mth 253 calculus (other topics) chapter 10 – conic sections and polar coordinates section 10.6 –...
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MTH 253Calculus (Other Topics)
Chapter 10 – Conic Sections and Polar Coordinates
Section 10.6 – Graphing in Polar Coordinates
Copyright © 2009 by Ron Wallace, all rights reserved.
Graphs of Polar Equations
polar axis
(r, )
r
pole
The angle may be expressed in degrees or radians.
If r = f(), then the graph of this equation consists of ALL of the points whose coordinates make this equation true.
Graphing Polar Equations
Reminder: How do you graph rectangular equations? Method 1:
Create a table of values. Plot ordered pairs. Connect the dots in order as x increases.
Method 2: Recognize and graph various common
forms. Examples: linear equations, quadratic
equations, conics, …
The same basic approach can be applied to polar equations.
Graphing Polar EquationsMethod 1: Plotting and Connecting Points
1. Create a table of values.2. Plot ordered pairs.3. Connect the dots in order as
increases.
NOTE: Since most of these equations involve periodic functions (esp. sine and cosine), at some point the graph will start repeating itself (but not always).
Graphing Polar Equations
wrt x-axis• Replacing with – doesn’t change the function• Replacing r with –r and with – doesn’t change
the function
Symmetry Tests
(r,)
(r,-)=(-r,–)
Graphing Polar Equations
wrt y-axis• Replacing r with –r and with – doesn’t change the
function• Replacing with – doesn’t change the function
Symmetry Tests
(r,)(-r, -)=(r,-)
Graphing Polar Equations
wrt the origin• Replacing r with –r doesn’t change the function.• Replacing with doesn’t change the function.
Symmetry Tests
(r,)
(-r,)(r, )
Slope of Polar Curves
To find the slope of …
… remember …
… therefore …
dym
dx
( )r f
cos
sin
x r
y r
( ) cos
( )sin
f
f
dy dy dm
dx dx d
'( ) sin ( )cos
'( ) cos ( )sin
f f
f f
Slope of Polar Curves
Example: Find the equation of the tangent line to the following curve when = /4
'( ) sin ( )cos
'( ) cos ( )sin
f fm
f f
1 sinr
4
Graphing Polar EquationsRecognizing Common Forms
Circles Centered at the origin: r = a
radius: a period = 360
Tangent to the x-axis at the origin: r = a sin center: (a/2, 90) radius: a/2 period = 180 a > 0 above a < 0 below
Tangent to the y-axis at the origin: r = a cos center: (a/2, 90) radius: a/2 period = 180 a > 0 right a < 0 left
r = 4
r = 4 sin
r = 4 cos
Note the Symmetries
Graphing Polar EquationsRecognizing Common Forms
Flowers (centered at the origin) r = a cos n or r = a sin n
radius: |a| n is even 2n petals
petal every 180/n period = 360
n is odd n petals petal every 360/n period = 180
cos 1st petal @ 0 sin 1st petal @ 90/n
r = 4 sin 2
r = 4 cos 3
Note the Symmetries
Graphing Polar EquationsRecognizing Common Forms
Spirals Spiral of Archimedes: r = k
|k| large loose |k| small tight
r = r = ¼
Graphing Polar EquationsRecognizing Common Forms
Heart (actually: cardioid if a = b … otherwise: limaçon)
r = a ± b cos or r = a ± b sin
r = 3 + 3 cos r = 2 - 5 cos r = 3 + 2 sin r = 3 - 3 sin
Note the Symmetries
Graphing Polar EquationsRecognizing Common Forms
Leminscate2 cos 2r a 2 sin 2r a
a = 16
Note the Symmetries
Polar Graphs w/ Technology
TI-84 WinPlot
Intersections of Polar Curves
As with Cartesian equations, solve by the substitution method.
Warning: 2 polar curves may intersect, but at different values of . i.e. Setting the two equations equal to
each other may not reveal ALL of the points of intersection.
Solution: Always graph the equations.
Intersections of Polar Curves
Example: Find the points of intersection of …
cos & 1 cosr r
Note that 2 of the points are found by substitution, the third by the graph.