Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 3 - Derivatives - 3.2 Working with Derivatives - 3.2 Exercises - Page 143: 29


$a.$ At $t=0$; $b.$ Positive; $c.$ Decreasing; $d.$ The graph is on the figure below.

Work Step by Step

$a.$ The rate of change of the charge $Q'$ is the greatest at the point where the tangent to the graph of $Q(t)$ has the steepest upward slope which is for $x=0$; $b.$ It is positive because the charge is only increasing. $c.$ $Q'$ is decreasing because we see that as time goes by (as $t$ becomes bigger) the slope of the tangent to the graph is smaller and smaller. $d.$ The general shape of the graph of $Q'$ can be deduced from the shape of the graph of $Q$. We see that it starts from some positive value and keeps decreasing and asymptotically approaches zero because the graph of $Q$ becomes increasingly horizontal as $t$ grows and thus the slope of its tangent tends to zero. The graph is on the figure below.
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