Calculus (3rd Edition)

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

Chapter 3 - Differentiation - 3.6 Trigonometric Functions - Exercises - Page 140: 3


$$ y=2 x+1-\frac{ \pi}{2} .$$

Work Step by Step

Since $ y=\tan x $, then $ y'=\sec^2 x $ and hence the slope at $ x=\frac{\pi}{4}$ is $ m=\sec^2 \frac{\pi}{4}=2.$ Now, the tangent line equation is given by $$ y=2 x+c.$$ Since the curve and line coincide at $ x=\frac{\pi}{4}$, then $$\tan \frac{\pi}{4}=2\frac{\pi}{4}+c \Longrightarrow c= 1-\frac{ \pi}{2} .$$ Hence the equation of the tangent line is given by $$ y=2 x+1-\frac{ \pi}{2} .$$
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