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: 65


(a) $ 1.28 \ \text{cm}/\text{year }$ (b) $t\approx 2.33\ \text{ years}$ (c) $ 32 \mathrm{cm}$

Work Step by Step

Given $$L(t)=32\left(1-e^{-0.37 t}\right)$$ (a) Since $$ L'(t)=32\left(0.37e^{-0.37 t}\right)= 11.84e^{-0.37 t}$$ at $t=6$, we get \begin{align*} L'(6)&= 11.84e^{-0.37 (6)}\\ &= 1.28 \ \text{cm}/\text{year } \end{align*} (b) Since \begin{align*} L'(t)&=11.84e^{-0.37 t}\\ 5&= 11.84e^{-0.37 t}\\ \ln (0.42229)&= -0.37t \end{align*} Then $$t\approx 2.33\ \text{ years}$$ (c) Since \begin{align*} L&=\lim _{t \rightarrow \infty}\left(32-32 e^{-0.37 t}\right)\\ &=32-32 \lim _{t \rightarrow \infty} e^{-0.37 t}\\ &=32-0\\ &=32 \mathrm{cm} \end{align*}
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