applying calculus concepts to parametric curves 11.2

Applying Calculus Conceptsto Parametric Curves

11.2

Basic ideas…

• Slopes and rates of change

What is the slope at the point (x,y) for the curve shown on the right if the curve represents the relation:

2

3 3

x t

y t t

• Motivating idea comes from…

0

dydy dxdt if

dxdx dtdt

We can develop a similar expression for a second or higher derivative…

2

2

( ) ( )dy d dy

dd y dx dt dxdxdx dxdt

What does this mean?

What does this mean?

Areas

• How can we apply our basic understanding of how to find areas to parametric equations?

• Start with x(t) = f(t), y(t) = g(t) and

( )b

a

A y x dx

Arc Length…

2( ) 1y x x 21 ( )

b

a

dyl dx

dx

This is not a “trvial” integral to do directly and the result (you may recall from Math 205) involves trig subs and arcsin!). Let’s try it using a change to parametric form…

Take-home message from 11.2…

• Most basic calculus operations can be re-written in parametric form

• Sometimes – changing to a parametric form makes life easier (but not always!)