11DIFFERENTIATION - Department of Mathematics

EXAMPLE 4
SOLUTION ✔
EXAMPLE 5
SOLUTION ✔
FIGURE 11.14
The CPI of a certain economy from year a
to year b is given by I(t).
I(t)
a b
t
11.5 HIGHER-ORDER DERIVATIVES 753
Refer to the example on page 621. The distance s (in feet) covered by a
maglev moving along a straight track t seconds after starting from rest is given
by the function s 4t 2 (0 t 10). What is the maglev’s acceleration at the
end of 30 seconds?
The velocity of the maglev t seconds from rest is given by
v ds d
dt dt (4t 2 ) 8t
The acceleration of the maglev t seconds from rest is given by the rate of
change of the velocity of t—that is,
a d d
v
dt dtds
dt d 2s d
(8t) 8
2 dt dt
or 8 feet per second per second, normally abbreviated, 8 ft/sec2 .
Aball is thrown straight up into the air from the roof of a building. The
height of the ball as measured from the ground is given by
s 16t 2 24t 120
where s is measured in feet and t in seconds. Find the velocity and acceleration
of the ball 3 seconds after it is thrown into the air.
The velocity v and acceleration a of the ball at any time t are given by
v ds d
dt dt (16t2 24t 120) 32t 24
and
a d 2t d
2 dt dtds
dt d
(32t 24) 32
dt
Therefore, the velocity of the ball 3 seconds after it is thrown into the air is
v 32(3) 24 72
That is, the ball is falling downward at a speed of 72 ft/sec. The acceleration
of the ball is 32 ft/sec2 downward at any time during the motion.
Another interpretation of the second derivative of a function—this time
from the field of economics—follows. Suppose the consumer price index (CPI)
of an economy between the years a and b is described by the function I(t)
(a t b) (Figure 11.14). Then, the first derivative of I, I(t), gives the rate
of inflation of the economy at any time t. The second derivative of I, I(t),
gives the rate of change of the inflation rate at any time t. Thus, when the
economist or politician claims that ‘‘inflation is slowing,’’ what he or she is
saying is that the rate of inflation is decreasing. Mathematically, this is equivalent
to noting that the second derivative I(t) is negative at the time t under
consideration. Observe that I(t) could be positive at a time when I(t) is
negative (see Example 6). Thus, one may not draw the conclusion from the
aforementioned quote that prices of goods and services are about to drop!