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The Difference Quotient

Let's do another one:

Given f( x ) = 2x^2 + 3 find the difference quotient:

          Do the blob thang:

f( blob ) = 2( blob )^2 + 3
 

f( x + h ) - f( x ) / h = [ 2( x + h )^2 + 3 ] - ( 2x^2 + 3 ) / h ... the [ 2( x + h )^2 + 3 ] is f( x + h) ... the ( 2x^2 + 3 ) is f( x )
 

( 2( x^2 + 2xh + h^2 ) + 3 - 2x^2 - 3 ) / h
 

= ( 2x^2 + 4xh + 2h^2 + 3 - 2x^2 - 3 ) / h ... at this point, everything without an h should subtract out!
 

= ( 4xh + 2h^2 ) / h = ( h ( 4x + 2h ) ) / h = 4x + 2h
* If you do these correctly,
this h will reduce out at the end!

YOUR TURN:

Given f( x ) = 5 - 3x
find the difference quotient.

Given f( x ) = 8 - 3x^2
find the difference quotient.