Calculus (3rd Edition)

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

Chapter 1 - Precalculus Review - 1.3 The Basic Classes of Functions - Exercises - Page 23: 39


$P(t+10)=2P(t)$ $g\left(t+\dfrac{1}{k}\right)=2g(t)$

Work Step by Step

We are given the function: $P(t)=30\cdot 2^{0.1t}$ Compute $P(t+10)$: $P(t+10)=30\cdot 2^{0.1(t+10)}=30\cdot 2^{0.1t+1}$ $=30\cdot 2^{0.1t}\cdot 2^1$ $=(30\cdot 2^{0.1t})\cdot 2$ $=2P(t)$ We got: $P(t+10)=2P(t)$ So the population doubles every 10 years. Generalisation: $g(t)=a2^{kt}$ Compute $g\left(t+\dfrac{1}{k}\right)$: $g\left(t+\dfrac{1}{k}\right)=a2^{k\left(t+\frac{1}{k}\right)}$ $=a2^{kt+1}$ $=a2^{kt}\cdot 2^1$ $=(a2^{kt})\cdot 2$ $=2g(t)$ We got: $g\left(t+\dfrac{1}{k}\right)=2g(t)$ So the function doubles each $\dfrac{1}{k}$ years.
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