04-New-Coordinates.tex

\noindent 
This unit covers the following ideas. In preparation for the quiz and exam, make sure you have a lesson plan containing examples that explain and illustrate the following concepts.  
\begin{enumerate}

\item Be able to convert between rectangular and polar coordinates in 2D.
\item Graph polar functions in the plane. Find intersections of polar equations, and illustrate that not every intersection can be obtained algebraically (you may have to graph the curves).
\item Find derivatives and tangent lines in polar coordinates.

\end{enumerate}
You'll have a chance to teach your examples to your peers prior to the exam.

\section{Polar Coordinates}
\larsonfive{\marginpar{See Larson section 10.4 for more reference material on the things we cover in this chapter.}}%
In the last section, we studied various plane and space transformations.  These were viewed as either transforming the plane (stretching, shrinking, folding, etc.), or as a way of giving different labels to the same physical location.  In this chapter, we look at the polar coordinate transformation more deeply.

Up to now, we most often give the location of a point (or coordiantes of a vector) by stating the $(x,y)$ coordinates.  These are called the Cartesian (or rectangular) coordinates.  Some problems are much easier to work with if we know how far a point is from the origin, together with the angle between the $x$-axis and a ray from the origin to the point.

\begin{problem}\marginpar{\bmw{See 11.3:5-10.}}
There are two parts to this problem.
\begin{enumerate}
\item Consider the point $P$ with Cartesian (rectangular) coordinates $(2,1)$.  Find the distance $r$ from $P$ to the origin. Consider the ray $\vec {OP}$ from the origin through $P$. Find an angle between $\vec{OP}$ and the $x$-axis. 

\item Suppose that a point $Q=(a,b)$ is 6 units from the origin, and the angle the ray $\vec {OP}$ makes with the $x$-axis is $\pi/4$ radians.  Find the Cartesian (rectangular) coordinates $(a,b)$ of $Q$.
\end{enumerate}

 
\end{problem}

\begin{definition}
Let $Q$ be be a point in the plane with Cartesian coordinates $(x,y)$.  Let $O=(0,0)$ be the origin. We define the polar coordinates of $Q$ to be the ordered pair $(r,\theta)$ where $r$ is the displacement from the origin to $Q$, and $\theta$ is an angle of rotation (counter-clockwise) from the $x$-axis to the ray $\vec {OP}$.
\end{definition}

\begin{problem}  \marginpar{\bmw{See 11.3:5-10.}}
The following points are given using their polar coordinates.  Plot the points in the Cartesian plane, and give the Cartesian (rectangular) coordinates of each point. The points are
$$
(1,\pi), 
\ds \left( 3,\frac{5\pi}{4}\right),
\ds \left( -3,\frac{\pi}{4}\right),\text{ and }
\ds \left( -2,-\frac{\pi}{6}\right).$$
\end{problem}

The next problem provides general formulas for converting between the Cartesian (rectangular) and polar coordinate systems.

\begin{problem}\label{polar coordinate equations}  \marginpar{\bmw{See page 647.}}
Suppose that $Q$ is a point in the plane with Cartesian coordinates $(x,y)$ and polar coordinates $(r,\theta)$.  
\begin{enumerate}
\item Write formulas for $x$ and $y$ in terms of $r$ and $\theta$: $x=?$, $y=?$
\item Write a formula to find the distance $r$ from $Q$ to the origin (in terms of $x$ and $y$): $r=?$
\item Write a formula to find the angle $\theta$ between the $x$-axis and a line connecting $Q$ to the origin, in terms of $x$ and $y$: $\theta = ?$. [Hint: A picture of a triangle will help here.]
\end{enumerate}
\end{problem}
 
In problem \ref{polar coordinate equations}, you should have obtained the polar coordinate transformation: $T(r,\theta)=(r\cos\theta, r\sin\theta)$ (or $x=r\cos\theta$, $y=r\sin\theta$).  

% The following problem will show you how to graph a coordinate transformation.  When you're done, you should essentially have polar graph paper.

\note{Delete the next problem}
\begin{problem}[Ignore] 
In the plane, graph the curve $y=\sin x$ for $x\in[0,2\pi]$ (make an $x,y$ table) and then graph the curve $r=\sin\theta$ for $\theta\in[0,2\pi]$ (an $r,\theta$ table).  The graph should look very different.  If one looks like a circle, you're on the right track.  
\end{problem}

\begin{problem}\marginpar{\bmw{See 11.3: 53-66.}}
Each of the following equations is written in the Cartesian (rectangular) coordinate system.  Convert each to an equation in polar coordinates, and then solve for $r$ so that the equation is in the form $r=f(\theta)$.
\begin{enumerate}
\item $x^2+y^2=7$
\item $2x+3y=5$
\item $x^2=y$
\end{enumerate}
\end{problem}

\begin{problem} \marginpar{\bmw{See 11.3: 27-52. I strongly suggest that you do many of these as practice.}}
Each of the following equations is written in the polar coordinate system.  Convert each to an equation in the Cartesian coordinates.
\begin{enumerate}
\item $r=9\cos\theta$
\item $\ds r=\frac{4}{2\cos\theta+3\sin\theta}$
\item $\theta = 3\pi/4$
\end{enumerate}
\end{problem}

\subsection{Graphing and Intersections}
To construct a graph of a polar curve, just create an $r,\theta$ table. Choose values for $\theta$ that will make it easy to compute any trig functions involved. Then connect the points in a smooth manner, making sure that your radius grows or shrinks appropriately as your angle increases.  

\begin{problem} \marginpar{\bmw{See 11.4: 1-20.}}\marginpar{See \href{http://sagecell.sagemath.org/?z=eJwr0ijJSC1J1FSwVTDSNtJKzi-GCnAV5OckFsUX5OSXaBTpKEBEdQx0jLQKMjU1uQD67RE_&lang=sage}{Sage}.}%
Graph the polar curve $r=2+2\cos\theta$.
\end{problem}

\begin{problem}\marginpar{Use Sage to check your answer.}
Graph the polar curve $r=2\sin 3\theta$.
\end{problem}

\begin{problem}\marginpar{Use Sage to check your answer.}
Graph the polar curve $r=3\cos 2\theta$.
\end{problem}


\begin{problem}
Find the points of intersection of $r=3-3\cos\theta$ and $r=3\cos\theta$. (If you don't graph the curves, you'll probably miss a few points of intersection.)
\end{problem}

% \begin{problem}
% Find the points of intersection of $r=2\cos 2\theta $ and $r=\sqrt 3$. (If you don't graph the curves, you'll probably miss a few points of intersection.)
% \end{problem}


\subsection{Calculus with Polar Coordinates}

Recall that for parametric curves $\vec r(t) = (x(t),y(t))$, to find the slope of the curve we just compute $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}.$$ A polar curve of the form $r=f(\theta)$ can be thought of as just the parametric curve $(x,y) = (f(\theta)\cos\theta,f(\theta)\sin\theta)$. So you can find the slope by computing
$$\frac{dy}{dx}=\frac{dy/d\theta}{dx/d\theta}.$$

\begin{problem}\marginpar{\bmw{See 11.2: 1-14.}}
Consider the polar curve $r=1+2\cos \theta$. (It wouldn't hurt to provide a quick sketch of the curve.)
\begin{enumerate}
\item Compute both $dx/d\theta$ and $dy/d\theta$.
\item Find the slope $dy/dx$ of the curve at $\theta=\pi/2$.
\item Give both a vector equation of the tangent line, and a Cartesian equation of the tangent line at $\theta=\pi/2$.
\end{enumerate}
\end{problem}

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