This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: Calculus- Stewart Dr. Berg Summer ‘09 Page 1 11.2 11.2 Calculus With Parametric Curves Tangents Suppose a curve is defined by the equation y = F ( x ) and can also be expressed as the parametric equations x = f ( t ) and y = g ( t ) where F , f , and g are all differentiable. Then when we differentiate y = g ( t ) = F ( x ) = F ( f ( t )) with respect to t and apply the chain rule, we get ′ g ( t ) = ′ F ( f ( t )) ′ f ( t ) = ′ F ( x ) ′ f ( t ) so that ′ F ( x ) = ′ g ( t ) ′ f ( t ) if ′ f ( t ) ≠ . Using Leibniz notation yields: Proposition dy dx = dy dt dx dt if dx dt ≠ . Note that a curve has a horizontal tangent when dy dt = (provided that dx dt ≠ ) and has a vertical tangent when dx dt = (provided that dy dt ≠ ). It can also be useful to consider d 2 y dx 2 . Corollary d 2 y dx 2 = d dt dy dx dx dt . Example A Find the x- y coordinates of the places where the curve given by x = t 3 − 3 t and y = 3 t 2 − 9 has vertical or horizontal tangents. Indeed dy dx = D t 3 t 2 − 9 ( ) D t t 3 − 3 t ( ) = 6 t 3 t 2 − 3 = 2 t ( t − 1)( t + 1) . Thus, we get a horizontal tangent when t = at (0, –9), and there are vertical tangents when t = ± 1 at (–2, –6) and (2, –6). Here is the graph of the curve: Calculus- Stewart Dr. Berg Summer ‘09 Page 2 11.2 Notice that the curve crosses itself at the origin. This corresponds to two values of Notice that the curve crosses itself at the origin....
View Full Document
This note was uploaded on 06/06/2010 for the course M 408 taught by Professor Hodges during the Spring '08 term at University of Texas.
- Spring '08