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\usepackage{amsmath, amssymb, amsthm}
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\title{Parametric equations}
\date{November 18, 2020}

\begin{document}

\ 

\vspace{-.25in}

\maketitle

\vspace{-.15in}

\prob{
This problem concerns two sets of parametric equations: $\displaystyle (x,y)= (\cos t , \sin^2 t)$ and $\displaystyle (x,y)= (1-s , 2s-s^2)$, both defined for all values of the parameters $t$ and $s$.
\begin{enumerate}[a)]
\item Plot both parametric curves (make a table; or do part b first if you're having trouble; or use \href{https://www.desmos.com/calculator}{Desmos}).

\begin{tikzpicture}
\draw[step=0.5cm,color=lightgray,very thin] (-4,-4) grid (4,4);
\end{tikzpicture}
\hspace{.2in}
\begin{tikzpicture}
\draw[step=0.5cm,color=lightgray,very thin] (-4,-4) grid (4,4);
\end{tikzpicture}

\item Eliminate the parameters to find Cartesian equations for the curves.
\item What is the difference between the paths traced out by the two pairs of parametric equations?
\end{enumerate}
}

\newpage

\prob{
The parametric equations $x(t) = x_0 + t(x_1 - x_0)$, $y(t) = y_0 + t(y_1 - y_0)$ with $0 \leq t \leq 1$ describe a line starting at the point $(x_0, y_0)$ and ending at the point $(x_1, y_1)$.
\begin{enumerate}[a)]
\item Sketch a graph of the line line segment from $(1,-1)$ to $(3, 4)$.

\begin{tikzpicture}
\draw[step=0.5cm,color=lightgray,very thin] (-3,-3) grid (3,3);
\end{tikzpicture}

\item Find a parametric description of the line from $(1,-1)$ to $(3, 4)$.

\item Your parametric equations should work just fine if $t$ is less than $0$ or greater than $1$. What happens for these values of $t$?

\item Find a parametric description of the same line using $x(t) = t^2$ (don't worry about getting just the segment between the two points).

\item What is the slope of your line (that is, find $\frac{dy}{dx}$)? How is the slope connected to the derivatives $\frac{dx}{dt}$ and $\frac{dy}{dt}$? Check that it works for both descriptions (in parts b and d).

\end{enumerate}
}

\vfill

\prob{
Use your formula for $\frac{dy}{dx}$ in terms of $\frac{dx}{dt}$ and $\frac{dy}{dt}$ to find $\frac{dy}{dx}$ for both sets of parametric equations in problem 1. 
Do your answers agree?
Should they agree?
}

\vfill


\newpage

\prob{
Find a parametric description of the curve traced out by a point on the outside of a wheel of radius 1  as the wheel rolls (this curve is called a \href{https://en.wikipedia.org/wiki/Cycloid}{cycloid}).
Hint: start with the point on the bottom of the wheel and then use the angle of rotation as the parameter; first locate the center of the wheel then work back to the point.

\vspace{.4in}

\begin{tikzpicture}[scale=2]
\draw[thick] (0,0)--(7,0);
\draw[color=gray] (2,2) circle (2);
\draw (2,0)--(2,2);
\filldraw (2,0) circle (.05);
\draw (2,1) node[anchor=east] {1};

\draw (4.0944,2) circle (2);
\draw[color=gray] (4.0944,0)--(4.0944,2);
\draw (4.0944,2)--(2.3623,1);
\filldraw (2.3623,1) circle (.05);
\draw[color=red] (3.748,1.8) arc (210:270:.4);
\draw[color=red] (3.9944,1.65) node[anchor=north east] {$\theta$};
\draw[color=red, thick]  (2.3623,1) arc (210:270:2);
\draw[very thick, color=red] (2,0)--(4.0944,0);


\draw[color=lightgray, ->>] (2,2)--(2.67,2);
\draw[color=lightgray, ->>] (2.67,2)--(3.33,2);
\draw[color=lightgray, ->>] (3.33,2)--(4.0944,2);
\end{tikzpicture}
}

\vfill

\chal{
Repeat the previous problem but follow a point on a circle of radius 1 rolling inside a circle of radius 3 (like a \href{https://en.wikipedia.org/wiki/Spirograph}{spirograph}). 
This produces a \href{https://en.wikipedia.org/wiki/Hypocycloid}{hypocycloid}
}


\end{document}