University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 3 - Section 3.4 - The Derivative as a Rate of Change - Exercises - Page 147: 34

Answer

The detailed explanations are below.
1537997325

Work Step by Step

$s=t^2-3t+2$, $0\le t\le5$ 1) Find $v$ and $a$: $$v=\frac{ds}{dt}=\frac{d}{dt}(t^2-3t+2)=2t-3$$ $$a=\frac{dv}{dt}=\frac{d}{dt}(2t-3)=2$$ The graphs of $s, v$ and $a$ are shown below. 2) We examine each element of the graph: The curve $s$: - The curve $s$ starts from $2$ at $t=0$, decreases constantly before reaching its minimum value at $t=1.5$. So from $t=0$ to $t=1.5$, the object moves backwards. - From $t=1.5$ to $t=5$, the curve increases constantly before reaching its maximum value, which is $12$ at $t=5$. So during these times, the object moves forward. That being said, at $t=1.5$, the object has changed its direction. Overall, the object is furthest from the origin at $t=5$, when its position reaches the value $12$. The line $v$: - The line $v$ increases constantly on $[0,5]$. From $t=0$ to $t=1.5$, $v$ is negative and increases constantly before reaching $0$ at $t=1.5$. The object was moving backwards at the highest speed at the starting time. Then as it continued to move backwards $(v\lt0)$, it went more and more slowly before stopping at $t=1.5$, which is also when it changed direction. - From $t=1.5$ to $t=5$, $v$ is positive and increases constantly. After the object stopped moving backwards at $t=1.5$, it started to move forward $(v\gt0)$ and its velocity rose constantly and reached its maximum at $t=5$. The line $a$: The line $a$ is a horizontal line $a=2$ throughout the whole interval. Since $a$ is positive, we know that the movement trend of velocity is increase on the whole interval, and since $a$ does not change its value at all, this increase trend is constant.
Small 1537997325
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.