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

Section 3.7 Rates of Change in the Natural and Social Sciences 227
while the density right at x − 1 is
− dm
dx
Z − Z 1 − 0.50 kgym
x−1 2sx x−1
■
FIGURE 6
ExamplE 3 A current exists whenever electric charges move. Figure 6 shows part
of a wire and electrons moving through a plane surface, shaded red. If DQ is the net
charge that passes through this surface during a time period Dt, then the average current
during this time interval is defined as
average current − DQ
Dt
− Q 2 2 Q 1
t 2 2 t 1
If we take the limit of this average current over smaller and smaller time intervals,
we get what is called the current I at a given time t 1 :
DQ
I − lim
Dt l 0 Dt
− dQ
dt
Thus the current is the rate at which charge flows through a surface. It is measured in
units of charge per unit time (often coulombs per second, called amperes).
■
Velocity, density, and current are not the only rates of change that are important in
physics. Others include power (the rate at which work is done), the rate of heat flow,
temperature gradient (the rate of change of temperature with respect to position), and the
rate of decay of a radioactive substance in nuclear physics.
Chemistry
ExamplE 4 A chemical reaction results in the formation of one or more substances
(called products) from one or more starting materials (called reactants). For instance,
the “equation”
2H 2 1 O 2 l 2H 2 O
indicates that two molecules of hydrogen and one molecule of oxygen form two molecules
of water. Let’s consider the reaction
A 1 B l C
where A and B are the reactants and C is the product. The concentration of a reactant
A is the number of moles (1 mole − 6.022 3 10 23 molecules) per liter and is denoted
by fAg. The concentration varies during a reaction, so fAg, fBg, and fCg are all functions
of time std. The average rate of reaction of the product C over a time interval
t 1 < t < t 2 is
DfCg
Dt
− fCgst 2d 2 fCgst 1 d
t 2 2 t 1
But chemists are more interested in the instantaneous rate of reaction, which is
obtained by taking the limit of the average rate of reaction as the time interval Dt
approaches 0:
rate of reaction − lim
Dt l 0
DfCg
Dt
− dfCg
dt
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