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.

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 .

Name | Equation |

cardioid | or |

limaçon | or |

rose | or |

- 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 any value.

- 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
coordinates.
- and for .
- and for .

- Plot and for 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.

2005-02-14