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\title{\vspace*{-1.5in}\textbf{Calculus of Parametric Eq. (\S1.7, 3.4, 6.4)}\\[3pt]\normalsize Thanks to Faan Tone Liu}
\date{}
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\maketitle
%\underline{Math 1300-400: Quiz 1}
\lhead{Math 2300--007: Calculus II}
\chead{April 24, 2018}
\rhead{Calculus of Parametric Eq.}
\vspace{-.7in}
\noindent
\textbf{Key Points:}
\begin{itemize}
\item Suppose $x(t)$ and $y(t)$ are parametric equations and $t$ represents time. Then\\[-8pt]
\begin{itemize}
\item[$\circ$] $\frac{dx}{dt}$ represents \rule{4.5in}{.5pt}.\\
\item[$\circ$] $\frac{dy}{dt}$ represents \rule{4.5in}{.5pt}.\\
\item[$\circ$] $\frac{dy}{dx}$ represents \rule{4.5in}{.5pt}. 
\end{itemize}
\item Formulae for $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ in terms of $t$ are
\begin{align*}
\frac{dy}{dx} &= \phantom{the answer is here hereerererer},\\\\
\frac{d^2y}{dx^2} &=  \phantom{the answer is here hereerererer}.
\end{align*}

\item The instantaneous speed (of the snail) along the curve as a function of $t$ is:\vskip1.5in
\item The arc length of the curve for $t$ in $[a,b]$ is the integral of \rule{1in}{.5pt}\ : \[\text{Arc Length} = \phantom{\int_a^b\sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}dt blahhhhh}.\]
\item Other Notes:
\end{itemize}

\newpage
\noindent
\textbf{Examples:}\vskip12pt
\begin{enumerate}
\item Consider the parametric curve given by $\begin{cases}x = 5\cos(3t)\\y = 5sin(3t).\end{cases}$\\[6pt] At $t= \frac{\pi}{4}$ and $t = \frac{\pi}{2}$, find the slope of the tangent line and the speed. \vskip2in
\item A half-line is parameterized by $\begin{cases}x = 2+ 3t\\y = -1 +5t,\end{cases}$ where $t \geq 0.$
\begin{enumerate}[(a)]
\item Does $(5,4)$ lie on the ray?\vfill
\item Does $(2,1)$ lie on the ray?\vfill
\item Does $(-1,-6)$ lie on the ray?\vfill
\item When does the line hit the $y$-axis?\vfill
\item What is the speed of motion along the line?\vfill
\item What is the slope of the line?\vfill
\end{enumerate}
\newpage
\item Use technology to graph \[\begin{cases}x = t^2\\y = t^3-3t\end{cases};\ t\in \mathbb{R}.\]
\begin{enumerate}[(a)] \item Find equations for the tangent lines to the curve at (3,0).\vfill
\item At $t=1$, find $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$. Check the concavity.\vfill
\end{enumerate}

\item Consider the parametric curve given by $\begin{cases}x = t\cos{t}\\y = t\sin{t}\end{cases}$, where $0 \leq t \leq 4\pi$.
\begin{enumerate}[(a)]
\item At time $t=3.5$, which direction is the particle moving? Find the speed of the particle at this time.\vfill
\item Find the average speed and the arc length of the particle on $[0,4]$. What do you notice?\vfill
\end{enumerate}
\newpage
\item A \textbf{cycloid} is a path traced by a point on the edge of a wheel. 
\begin{enumerate}[(a)]
\item Find parametric equations for the cycloid generated by the wheel of radius $1$ shown. Suppose $t$ is measured in radians.\\\\
\begin{tikzpicture}
\draw (0,0) circle (1);
\draw (-1.5,-1) -- (2.5,-1);
\end{tikzpicture}\vfill

\item For what value(s) of $t$ is the tangent line horizontal?\vfill
\item For what value(s) of $t$ is the point stopped?\vfill
\end{enumerate}
\end{enumerate}
\end{document}
