Areas, Volumes, and Arc Lengths 261
0
2
4
6
8
10
12
14
p(x)
y
x
12 34 56 7 8
Figure 12.1-5
TIP • Remember differentiability implies continuity, but the converse is not true, i.e.,
continuity does not imply differentiability, e.g., as in the case of a cusp or a corner.
Example 3
The position function of a moving particle on a coordinate axis is:
s=
∫t
0
f(x)dx, wheretis in seconds andsis in feet.
The functionfis a differentiable function and its graph is shown below in Figure 12.1-6.
–8
0
10
x
y
f(x)
12 3 4 5 6 7 8
(3,–5)
(4,–8)
Figure 12.1-6
(a) What is the particle’s velocity att=4?
(b) What is the particle’s position att=3?
(c) When is the acceleration zero?
(d) When is the particle moving to the right?
(e) Att=8, is the particle on the right side or left side of the origin?