To find the difference quotient for the given function [tex]\( f(x) = x^2 - 2x + 1 \)[/tex], we need to follow these steps:
1. Compute [tex]\( f(4+h) \)[/tex].
2. Compute [tex]\( f(4) \)[/tex].
3. Formulate the difference quotient [tex]\(\frac{f(4+h) - f(4)}{h}\)[/tex].
Let's start with the calculations.
### Step 1: Compute [tex]\( f(4+h) \)[/tex]
Substitute [tex]\( 4+h \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(4+h) = (4+h)^2 - 2(4+h) + 1
\][/tex]
Expand and simplify the expression:
[tex]\[
(4+h)^2 = 16 + 8h + h^2
\][/tex]
[tex]\[
- 2(4+h) = -8 - 2h
\][/tex]
So,
[tex]\[
f(4+h) = 16 + 8h + h^2 - 8 - 2h + 1
\][/tex]
Simplify further:
[tex]\[
f(4+h) = h^2 + 6h + 9
\][/tex]
### Step 2: Compute [tex]\( f(4) \)[/tex]
Substitute [tex]\( 4 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(4) = (4)^2 - 2(4) + 1
\][/tex]
[tex]\[
f(4) = 16 - 8 + 1
\][/tex]
[tex]\[
f(4) = 9
\][/tex]
### Step 3: Formulate the difference quotient
Now, construct the difference quotient:
[tex]\[
\frac{f(4+h) - f(4)}{h}
\][/tex]
Substitute the values calculated in Steps 1 and 2:
[tex]\[
\frac{h^2 + 6h + 9 - 9}{h}
\][/tex]
Simplify the numerator:
[tex]\[
\frac{h^2 + 6h}{h}
\][/tex]
Since [tex]\( h \neq 0 \)[/tex], we can factor out [tex]\( h \)[/tex] from the numerator:
[tex]\[
\frac{h(h + 6)}{h}
\][/tex]
Cancel out [tex]\( h \)[/tex] in the numerator and the denominator:
[tex]\[
h + 6
\][/tex]
Therefore, the simplified difference quotient is:
[tex]\[
\boxed{h + 6}
\][/tex]
