Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 12 - Parametric Equations, Polar Coordinates, and Conic Sections - 12.1 Parametric Equations - Exercises - Page 604: 60


The answer is $y=x-2 \pi+8$.

Work Step by Step

According to Eq.(7) the cycloid generated by a circle of radius 4 has parametric equations $x(t)=4 (t-\sin t)$, $y(t)=4(1-\cos t)$. Using Eq.(8) the slope of the tangent line to the cycloid is $dy/dx=y'(t)/x'(t)=4\sin t/4(1-\cos t)=\sin t/(1-\cos t)$. Since $c(\pi/2)=(2\pi-4,4)$ and $\frac{dy}{dx}|_{t=\pi /2 }=1$. The equation of the tangent line at $t=\pi/2$ is $y-4=x-(2 \pi-4)$, $y=x-2 \pi+8$.
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