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Comparing Average and Instantaneous Rates of Change
Velocity is only one example of the concept of rate of change. In general, suppose
that a quantity y depends on x according to the equation y f 1x2. As the
independent variable changes from a to a h 1¢x a h a h2, the
corresponding change in the dependent variable y is ¢y f 1a h2 f 1a2.
Average Rate of Change
¢y f 1a h2 f 1a2
The difference quotient is called the average rate of
¢x h
change in y with respect to x over the interval from x a to x a h.
From the diagram, it follows that the average rate of change equals the slope of
the secant PQ of the graph of f 1x2 where P1a, f 1a22 and Q1a h, f 1a h22.
The instantaneous rate of change in y with respect to x when x a is defined to
be the limiting value of the average rate of change as h S 0.
f(a + h)
y
Q y = f(x)
f(a)
P
a
x
a + h
y
x
Instantaneous Rates of Change
Therefore, we conclude that the instantaneous rate of change in y f 1x2
with respect to x when x a ¢y f 1a h2 f 1a2
is lim lim
, provided that
¢xS0
¢x
hS0 h
the limit exists.
It should be noted that, as with velocity, the instantaneous rate of change in y
with respect to x at x a equals the slope of the tangent to the graph of y f 1x2
at x a.
EXAMPLE 3
Selecting a strategy to calculate instantaneous rate of change
The total cost, in dollars, of manufacturing x calculators is given by
C1x2 10Vx 1000.
a. What is the total cost of manufacturing 100 calculators?
b. What is the rate of change in the total cost with respect to the number of
calculators, x, being produced when x 100?
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1.3 RATES OF CHANGE
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