Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 3 - Review - Exercises: 57

Answer

$y=2\sqrt{3}x+\dfrac{3-\pi\sqrt{3}}{3}$

Work Step by Step

$y=4\sin^{2}x$ $,$ $(\pi/6,1)$ Use the chain rule to evaluate the derivative of the function: $y'=8\sin x(\sin x)'=8\sin x\cos x$ Evaluate the derivative in $\dfrac{\pi}{6}$ and simplify to obtain the slope of the tangent line at the given point: $m=8\sin\dfrac{\pi}{6}\cos\dfrac{\pi}{6}=8\Big(\dfrac{1}{2}\Big)\Big(\dfrac{\sqrt{3}}{2}\Big)=2\sqrt{3}$ Both the slope of the tangent line and a point through which it passes are known. Use the point-slope form of the equation of a line, which is $y-y_{1}=m(x-x_{1})$ to obtain the equation of the tangent line: $y-1=2\sqrt{3}\Big(x-\dfrac{\pi}{6}\Big)$ $y-1=2\sqrt{3}x-\dfrac{\pi\sqrt{3}}{3}$ $y=2\sqrt{3}x-\dfrac{\pi\sqrt{3}}{3}+1$ $y=2\sqrt{3}x+\dfrac{3-\pi\sqrt{3}}{3}$
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