Calculus (3rd Edition)

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

Chapter 7 - Exponential Functions - 7.1 Derivative of f(x)=bx and the Number e - Exercises - Page 327: 62


$$L(x) = -5x +12$$

Work Step by Step

Given $$f(x) =x e^{6-3 x} ,\ \ a=2$$ Since $f(a) =2$ and \begin{align*} f'(x)&= \frac{d}{dx}\left(x\right)e^{6-3x}+\frac{d}{dx}\left(e^{6-3x}\right)x\\ &= (1-3x ) e^{6-3 x}\\ f'(2)&= -5 \end{align*} Then the linearization is given by \begin{align*} L(x)&=f^{\prime}(a)(x-a)+f(a)\\ &= -5 (x-2)+2\\ &= -5x +12 \end{align*}
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