solution}
    \newpage

  \question A particle travels on the path $\vec{c}(t) = e^t\hat\imath
    + e^{-t}\hat\jmath - \sqrt{2} t \hat{k}$. Find the arc length
    between $e^{-3}\hat\imath + e^3\hat\jmath + 3\sqrt{2}\hat{k}$
    and $e^2\hat\imath + e^{-2}\hat\jmath - 2\sqrt{2} \hat{k}$.
    (Hint: in computing the speed, you will get the square root of a perfect
    square of a function. Simplify to get a function that's easy to integrate.)

    \begin{solution}
      
    \end{solution}
    \newpage
\end{questions}
\end{document}