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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_B08.mws My_Documents
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
The relationship between area and integrals in polar coordinates is a little strange; the area inside a circle given (in polar coordinates) by is NOT just
. Here is the rule: Area inside is given by
. Note that this gives you the right answer for a circle: . So to find the area of the cardiod use the following command.
- For each of the following polar equations, plot the graph in polar coordinates using the plot command and identify the graph as a
cardioid, limaçon, or rose.
- Find all points of intersection for each pair of curves in polar
- Plot the polar equation
. Find the angles that create the the inner loop of
. Plot only the inner loop and then find the area inside the inner loop.
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Dina J. Solitro-Rassias