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Lab 1: Polar Equations
Seattle Pacific University, MAT 1236, Calculus III
Objectives
To match elements of the polar graph
r r
with corresponding values of
To develop mathematical formulas for slope and angle in polar coordinates, and to use
them to address questions that lie beyond the reach of strictly graphical analysis.
To match aspects of the graph
y f x
in rectangular coordinates against aspects of
r f
in polar coordinates.
Due Date: Friday 04/03 2:01 p.m.
Do not wait until the last minute to finish the lab. You never know what technical
problems you may encounter (no papers in the printer, electricity is out, your best friend
call and talk for 4 hours, alien attack…etc).
Name 1:
Name 2:
Date:
Instructions
Do not reformat the pages. Rescale your diagrams if necessary to fit into the space
given to each problem.
Label all your diagrams.
The original version of this lab was adapted by Brian Gill and the SPU mathematics department from Lab 24 (Polar
Equations, by John Fink) in Learning by Discovery: A Lab Manual for Calculus , Anita Solow, editor, 1993. The
book is volume 27 of the MAA Notes series published by the Mathematical Association of America.
Lab Exercises
The purpose of this lab is to give you some experience in working with the graphs of equations
defined by polar functions. We will use the fact that any polar function can be written in
parametric form. If the polar coordinates of a point P are
r ,^ ^
, then the rectangular coordinates
of P will be
x y ,^^ ^ r^ cos^ ^ ,^ r sin
. This leads to the standard parametrization of a polar curve
r r (^)
by
x (^) (^) r (^) cos
and
y (^) (^) r (^) (^) sin
y
x
r
( , x y ) ( cos r , r sin )
In the lab, a polar curve will be given as the graph of the function
r
in polar coordinates.
You will need to be able to match points on the curve with their corresponding values of
in
the interval
a b ,^
. For example, if
r (^)
is positive on the interval
2
, then values of
in
this interval will yield points in the first quadrant.
- What condition on r will force values of
in the interval
3
2
to yield points on the graph
in the first quadrant? Explain.
Maple Commands
Here is the Maple commands that plot the graphs of polar equations
r ( )
.
Load the graphing package
with(plots):
Plot the polar equation
r ( ) sin(3 )
for
polarplot(sin(3theta),theta=0..5Pi/6);
(iii) What values of
would correspond to the petal in the first quadrant? Explain and provide
evidence.
b. (i) Plot the polar graph of
r sin 2
for
in
0, 2
(ii) How many petals do you see?
(iii) Which values of
correspond to the petal in the first quadrant?
(iv) Could you have gotten the same graph from a smaller domain for
this time?
- Limaçons. Polar graphs of the functions
r (^) 1 k sin
are called limaçons , which is
French for snails, possibly because of their vague resemblance to the shape of these animals
for certain values of k. See what you think of the resemblance by plotting the polar graph of
r 1 2sin
for
in
0, 2^ ^
.
a. Your plot for k = 2 should show a smaller loop inside a larger loop. Which values of
correspond to this smaller loop? Explain how you get the answer.
(ii) Sketch the limaçons for several additional values of k. Besides being looped, dented, or
egg-like, did you find any other shapes occurring as limaçons?
