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Chap10_Sec2

Chap10_Sec2 - PARAMETRIC EQUATIONS POLAR COORDINATES 10.2...

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10.2 Calculus with arametric Curves Parametric Curves In this section, we will: Use parametric curves to obtain formulas for tangents, areas, arc lengths, and surface areas.

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TANGENTS In Section 10.1, some curves defined by parametric equations x = f ( t ) and y = g ( t ) can also be expressed in the form y = F ( x ), by eliminating the parameter t . If we substitute x = f ( t ) and y = g ( t ) in the equation y = F ( x ), we get: g ( t ) = F ( f ( t )) So, if g , F , and f are differentiable, the Chain Rule gives: g’ ( t ) = F’ ( f ( t )) f’ ( t ) = F’ ( x ) f’ ( t ) If f’ ( t ) 0, we can solve for F’ ( x ): (1) The slope of the tangent to the curve y = F ( x ) at ( x , F ( x )) is F’ ( x ). Thus, Equation 1 enables us to find tangents to parametric curves without having to eliminate the parameter. '( ) '( ) '( ) g t F x f t =
Using Leibniz notation, we can rewrite Equation 1 in an easily remembered form: (2) If we think of a parametric curve as being traced out by a moving particle, then

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Chap10_Sec2 - PARAMETRIC EQUATIONS POLAR COORDINATES 10.2...

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