To express [tex]\(\frac{f(x+h) - f(x)}{h}\)[/tex] in its simplest form for the function [tex]\(f(x) = 2x^2 - x + 1\)[/tex], we can follow these steps:
1. Determine [tex]\(f(x+h)\)[/tex]:
Given [tex]\(f(x) = 2x^2 - x + 1\)[/tex], we need to find [tex]\(f(x+h)\)[/tex].
[tex]\[
f(x+h) = 2(x+h)^2 - (x+h) + 1.
\][/tex]
2. Expand [tex]\(f(x+h)\)[/tex]:
[tex]\[
f(x+h) = 2(x^2 + 2xh + h^2) - x - h + 1.
\][/tex]
[tex]\[
f(x+h) = 2x^2 + 4xh + 2h^2 - x - h + 1.
\][/tex]
3. Compute [tex]\(f(x+h) - f(x)\)[/tex]:
Subtract [tex]\(f(x)\)[/tex] from [tex]\(f(x+h)\)[/tex]:
[tex]\[
f(x+h) - f(x) = (2x^2 + 4xh + 2h^2 - x - h + 1) - (2x^2 - x + 1).
\][/tex]
Simplify the expression:
[tex]\[
f(x+h) - f(x) = 4xh + 2h^2 - h.
\][/tex]
4. Form [tex]\(\frac{f(x+h) - f(x)}{h}\)[/tex]:
[tex]\[
\frac{f(x+h) - f(x)}{h} = \frac{4xh + 2h^2 - h}{h}.
\][/tex]
5. Simplify by dividing each term by [tex]\(h\)[/tex]:
[tex]\[
\frac{4xh + 2h^2 - h}{h} = \frac{4xh}{h} + \frac{2h^2}{h} - \frac{h}{h}.
\][/tex]
[tex]\[
\frac{4xh}{h} + \frac{2h^2}{h} - \frac{h}{h} = 4x + 2h - 1.
\][/tex]
Hence, the simplified form of [tex]\(\frac{f(x+h) - f(x)}{h}\)[/tex] is:
[tex]\[
\boxed{2h + 4x - 1}.
\][/tex]
