MA 15800
Lesson 6
The Difference Quotient and Average Rate of Change
Summer 2016
Average Rate of Change:
Given a function 𝑓(𝑥) defined on a closed interval [𝑎, 𝑏], the average rate of change is defined
as…
f (b)  f (a)
Average Rate of Change =
ba
Notice the similarity to the formula for slope of a line.
For each function, find the average rate of change for the given interval.
Ex 1:
f ( x)  x 2
[8,10]
Ex 3: h( x) 
x5
x 1
[3,11]
Ex 5: G( x) 
1
2x
[ x, x  h]
Ex 2: g ( x)  3x2  2 x
Ex 4: F ( x)  x 2  2 x
[1,5]
[ x, x  h]
Many of you may be taking a calculus or an applied calculus class here at Purdue. One of the
most important functions in calculus is a differential function or a derivative. The definition of a
derivative given in textbooks is based on a limit of an algebraic expression called the difference
quotient. It is important to be able to evaluate or simplify the difference quotient for a given
function. Below is the definition of the difference quotient.
Difference Quotient:
f ( x  h)  f ( x )
h
where (𝑥, 𝑓(𝑥)) and (𝑥 + ℎ, 𝑓(𝑥 + ℎ)) are two points or ordered pairs of the function. The
value of h is the small difference between the x-coordinates of the two point (ordered pairs).
If there is a function f (x), the difference quotient for that function is defined as
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MA 15800
Lesson 6
The Difference Quotient and Average Rate of Change
Summer 2016
Find the simplified difference quotient for each function. Evaluate each difference quotient if
ℎ = 0. (Evaluating the difference quotient when ℎ = 0 yields what is known as a derivative in
Calculus.
Ex 1: 𝑓(𝑥) = 𝑥 2 − 3𝑥 + 1
Ex 2: g ( x) 
2
x
2
MA 15800
Lesson 6
The Difference Quotient and Average Rate of Change
Summer 2016
Ex 3: h( x)  (3  5x)2
Ex 4: y  x  5
Hint: Rationalize the numerator in the difference quotient in order to simplify. This is the major
reason why we rationalized numerators in an earlier lesson.
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MA 15800
Ex 5: F ( x) 
Ex 6:
Lesson 6
The Difference Quotient and Average Rate of Change
Summer 2016
x
x2
f ( x)  x 3  x 2  2 x  9
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MA 15800
Ex 7: G ( x) 
Lesson 6
The Difference Quotient and Average Rate of Change
Summer 2016
x2
x
Alternate Definition of the Difference Quotient:
If there is a function f (x), the difference quotient for that function can be defined as
where 𝑎 is a value in the domain of function f.
f ( x)  f (a)
xa
Calculus textbooks use both definitions for the difference quotient given in this lesson.
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MA 15800
Lesson 6
The Difference Quotient and Average Rate of Change
Ex 8: Simplify the difference quotient of the form
Summer 2016
f ( x)  f (a)
for the function 𝑓(𝑥) = 2𝑥 3 .
xa
Ex 9: Use the alternate definition of the difference quotient and evaluate for the function
f ( x)  4 x2  2a  1 .
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MA 15800
Lesson 6
The Difference Quotient and Average Rate of Change
Summer 2016
Ex 10:
Find the simplify the following ‘difference quotient’ where 𝑓(𝑥) = 𝑥 2 + 3𝑥.
f (12  h)  f (12)
h
Ex 11:
Find and simplify using the first definition of the difference quotient for the function 𝑔(𝑥) =
√3𝑥.
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