1120_7 - Week 7 Outline The Slope of a Parametric Curve Arc...

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Week 7 Outline The Slope of a Parametric Curve Arc Lengths for Parametric Curves Surface Areas for Parametric Curves Areas Bounded by Parametric Curves The Slope of a Parametric Curve Let C be the parametric curve x = f ( t ), y = g ( t ), where f 0 ( t ) and g 0 ( t ) are continuous on an interval I . If f 0 ( t ) 6 = 0 on I , then d y d x = g 0 ( t ) f 0 ( t ) . Remark We can rewrite the above equation in an easily remembered form: d y d x = d y d t d x d t , if d x d t 6 = 0 It can be seen from this equation that the curve has a horizontal tangent when d y d t = 0 (provided d x d t 6 = 0) and it has a vertical tangent when d x d t = 0 (provided d y d t 6 = 0). This information is useful for sketching parametric curves. It is also useful to consider d 2 y d x 2 , which can be found by d 2 y d x 2 = d d x ± d y d x = d d t ( d y d x ) d x d t . Examples 1. A curve C is defined by the parametric equations x = t 2 , y = t 3 - 3 t . (a) Show that C has two tangents at the point (3 , 0) and find their equations. (b) Find the points on C where the tangent is horizontal or vertical. (c) Determine where the curve is concave upward or downward. (d) Sketch the curve.
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This note was uploaded on 12/09/2010 for the course MATH 1120 taught by Professor Gross during the Fall '06 term at Cornell University (Engineering School).

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1120_7 - Week 7 Outline The Slope of a Parametric Curve Arc...

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