Hi, there--

The Problem: 
Find and simplify the difference quotient of f,
f(x) = x^2+ 8x + 5 
F(x+h)-F(x)
 all over h

A Solution:
Difference quotients can seem VERY tedious. In addition, it is easy to make a computation 
error when you simplify them. When I first learned about difference quotients, my teacher 
didn't tell me why they were such a big deal. I now know that they are helpful later in 
understanding and calculating derivatives in calculus. (I personally don't mind doing tedious 
math zoo much if I know it will be helpful later.) Anyway...

f(x) = x^2 + 8x + 5
f(x+h) = (x+h)^2 + 8(x+h) + 5

[f(x+h) - f(x)]/h = {[(x+h)^2 + 8(x+h) + 5] - [x^2 + 8x + 5]}/h
 = {[x^2 + 2xh + h^2 +8x + 8h + 5] - [x^2 + 8x + 5]}/h
 = {x^2 + 2xh + h^2 +8x + 8h + 5 - x^2 - 8x - 5]}/h
 = {2xh + h^2 + 8h]}/h
 = 2x + h + 8

If this is unclear, or you still have questions, feel free to email.

Mrs.Figgy
math.in.the.vortex@gmail.com