Calculus (3rd Edition)

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

Chapter 3 - Differentiation - 3.1 Definition of the Derivative - Exercises - Page 103: 29


$f'(a)=f'(3)=22$ Equation of the tangent line is $y=22x-18$

Work Step by Step

Recall that $f'(a)=\lim\limits_{x \to a}\frac{f(x)-f(a)}{x-a}$ $\implies f'(3)=\lim\limits_{x \to 3}\frac{f(x)-f(3)}{x-3}=\lim\limits_{x \to 3}\frac{(2x^{2}+10x)-(2\times3^{2}+10\times3)}{x-3}$ $=\lim\limits_{x \to 3}\frac{(2x+16)(x-3)}{x-3}=\lim\limits_{x \to 3}(2x+16)=2\times3+16=22$ Equation of the tangent line is of the form $y-f(a)=f'(a)(x-a)$ Knowing that $a=3$ and $f(3)=48$, we have $y-48=22(x-3)$ $\implies y=22x-18$
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