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The Difference Quotient

( f( x + h ) - f( x ) ) / h

You probably saw this semi-obnoxious thing in Algebra...  And I know you saw it in Precalculus.  Go back and look at the Slopes of Lines lesson again...  This thing is just the slope of a line through the points ( x, f(x) ) and ( x + h, f(x + h) ).  It's going to be used in the most important Calculus theorems, so you really need to get comfortable with it.  I know right where students get messed up on these, so let's back up a bit and start with some basics.  We'll build up from there.

You're probably fine with the easy stuff:

Givenf( x ) = x^2 + 5

f( 3 ) = ( 3 )^2 + 5 = 9 + 5 = 14

f( -2 ) = ( -2 )^2 + 5 = 4 + 5 = 9
 

* Remember that parentheses (  ) are REALLY important on these!
 

f( k ) = k^2 + 5

 


TRY IT:

Same function:f( x ) = x^2 + 5
findf( 7 )
findf( -3 )
findf( booger )

 

Ha ha...  I made you write "booger."  I'm so mature.

Now dof( blob )

(This is important!  Use those parentheses!)