Calculus (3rd Edition)

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

Chapter 7 - Exponential Functions - Chapter Review Exercises - Page 386: 43


$$y=-\frac{1}{2} x+6$$

Work Step by Step

Since $f(x) $ has the tangent line $y=-2x+12 $ at $x=4$, then $f(x)$ passes through $$(4,-2(4)+12)=(4,4)$$ Since $g(x) $ is the inverse of $f(x)$, and $f, g$ are symmetric at $x=y$, then \begin{align*} g^{\prime}(4)&=\frac{1}{f^{\prime}(g(4))}\\ &=\frac{1}{f^{\prime}(4)}\\ &=\frac{1}{-2} \end{align*} Hence, the tangent line of $g(x)$ at $(4,4)$ is given by \begin{align*} \frac{y-y_1}{x-x_1}&=m\\ \frac{y-4}{x-4}&=\frac{-1}{2} y&= -\frac{1}{2} x+6 \end{align*}
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