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A difference quotient is an expression that represents the difference
between two function values divided by the difference between two
inputs. This is an extension of the slope formula from Lessons 16 and 17
โ๐ฆ
โ๐ฅ
), when we found the change in ๐ฆ (or the difference between two
๐ฆ values) and divided by the change in ๐ฅ. Now we will find the difference
between two function values, divided by the difference between two
inputs:
By combining like terms in the denominator, we get the following
simplified form of a difference quotient:
Difference Quotient :
- a fraction (or quotient) containing the difference of two functions
values in the numerator, and the difference of two inputs in the
denominator
o
๐
( ๐ฅ+โ
) โ๐
( ๐ฅ
)
โ
o the input ๐ฅ could be replaced with a numeric value or another
expression
Our focus when working with difference quotients in this class is to
simplify them. To do so, I prefer to follow the step-by-step procedure
which is demonstrated on the next page, but you are welcome to use
another method if you choose.
Example 1: Given ๐
= 5 ๐ฅ โ 2 , find the difference quotient
๐(๐+โ)โ๐(๐)
โ
Steps for Simplifying a Difference Quotient :
- find the first function value
o in Example 1 I find ๐
by replacing ๐ฅ in the function
๐(๐ฅ) = 5 ๐ฅ โ 2 with the expression ๐ + โ
- find the second function value
o I find ๐
by replacing ๐ฅ in the function ๐
= 5 ๐ฅ โ 2 with
the expression ๐
- find the difference between the two function values
o I find ๐(๐ + โ) โ ๐(๐) by taking the two function values from
steps 1 and 2 and subtracting them
- divide the difference by the expression โ
o since a difference quotient is a fraction, be sure to simplify
completely by factoring and canceling common factors
๐(๐+โ)โ๐(๐)
โ
5โ
โ
Example 3 : Given the function ๐
= 2 ๐ฅ โ 3 , find the difference
quotient
๐
( ๐+โ
) โ๐
( ๐
)
โ
Steps for Determining the Value of a Difference Quotient :
- find the first function value
- find the second function value
- find the difference between the two function values
- divide the difference by the expression โ
Again, I prefer to break difference quotients into smaller pieces in order to
simplify them, but you do not have to. You can go through and simplify
difference quotients by leaving them as one single expression the entire
time, as demonstrated in the next example.
Example 4 : Given the function ๐
2
- 10 ๐ฅ, find the difference
quotient
๐
( ๐ฅ+โ
) โ๐
( ๐ฅ
)
โ
2
2
2
2
2
2
2
2
2
2
When simplifying difference quotients, use whichever procedure makes
the most sense to you.
Example 6 : Given the function ๐
2
, find the difference
quotient
๐(๐+โ)โ๐(๐)
โ
Example 7 : Given the function ๐(๐ฅ) =
1
๐ฅ
, find the difference quotient
๐
( ๐ฅ+โ
) โ๐
( ๐ฅ
)
โ
รท โ
Answers to Examples:
1.
๐(๐+โ)โ๐(๐)
โ
= 5 ; 2 a. ๐
( 3 + โ
) = โโ
2
โ 7โ โ 5 ; 2b. ๐
( 3 + โ
) = โ 5 ;
2 c. ๐( 3 + โ) โ ๐( 3 ) = โโ
2
โ 7โ ; 2 d.
๐
( 3 +โ
) โ๐
( 3
)
โ
= โโ โ 7 ;
3.
๐(๐+โ)โ๐(๐)
โ
= 2 ; 4.
๐(๐ฅ+โ)โ๐(๐ฅ)
โ
= โ 10 ๐ฅ โ 5โ + 10 ;
5.
๐
( โ 2 +โ
) โ๐
( โ 2
)
โ
=
1
3
โ +
11
3
; 6.
๐
( ๐+โ
) โ๐
( ๐
)
โ
= โ 2 + โ + 2 ๐ ;
7.
๐(๐ฅ+โ)โ๐(๐ฅ)
โ
=
โ 1
๐ฅ
( ๐ฅ+โ
)
;
