3.3 Rates of change
• Find Average Rate of Change
• Determine if a Function is Continuous at a Point
• Skip Instantaneous Rate of Change until section
3.4
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)
ba
Example 1
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
x
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
2000.
Example 1
f(60)=29.4829
f(100)=16.7744
Average rate of change = f(100) – f(60)
100 – 60
= 16.7744 - 29.4829
100 – 60
= -0.3177
Example 2
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.
Example 2
(1995, 21) (2003, 721)
Average Rate of Change =
721 – 21
2003 – 1995
= 87.5
Find the average rate of
change for the function over the
given interval.
5
A) y 
between x  2 and x  4
2x  3
B) y  4 x  6 between x  2 and x  5
2
Answers
A) 2
B) -28
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.
Requirements for Function to be Continuous
A function is continuous at x = c if
the following conditions are
satisfied:
1) f(c) is defined
2) lim f(x) - does exist
x c
3)lim f(x) = f(c)
x c
If a function is not continuous at
c, it is discontinuous there.
Give the x-values where the
graph is discontinous.
The graph is discontinous where
x = 1 and where x = 3
Come back and work these after
doing section 3.4 or even 4.1.
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 )
lim
h 0
h
Find the instantaneous rate of
change for the function at the
given value.
s(t )  4t  6
2
at t = 2
Problem #30 p.191
The revenue (in thousands of dollars) from
producing x units of an item is
R( x)  10 x  0.002 x 2
a) Find the average rate of change of revenue
when production is increased from 1000 to
1001 units.
b) Find and interpret the instantaneous rate of
change of revenue when 1000 units are
produced.
Answers
a) $5.998 Thousand or
$5998
b) $6 Thousand or
$6000