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Parametric Equations. Dr. Dillon Calculus II Spring 2000. Introduction. Some curves in the plane can be described as functions. Others. cannot be described as functions. Example:. Ways to Describe a Curve in the Plane. An equation in two variables. This equation describes a circle.

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parametric equations

Parametric Equations

Dr. Dillon

Calculus II

Spring 2000

introduction
Introduction

Some curves in the plane can be described as functions.

others
Others...

cannot be described as functions.

ways to describe a curve in the plane

Example:

Ways to Describe a Curve in the Plane

An equation in two variables

This equation describes a circle.

a polar equation
A Polar Equation

This polar equation describes a double spiral.

We’ll study polar curves later.

parametric equations1
Parametric Equations

Example:

The “parameter’’ is t.

It does not appear in the graph of the curve!

slide7
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.

two functions one curve

If

Two Functions, One Curve?

Yes.

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

why use parametric equations
Why use parametric equations?
  • Use them to describe curves in the plane when one function won’t do.
  • Use them to describe paths.
paths
Paths?

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

slide12
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.

from that point of view
From that point of view...

The path described by

is a particular route along the curve.

slide15

As t increases from 0, x first decreases,

then increases.

Path moves right!

Path moves left!

more paths
More Paths

To designate one route around the unit circle use

slide17

That Takes Us...

counterclockwise from (1,0).

where do you get that
Where do you get that?

Think of t as an angle.

If it starts at zero, and increases to

then the path starts at t=0, where

how do you find the path
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
plotting points
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
consider
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?)
note further
Note Further
  • x has a minimum when t=0 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).
eliminate the parameter

Still use

Eliminate the Parameter

Solve one of the equations for t

Here we get t=y/2

Substitute into the other equation

Here we get

using the ti82
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
some questions
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.)
web resources
Web Resources
  • MathView Notebook on your instructor’s site (Use Internet Explorer to avoid glitches!)
  • IES Web
summary
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...