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

abafbf

)()(

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 where x is the number of years since 1900. Find the average rate of change of this percent from 1960 to 2000.

( ) 68.7(.986)xf x

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 1

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

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.

2) 4 6 between x 2 and x 5B y x

5) between x 2 and x 4

2 3A y

x

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.

A function is continuous at x = c if the following conditions are satisfied:1) f(c) is defined2) lim f(x) - does exist

3)lim f(x) = f(c)

If a function is not continuous at c, it is discontinuous there.

cx

cx

Requirements for Function to be Continuous

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

0

( ) ( )limh

f a h f a

h

Find the instantaneous rate of change for the function at the given value.

64)( 2 ttsat t = 2

Problem #30 p.191

The revenue (in thousands of dollars) from producing x units of an item is

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.

2( ) 10 0.002R x x x

Answersa) $5.998 Thousand or $5998b) $6 Thousand or $6000