Calculus (3rd Edition)

Published by W. H. Freeman

Chapter 7 - Exponential Functions - 7.1 Derivative of f(x)=bx and the Number e - Exercises - Page 328: 66

Answer

$$0.3567\ \text{kg}/\text{year}$$

Work Step by Step

Given $$W(t)=\left(3.46293-3.32173 e^{-0.03456 t}\right)^{3.4026}$$ Since \begin{align*} W'(t)& =3.4026\left(3.46293-3.32173 e^{-0.03456 t}\right)^{2.4026} \left(0.1147989 e^{-0.03456 t}\right) \\ &=0.3906147 e^{-0.03456 t}\left(3.46293-3.32173 e^{-0.03456 t}\right)^{2.4026} \end{align*} Then the rate of change at $t=10$ is \begin{align*} W'(10) &=0.3906147 e^{-0.03456(10)}\left(3.46293-3.32173 e^{-0.03456 (10)}\right)^{2.4026}\\ &\approx 0.3567\ \text{kg}/\text{year} \end{align*}

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