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\newcommand{\dueday}{Friday}
\newcommand{\duewhen}{11/13/20 at 11:59pm}
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\newcommand{\classnumber}{Math 20C -- Swanson }
\newcommand{\classiteration}{Fall 2020}

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{Due \duewhen}
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\begin{document}
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  \classnumber -- \classiteration \\
  Homework \hwnum \\
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\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 Sketch these parametric curves by hand, using methods discussed in class
    for Chapter 2.4:
    \begin{enumerate}[(a)]
      \item $\vec{c}(t) = (t \cos(t), t \sin(t))$, where $0 \leq t \leq 4\pi$.
      \item $x =-3t \sin(t)$, $y = 4t \cos(t)$, where $0 \leq t \leq 4\pi$.
      \item $\vec{c}(t) = -\sin^2(t) \hat\imath + \cos^2(t) \hat\jmath$.
      \item $\vec{c}(t) = 2t \hat\imath + 3 \cos(t) \hat\jmath + 4 \sin(t) \hat{k}$.
      \item $x = \cos(t), y = \sin(t), z = \sin(3t)$.
      \item $x = 5e^{-t}, y=e^t$, where $0 \leq t \leq \ln(10)$.
    \end{enumerate}
    After sketching them by hand, check them with a computer.
    
    \begin{solution}
      
    \end{solution}
    \newpage
  
  \question
    Recall the first part of the Fundamental Theorem of Calculus:
    if $F(x) = \int_a^x f(t)\,dt$, then $F'(x) = f(x)$. For example,
    if $F(x) = \int_a^x \sin(\sqrt{t+8})\,dt$, then $F'(x) = \sin(\sqrt{x + 8})$.
    The integral $\int_0^x e^{-t^2}\,dt$ arises in probability and statistics.
    It can't be evaluated in terms of ordinary functions. Nonetheless,
    we may compute:
    
    \begin{enumerate}[(a)]
      \item $\frac{d}{dx} \int_0^x e^{-t^2}\,dt$.
      \item $\frac{d}{dx} \int_0^{x^3} e^{-t^2}\,dt$ by completing the
        following outline. Let $G(u) = \int_0^u e^{-t^2}\,dt$.
        Note that $\int_0^{x^3} e^{-t^2}\,dt = G(x^3)$.
        Use the chain rule to compute $\frac{d}{dx} G(x^3)$.
      \item Compute $\frac{\partial}{\partial x} \int_{10y}^{x^3} e^{-t^2}\,dt$
        and $\frac{\partial}{\partial y} \int_{10y}^{x^3} e^{-t^2}\,dt$.
        Note that $\int_{10y}^{x^3}e^{-t^2}\,dt = G(x^3) - G(10y)$,
        using the same $G(u)$ defined in the previous part.
    \end{enumerate}
  
    \begin{solution}
      
    \end{solution}
    \newpage
\end{questions}
\end{document}