Applying Calculus Concepts to Parametric Curves 11.2.
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Transcript of 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!)