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: 28

Answer

The equation of the tangent line at $ x=0$ is given by $$ y=x+3.$$

Work Step by Step

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