Math 135 Business Calculus Spring 2009Class Notes1.3 Average Rates of Change

Consider a function y = f(x) and two input values x1 and x2. The change in input , or the change inx, is

x2 − x1.The corresponding change in output , or the change in y, is

y2 − y1 = f(x2)− f(x1).

DEFINITION OF AVERAGE RATE OF CHANGE

The average rate of change of y = f(x) with respect to x, as x changes from x1 to x2, is the ratio ofthe change in output to the change in input:

y2 − y1

x2 − x1=

f(x2)− f(x1)x2 − x1

where x1 6= x2.

If we look at the graph of the function, thenthe average rate of change will equal the slopeof the line passing through the points P (x1, y1)and Q(x2, y2). The line passing through P andQ is called a secant line.

EXAMPLE The graph in the figure shows a typical response to adversing. Aftr an amount a is spenton advertising, the company sells N(a) units of a product. Find the average rate of change of N as achangesa) from 0 to 1

b) from 1 to 2

c) from 2 to 3

0 1 2 3 4a0

100

200

300

400

500

600

700

N

11

12 Chapter 1 Differentiation

DIFFERENCE QUOTIENTS AS AVERAGE RATES OF CHANGE

We can rewrite the average rate ofchange of a function in a different no-tation as follows. Instead of using x1

for an initial input, use x. From x,we move h units to the second inputx2 = x + h. Then the average rate ofchange is

f(x2)− f(x1)x2 − x1

=f(x + h)− f(x)

(x + h)− x

=f(x + h)− f(x)

h

DEFINITION OF DIFFERENCE QUOTIENT

The average rate of change of a function f with respect to x is also called the difference quotient. Itis given by

f(x + h)− f(x)h

The difference quotient is equal to the slope of the line from°x, f(x)

¢to

°x + h, f(x + h)

¢.

EXAMPLE Let f(x) = x2. Find the difference quotient when:a) x = 5 and h = 3

b) x = 5 and h = 0.1

EXAMPLE Let f(x) = x2. Find a simplified form of the difference quotient. Then find the value ofthe difference quotient when x = 5 and h = 0.1.

1.3 Average Rates of Change 13

EXAMPLE Let f(x) = x3. Find a simplified form of the difference quotient.

EXAMPLE Let f(x) = 3/x. Find a simplified form of the difference quotient.