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: 25

Answer

The tangent line at $ x=0$ is given by $$ y=1.$$

Work Step by Step

Since $ y=x^3+\cos x $, then the slope at $ x=0$ is given as follows $$ y'=3x^2-\sin x\Longrightarrow m=y'(0)=0.$$ The equation of the tangent line at $ x=0$ is given by $$ y=(0)x+c=c.$$ To find $ c $ we use the fact that the tangent line and the curve coincide at $ x=0$, so we have $ y(0)=1=c $. Hence the equaion of the tangent line at $ x=0$ is given by $$ y=1.$$
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