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

146 Chapter 2 Limits and Derivatives
We know that one interpretation of the derivative f 9sad is as the slope of the tangent
line to the curve y − f sxd when x − a. We now have a second interpretation:
y
Q
The derivative f 9sad is the instantaneous rate of change of y − f sxd with respect
to x when x − a.
P
FIGURE 9
The y-values are changing rapidly
at P and slowly at Q.
x
The connection with the first interpretation is that if we sketch the curve y − f sxd,
then the instantaneous rate of change is the slope of the tangent to this curve at the point
where x − a. This means that when the derivative is large (and therefore the curve is
steep, as at the point P in Figure 9), the y-values change rapidly. When the derivative is
small, the curve is relatively flat (as at point Q) and the y-values change slowly.
In particular, if s − f std is the position function of a particle that moves along a
straight line, then f 9sad is the rate of change of the displacement s with respect to the
time t. In other words, f 9sad is the velocity of the particle at time t − a. The speed of the
particle is the absolute value of the velocity, that is, | f 9sad | .
In the next example we discuss the meaning of the derivative of a function that is
defined verbally.
ExamplE 6 A manufacturer produces bolts of a fabric with a fixed width. The cost of
producing x yards of this fabric is C − f sxd dollars.
(a) What is the meaning of the derivative f 9sxd? What are its units?
(b) In practical terms, what does it mean to say that f 9s1000d − 9?
(c) Which do you think is greater, f 9s50d or f 9s500d? What about f 9s5000d?
SOLUtion
(a) The derivative f 9sxd is the instantaneous rate of change of C with respect to x; that
is, f 9sxd means the rate of change of the production cost with respect to the number of
yards produced. (Economists call this rate of change the marginal cost. This idea is
discussed in more detail in Sections 3.7 and 4.7.)
Because
f 9sxd − lim
Dx l 0
DC
Dx
Here we are assuming that the cost
function is well behaved; in other
words, Csxd doesn’t oscillate rapidly
near x − 1000.
the units for f 9sxd are the same as the units for the difference quotient DCyDx. Since
DC is measured in dollars and Dx in yards, it follows that the units for f 9sxd are dollars
per yard.
(b) The statement that f 9s1000d − 9 means that, after 1000 yards of fabric have been
manufactured, the rate at which the production cost is increasing is $9yyard. (When
x − 1000, C is increasing 9 times as fast as x.)
Since Dx − 1 is small compared with x − 1000, we could use the approximation
f 9s1000d < DC
Dx − DC
1 − DC
and say that the cost of manufacturing the 1000th yard (or the 1001st) is about $9.
(c) The rate at which the production cost is increasing (per yard) is probably lower
when x − 500 than when x − 50 (the cost of making the 500th yard is less than the
cost of the 50th yard) because of economies of scale. (The manufacturer makes more
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