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 - Page 108: 24


a. $0\lt t \lt 3$ b. The derivative is decreasing an hour before.

Work Step by Step

a. From what can be seen on the graph, the function appears to be sloping upwards from the time $t=0$ to $t=3$ indicating that the effectiveness of the medicinal drug is increasing with time. b. The drug reaches a maximum when the function has a peak, which takes place at $t=3$. This is when the effectiveness is maximized. Drawing a tangent line would give us a horizontal line where the derivative will be equal to 0. As the time increases an hour before the 3 hour mark, the derivative appears to be diminishing. While the derivative from $t=2$ to $t=3$ is positive, it appears that the drug is becoming more effective more and more slowly, indicating that the value of the derivative itself is decreasing. ( Second derivative is negative )
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