Expert: Paul Klarreich - 9/10/2009

Question
Hi. All my attempts to solve this problem have failed so I hope you can assist me.

If f(x) = x^3, evaluate the difference quotient  (f(2+h)- f(2))/h and simplify your answer

Thank You

Answer
Questioner: Ntokozo
Country: Canada
Category: Calculus
Private: No
Subject: Calculus Functions
Question: Hi. All my attempts to solve this problem have failed so I hope you can assist me.

If f(x) = x^3, evaluate the difference quotient  (f(2+h)- f(2))/h and simplify your answer

Thank You
.........................................
Here's the key to success in this stuff.  See those things above the '9' and '0' on the keyboard?  PARENTHESES!  Love them.

You have four things to do:
  Substitute
  Substitute
  Subtract
  Divide
(do whatever simplification is right at each step.)

Substitute:

Always do it with parentheses.
Always do it with parentheses.
Always do it with parentheses.
Always do it with parentheses.
Always do it with parentheses.
Always do it with parentheses.
Always do it with parentheses.
Always do it with parentheses.
Always do it with parentheses.
Always do it with parentheses.
Always do it with parentheses.
Always do it with parentheses.
Always do it with parentheses.
Always do it with parentheses.

Yes, I mean it.

f() = ()^3
f(2 + h) = (2 + h)^3
Now remember your algebra. (Binomial expansion.)

f(2 + h) = 8 + 12h + 6h^2 + h^3


Substitute:

f(2) = (2)^3
f(2) = 8

Subtract.

f(2 + h) - f(2) = 8 + 12h + 6h^2 + h^3 - 8

f(2 + h) - f(2) = 12h + 6h^2 + h^3

Divide.

f(2 + h) - f(2)   12h + 6h^2 + h^3
--------------- = -----------------
     h                  h  

f(2 + h) - f(2)   
--------------- = 12 + 6h + h^2
     h

End.