Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 3: Derivatives - Section 3.1 - Tangents and the Derivative at a Point - Exercises 3.1: 23

Answer

a. The value of the derivative $P'(5)$ indicates the number of yeast cells that the culture will grow by at the time, $t=5$. The units are cells/second. b. By observing the graphs, sketching a tangent at both values of $t=2$ and $t=3$, the slope at $t=3$ will be greater indicating a greater growth rate of yeast cells. c. To find the instantaneous rate of growth at the time of 5 hours, we need to find the derivative function of the population function. Given that $P(t)=6.10t^2 - 9.28t + 16.43$ then, $P'(t)=2(6.10)t^{2-1}-9.28t^{1-1}=12.20t-9.28$ Thus, $P'(5)=12.20(5)-9.28=51.72$ cells per second

Work Step by Step

a. The value of the derivative $P'(5)$ indicates the number of yeast cells that the culture will grow by at the time, $t=5$. The units are cells/second b. By observing the graphs, sketching a tangent at both values of $t=2$ and $t=3$, the slope at $t=3$ will be greater indicating a greater growth rate of yeast cells. c. To find the instantaneous rate of growth at the time of 5 hours, we need to find the derivative function of the population function. Given that $P(t)=6.10t^2 - 9.28t + 16.43$ then, $P'(t)=2(6.10)t^{2-1}-9.28t^{1-1}=12.20t-9.28$ Thus, $P'(5)=12.20(5)-9.28=51.72$ cells per second
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