Parametric Equations

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Parametric Equations

• Dr. Dillon
• Calculus II
• Spring 2000

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Introduction

• Some curves in the plane can be described as
  functions.

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Others...

• cannot be described as functions.

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Ways to Describe a Curve in the Plane
An equation in two variables
This equation describes a circle.
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A Polar Equation
This polar equation describes a double spiral.
Well study polar curves later.
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Parametric Equations

• Example

The parameter is t.
It does not appear in the graph of the curve!
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Why?

• The x coordinates of points on the curve are
  given by a function.

The y coordinates of points on the curve are
given by a function.
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Two Functions, One Curve?

• Yes.

then in the xy-plane the curve looks like this,
for values of t from 0 to 10...
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Why use parametric equations?

• Use them to describe curves in the plane when one
  function wont do.
• Use them to describe paths.

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Paths?

• A path is a curve, together with a journey traced
  along the curve.

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Huh?

• When we write

we might think of x as the x-coordinate of the
position on the path at time t
and y as the y-coordinate of the position on the
path at time t.
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From that point of view...

• The path described by

is a particular route along the curve.
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As t increases from 0, x first decreases,
then increases.
Path moves right!
Path moves left!
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More Paths

• To designate one route around the unit circle use

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That Takes Us...
counterclockwise from (1,0).
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Where do you get that?
Think of t as an angle.
If it starts at zero, and increases to
then the path starts at t0, where
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To start at (0,1)...
Use
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That Gives Us...
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How Do You Find The Path

• Plot points for various values of t, being
  careful to notice what range of values t should
  assume
• Eliminate the parameter and find one equation
  relating x and y
• Use the TI82/83 in parametric mode

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Plotting Points

• Note the direction the path takes
• Use calculus to find
• maximum points
• minimum points
• points where the path changes direction
• Example Consider the curve given by

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Consider

• The parameter t ranges from -5 to 5 so the first
  point on the path is (26, -10) and the last point
  on the path is (26, 10)
• x decreases on the t interval (-5,0) and
  increases on the t interval (0,5). (How can we
  tell that?)
• y is increasing on the entire t interval (-5,5).
  (How can we tell that?)

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Note Further

• x has a minimum when t0 so the point farthest to
  the left on the path is (1,0).
• x is maximal at the endpoints of the interval
  -5,5, so the points on the path farthest to the
  right are the starting and ending points, (26,
  -10) and (26,10).
• The lowest point on the path is (26,-10) and the
  highest point is (26,10).

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Eliminate the Parameter
Solve one of the equations for t
Here we get ty/2
Substitute into the other equation
Here we get
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Using the TI82

• Change mode to PAR (third row)
• Mash y button
• Enter x as a function of t, hit enter
• Enter y as a function of t, hit enter
• Check the window settings, after determining the
  maximum and minimum values for x and y

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Some Questions

• What could you do in the last example to reverse
  the direction of the path?
• What could you do to restrict or to enlarge the
  path in the last example?
• How can you cook up parametric equations that
  will describe a path along a given curve? (See
  the cycloid on the Web.)

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Web Resources

• MathView Notebook on your instructors site (Use
  Internet Explorer to avoid glitches!)
• IES Web

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Summary

• Use parametric equations for a curve not given by
  a function.
• Use parametric equations to describe paths.
• Each coordinate requires one function.
• The parameter may be time, angle, or something
  else altogether...