Explanation The parametric equation of the curve is given as follows. x = − 2 cos t (1) y = sin t + sin 2 t (2) Use the online graphing calculator to graph the curve with the parametric equation x = − 2 cos t and y = sin t + sin 2 t as shown below in Figure 1. From Figure 1, it is observed that the curve crosses itself at the point ( 1 , 0 ) . It is enough to substitute either x = 1 in equation (1) or y = 0 in equation (2), since both will yield same solution for t. Substitute x = 1 in the equation (1). x = − 2 cos t − 2 cos t = 1 cos t = − 1 2 t = 2 π 3 , 4 π 3 for 0 ≤ t ≤ 2 π Differentiate the parametric equation for the variable x with respect to t . x = − 2 cos t d x d t = 2 sin t Differentiate the parametric equation for the variable y with respect to t . y = sin t + sin 2 t d y d t = cos t + 2 cos 2 t Use the chain rule for the expression d y d x . d y d x = ( d y d t ) ( d x d t ) Substitute ( cos t + 2 cos 2 t ) for d y d t and ( 2 sin t ) for d x d t in the above equation. d y d x = ( cos t + 2 cos 2 t ) ( 2 sin t ) (3) Substitute 2 π 3 for t in equation (3). d y d x = ( cos t + 2 cos 2 t ) ( 2 sin t ) m = ( cos ( 2 π 3 ) + 2 cos 2 ( 2 π 3 ) ) ( 2 sin ( 2 π 3 ) ) = − 1 2 + 2 ( − 1 2 ) 2 ( 3 2 ) = ( − 3 2 ) 3 = − 3 2 The equation for tangent line is given by, y − y 1 = m ( x − x 1 )

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ISBN: 9781285741550

Author: James Stewart

Publisher: Cengage Learning

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Chapter 10.2, Problem 26E

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To graph: The curve for the parametric equations x=−2cost, y=sint+sin2t and find the equation of tangents at the point where the curve crosses itself.

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Chapter 10.2, Problem 25E

Chapter 10.2, Problem 27E

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Chapter 10 Solutions

Calculus: Early Transcendentals

Ch. 10.1 - Sketch the curve by using the parametric equations...Ch. 10.1 - Sketch the curve by using the parametric equations...Ch. 10.1 - Prob. 3E Ch. 10.1 - Prob. 4E Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Prob. 10E

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To graph: The curve for the parametric equations x = − 2 cos t , y = sin t + sin 2 t and find the equation of tangents at the point where the curve crosses itself.

Solution Summary: The author explains how to graph a curve with the parametric equations x=-2mathrmcost, and y=

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To graph: The curve for the parametric equations x=−2cost, y=sint+sin2t and find the equation of tangents at the point where the curve crosses itself.

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Chapter 10 Solutions