PP 11.2 - Calculus with Parametric Curves Some curves...

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Unformatted text preview: Calculus with Parametric Curves Some curves defined by parametric equations x = f ( t ) and y = g ( t ) can also be expressed in the form y = F ( x ). This can be accomplish by eliminating the parameter. 2 , 1 : Example 2- =- = t y t x 2 ) 1 ( 1 1 2- + = + = - = x y x t t x If we substitute x = f ( t ) and y = g ( t ) in the equation y = F ( x ), we get: g ( t ) = F ( f ( t )) If g , F , and f are differentiable, the Chain Rule gives: g ( t ) = F ( f ( t )) f ( t ) = F ( x ) f ( t ) ) ( , ) ( ) ( ) ( = t f t f t g x F But y = F ( x ), x = f ( t ) and y = g ( t ), so have found a formula for the slope of the tangent line. , to equivalent is ) ( , ) ( ) ( ) ( = = dt dx dt dx dt dy dx dy t f t f t g x F Tangent Lines curve. parametric a be ) ( ), ( : Let t y y t x x C = = . provided is to ing correspond point the o tangent t line the of slope The a t = = = a t dt dx dt dx dt dy a t Comments From the formula for the slope, it can be deduced that tangent lines are horizontal when = dt dy Also, tangent lines are vertical when = dt dx Example...
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This note was uploaded on 01/21/2011 for the course PHYS 4A 60865 taught by Professor L. oldewurtel during the Fall '09 term at Irvine Valley College.

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PP 11.2 - Calculus with Parametric Curves Some curves...

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