SOLUTION: Construct and Simplify the difference quotient {{{ (f(x+h)-f(x))/h }}} for the function given by: {{{ f(x)=2x^2-x }}} I am really having trouble with this type of problem

Algebra ->  Rational-functions -> SOLUTION: Construct and Simplify the difference quotient {{{ (f(x+h)-f(x))/h }}} for the function given by: {{{ f(x)=2x^2-x }}} I am really having trouble with this type of problem      Log On


   

Question 287735: Construct and Simplify the difference quotient +%28f%28x%2Bh%29-f%28x%29%29%2Fh+
for the function given by: +f%28x%29=2x%5E2-x+
I am really having trouble with this type of problem
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
Construct and Simplify the difference quotient +%28f%28x%2Bh%29-f%28x%29%29%2Fh+
for the function given by: +f%28x%29=2x%5E2-x+
I am really having trouble with this type of problem.

Let's do it piece by piece. 

+%28highlight%28f%28x%2Bh%29%29-f%28x%29%29%2Fh+

Let's first find highlight%28%22f%28x%2Bh%29%22%29%29 using

+f%28red%28x%29%29=2red%28x%29%5E2-red%28x%29+

In place of those red red%28x%29's, let's put red%28%28x%2Bh%29%29

+f%28red%28x%2Bh%29%29=2red%28%28x%2Bh%29%29%5E2-red%28%28x%2Bh%29%29+

Simplify the right side:

+f%28x%2Bh%29=2%28x%2Bh%29%5E2-%28x%2Bh%29+
+f%28x%2Bh%29=2%28x%2Bh%29%28x%2Bh%29-%28x%2Bh%29+
+f%28x%2Bh%29=2%28x%5E2%2B2hx%2Bh%5E2%29-x-h+
+f%28x%2Bh%29=2x%5E2%2B4hx%2B2h%5E2-x-h+
+f%28x%2Bh%29=2x%5E2%2B4hx%2B2h%5E2-x-h+

Replace the f%28x%2Bh%29 in

+%28highlight%28f%28x%2Bh%29%29-f%28x%29%29%2Fh+

by highlight%282x%5E2%2B4hx%2B2h%5E2-x-h%29+

+%28highlight%282x%5E2%2B4hx%2B2h%5E2-x-h%29-f%28x%29%29%2Fh+

Next we need to replace replace the highlight%28f%28x%29%29 in 

+%282x%5E2%2B4hx%2B2h%5E2-x-h-highlight%28f%28x%29%29+%29%2Fh+

by what we are given for %22f%28x%29%22, namely f%28x%29=highlight%28%282x%5E2-x%29+%29+

+%282x%5E2%2B4hx%2B2h%5E2-x-h-highlight%28%282x%5E2-x%29%29+%29%2Fh+

Now we simplify:

+%282x%5E2%2B4hx%2B2h%5E2-x-h-%282x%5E2-x%29+%29%2Fh+

+%282x%5E2%2B4hx%2B2h%5E2-x-h-2x%5E2%2Bx+%29%2Fh+

Cancel the 2x%5E2 and the -2x%5E2

+%28cross%282x%5E2%29%2B4hx%2B2h%5E2-x-h-cross%282x%5E2%29%2Bx+%29%2Fh+

+%284hx%2B2h%5E2-x-h%2Bx+%29%2Fh+

Cancel the -x and the %22%22%2Bx

+%284hx%2B2h%5E2-cross%28x%29-h%2Bcross%28x%29+%29%2Fh+

+%284hx%2B2h%5E2-h+%29%2Fh+

Factor h out of the numerator:

+%28h%284x%2B2h-1%29+%29%2Fh+

Cancel the h's

+%28cross%28h%29%284x%2B2h-1%29+%29%2Fcross%28h%29+

4x%2B2h-1

That's it!

Edwin