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:
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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
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a) Find the difference quotient using the original notation from above.
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Note that the slope of the
secant line depends on x and h.
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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.
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1.3-1
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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
=
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Thus the answers agree.
Another example:
Consider the function:
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a) Find the difference quotient:
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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):
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which gives that
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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.