microbook_3e-page19 - to denote a change and start using...

columns because we are looking at “large” changes in x as opposed to the infinitesimally small changes
described in the notes entitled: “What’s the Difference between Marginal Cost and Average Cost?”
Why does it make a difference whether we look at small or large changes? Consider the following
derivation of the slope of
+ ,
x f :
+ ,
+ , + , + ,
+ , + , + , + ,
+ ,
+ , + , + ,
+ ,
x 3 x 6 x f
x
x 3
x
x x 6
x
x 3 x x 6
x
x 3 x 3 x x 6 x 3
x
x 3 x x x 2 x 3
x
x 3 x x x x 3
x
x 3 x x 3
x
x f x x f
x
x f
x f
2 2
2
2
2
2 2 2 2 2 2
' .
c
'
'
.
'
'
'
' . '
'
0 ' . ' .
'
0 ' . ' .
'
0 ' . ¡ ' .
'
0 ' .
'
0 ' .
'
'
c
If we look at one unit changes in the value of x – i.e. 1 x ' – then the slope of
+ ,
x f evaluated at each
value of x is equal to x 3 x 6 ' . which equals 3 x 6 . since 1 x ' .
If we look at changes in x that are so small that the changes are approximately zero – i.e.: 0 x | ' – then
the slope of
+ ,
x f evaluated at each value of x is approximately equal to x 6 and gets closer and closer to
x 6 as the change in x goes to zero.
So if
+ , 2
x 3 x f , then
+ ,
x 6 x f
c
.
Since we’ll be looking at infinitesimally small changes in x , we’ll stop using the symbol