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

Section 2.8 The Derivative as a Function 159
ExamplE 6 If f sxd − x 3 2 x, find and interpret f 0sxd.
SOLUTION In Example 2 we found that the first derivative is f 9sxd − 3x 2 2 1. So the
second derivative is
f 99sxd − s f 9d9sxd − lim
h l0
f 9sx 1 hd 2 f 9sxd
h
_1.5
2
f·
fª
f
1.5
− lim
h l0
f3sx 1 hd 2 2 1g 2 f3x 2 2 1g
h
− lim
h l0
3x 2 1 6xh 1 3h 2 2 1 2 3x 2 1 1
h
_2
− lim
h l0
s6x 1 3hd − 6x
FIGURE 10
TEC In Module 2.8 you can see how
changing the coefficients of a polynomial
f affects the appearance of the
graphs of f , f 9, and f 99.
The graphs of f , f 9, and f 0 are shown in Figure 10.
We can interpret f 0sxd as the slope of the curve y − f 9sxd at the point sx, f 9sxdd. In
other words, it is the rate of change of the slope of the original curve y − f sxd.
Notice from Figure 10 that f 0sxd is negative when y − f 9sxd has negative slope
and positive when y − f 9sxd has positive slope. So the graphs serve as a check on our
calculations.
n
In general, we can interpret a second derivative as a rate of change of a rate of change.
The most familiar example of this is acceleration, which we define as follows.
If s − sstd is the position function of an object that moves in a straight line, we know
that its first derivative represents the velocity vstd of the object as a function of time:
vstd − s9std − ds
dt
The instantaneous rate of change of velocity with respect to time is called the acceleration
astd of the object. Thus the acceleration function is the derivative of the velocity
function and is therefore the second derivative of the position function:
or, in Leibniz notation,
astd − v9std − s0std
a − dv
dt − d 2 s
dt 2
Acceleration is the change in velocity you feel when speeding up or slowing down in
a car.
The third derivative f - is the derivative of the second derivative: f -− s f 0d9. So
f -sxd can be interpreted as the slope of the curve y − f 0sxd or as the rate of change of
f 0sxd. If y − f sxd, then alternative notations for the third derivative are
y- − f -sxd − S 2D
d d 2 y
− d 3 y
dx dx dx 3
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