MA 109

The Difference Quotient

General Explanation:

The difference quotient at x for a function f is given by:

Sometimes this is written using Dx for the change in x and Dy for the change in y:

=

=

The difference quotient thus becomes simply:

Geometric meaning:

The difference quotient gives the slope of the "secant line", the line shown in red in the

following graph:

Example from class:

Consider the function

=

a) Find the difference quotient using the original notation from above.

=

=

Note that the slope of the

secant line depends on x and h.

=

=

=

b) Using the alternate notation, find Dy and Dx if x changes from 1 to 1.3. Then find the value

of the difference quotient in this situation.

=

1.3-1

=

=

=

=

=

=

=

=

This is the slope of the secant line when we have x change from

1 to 1.3.

c) Show that the answer to part b) is consistent with the formula found in part a).

Part a) showed that the difference quotient for this function simplifies to

Using

x = 1

and

h

=

we get a difference quotient value of

=

=

Thus the answers agree.

Another example:

Consider the function:

=

a) Find the difference quotient:

=

=

=

=

The difference quotient is always 5. This makes sense since our function f is a linear function.

A line has a constant slope, in this case 5.

b) Find Dy if we use Dx = 0.2,

The fastest method is to use what we found in part a):

=

which gives that

=

=

=

Notice that in this problem, unlike the previous example, Dy does not depend at all on the

starting x value. It is always 5 times Dx.