Explanation Given: c ( t ) = ( t + sin t , t − 2 sin t ) where the tangent is vertical or horizontal . Calculation: It is given that c ( t ) = ( t + sin t , t − 2 sin t ) that means we have x ( t ) = t + sin t , y ( t ) = t − 2 sin t Now, differentiate both the equations x ( t ) = t + sin t , y ( t ) = t − 2 sin t with respect to t and we get: x ( t ) = t + sin t , y ( t ) = t − 2 sin t d x d t = 1 + cos t , d y d t = 1 − 2 cos t Now, we have to find out d y d x So, d y d x = d y d t d x d t = 1 − 2 cos t 1 + cos t Now, the tangent will be horizontal at points where: 1 − 2 cos t = 0 and 1 + cos t ≠ 0 cos t = 1 2 and cos t ≠ − 1 t = π 3 , 5 π 3 and t ≠ π Thus, the tangent is horizontal to the given curve at the points: ( t + sin t

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Author: Rogawski

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Chapter 12, Problem 18CRE

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The points on $(t + \sin t, t - 2 \sin t)$ where the tangent is vertical or horizontal

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Chapter 12, Problem 17CRE

Chapter 12, Problem 19CRE

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