# The Slope of a Parametric Curve Arc Lengths for Parametric...

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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. 2. Find the coordinates of the points at which the given parametric curve has (i) a horizontal tangent and (ii) a vertical tangent. (a)