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The purpose of this lab is to help you become familiar with graphs in
To assist you, there is a worksheet associated with this lab that
contains examples and even solutions to some of the exercises. You can
copy that worksheet to your home directory with the following command,
which must be run in a terminal window, not in Maple.
cp /math/calclab/MA1023/Polar_start.mws ~
You can copy the worksheet now, but you should read through the lab
before you load it into Maple. Once you have read to the exercises,
start up Maple, load
the worksheet Polar_start.mws, and go through it
carefully. Then you can start working on the exercises.
There are many times in math, science, and engineering that coordinate
systems other than the familiar one of Cartesian coordinates are
convenient. In this lab, we consider one of the most common and useful
such systems, that of polar coordinates.
The main reason for using polar coordinates is that they can be used
to simply describe regions in the plane that would be very difficult
to describe using Cartesian coordinates. For example, graphing the
circle in Cartesian coordinates requires two functions -
one for the upper half and one for the lower half. In polar
coordinates, the same circle has the very simple representation .
These are three types of well-known graphs in polar coordinates. The
table below will allow you to identify the graphs in the exercises.
Finding where two graphs in Cartesian coordinates intersect is
straightforward. You just set the two functions equal and solve for
the values of . In polar coordinates, the situation is more
difficult. Most of the difficulties are due to the following considerations.
These considerations can make finding the intersections of two graphs in polar
coordinates a difficult task. As the exercises demonstrate, it
usually requires a combination of plots and solving equations to find
all of the intersections.
- A point in the plane can have more than one representation in
polar coordinates. For example, , is the same
point as , . In general a point in the plane can have
an infinite number of representations in polar coordinates, just by
adding multiples of to . Even if you restrict
a point in the plane can have several different representations.
- The origin is determined by . The angle can have
- For each of the following polar equations, plot the graph in two
ways. First, use the plot command and identify the graph as a
cardioid, limaçon, or rose. Second, use the
ParamPlot command to obtain an animated plot. Use the
animated plot to help you understand the graph of this function.
- Find all points of intersection for each pair of curves in polar
using a polar corrdinate plot and count the number of intersection points you see. Then make a second plot of the same two functions using a rectangular plot. Can you explain why one of the intersection points did not show up on the rectangular plot? Find all intersection points.
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