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3.3 Rates of change
One of the main applications of
calculus is determining how one
variable changes in relation to
another. For example, a
manager would want to know
how much profit changes with
respect to the amount of money
spent on advertising.
The average rate of change
(just like slope of line)in a
function f(x) with respect to x
for a function f as x changes
from a to b is given by
f (b)  f (a)
The percentage of men aged 65 and
older in the workforce has been
declining over the last century. The
percent can be approximated by the
function f ( x)  68.7(.986)
where x is the number of years since
1900. Find the average rate of
change of this percent from 1960 to
The graph below gives the Annual Numbers
of New Nonmedical Users of OxyContin®:
1995-2003. Find the average rate of change
from 1995 to 2003.
Finding the average rate of
change of a function over a large
interval can lead to answers that
are not very helpful. The results
are often more useful if the
average is found over a fairly
small interval. Finding the exact
rate of change at a given x-value
requires a continuous function.
A function is continuous at x = c if
the following conditions are
1) f(c) is defined
2) lim f(x) exist
x c
3)lim f(x) = f(c)
x c
If a function is not continuous at
c, it is discontinuous there.
The exact rate of change
of f at x = a, called the
instantaneous rate of change
of f at x = a is
f ( a  h)  f ( a )
h 0
Page 185 ex 3b
Find the average rate of
change for the function over the
given interval.
between x  2 and x  4
2x  3
y  4 x  6 between x  2 and x  5
Find the instantaneous rate of
change for the function at the
given value.
s(t )  4t  6
at t = 2
Problem #26 p.190
Problem #30 p.191