James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 3.7 Rates of Change in the Natural and Social Sciences 225
(h) Graph the position, velocity, and acceleration functions for 0 < t < 5.
(i) When is the particle speeding up? When is it slowing down?
SOLUTION
(a) The velocity function is the derivative of the position function.
s − f std − t 3 2 6t 2 1 9t
vstd − ds
dt − 3t 2 2 12t 1 9
(b) The velocity after 2 s means the instantaneous velocity when t − 2, that is,
vs2d − ds
dt
Z − 3s2d 2 2 12s2d 1 9 − 23 mys
t−2
The velocity after 4 s is
vs4d − 3s4d 2 2 12s4d 1 9 − 9 mys
(c) The particle is at rest when vstd − 0, that is,
3t 2 2 12t 1 9 − 3st 2 2 4t 1 3d − 3st 2 1dst 2 3d − 0
and this is true when t − 1 or t − 3. Thus the particle is at rest after 1 s and after 3 s.
(d) The particle moves in the positive direction when vstd . 0, that is,
3t 2 2 12t 1 9 − 3st 2 1dst 2 3d . 0
t=3
s=0
t=0
s=0
t=1
s=4
s
This inequality is true when both factors are positive st . 3d or when both factors are
negative st , 1d. Thus the particle moves in the positive direction in the time intervals
t , 1 and t . 3. It moves backward (in the negative direction) when 1 , t , 3.
(e) Using the information from part (d) we make a schematic sketch in Figure 2 of the
motion of the particle back and forth along a line (the s-axis).
(f) Because of what we learned in parts (d) and (e), we need to calculate the distances
traveled during the time intervals [0, 1], [1, 3], and [3, 5] separately.
The distance traveled in the first second is
FIGURE 2
| f s1d 2 f s0d | − | 4 2 0 | − 4 m
From t − 1 to t − 3 the distance traveled is
25
√
a
s
0 5
-12
FIGURE 3
From t − 3 to t − 5 the distance traveled is
| f s3d 2 f s1d | − | 0 2 4 | − 4 m
| f s5d 2 f s3d | − | 20 2 0 | − 20 m
The total distance is 4 1 4 1 20 − 28 m.
(g) The acceleration is the derivative of the velocity function:
astd − d 2 s
dt − dv − 6t 2 12
2 dt
as4d − 6s4d 2 12 − 12 mys 2
(h) Figure 3 shows the graphs of s, v, and a.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.