To find the difference quotient for the function [tex]\( f(x) = x^2 - 2x + 1 \)[/tex], we need to follow a series of steps. Let's go through them in detail.
### Step 1: Compute [tex]\( f(4) \)[/tex]
First, we substitute [tex]\( x = 4 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(4) = 4^2 - 2 \cdot 4 + 1 = 16 - 8 + 1 = 9
\][/tex]
### Step 2: Compute [tex]\( f(4 + h) \)[/tex]
Next, we substitute [tex]\( x = 4 + h \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(4 + h) = (4 + h)^2 - 2(4 + h) + 1
\][/tex]
Expanding the expression, we get:
[tex]\[
(4 + h)^2 = 16 + 8h + h^2
\][/tex]
[tex]\[
-2(4 + h) = -8 - 2h
\][/tex]
Combining these terms, we obtain:
[tex]\[
f(4 + h) = 16 + 8h + h^2 - 8 - 2h + 1 = h^2 + 6h + 9
\][/tex]
### Step 3: Form the difference quotient [tex]\( \frac{f(4 + h) - f(4)}{h} \)[/tex]
Now, we substitute [tex]\( f(4 + h) \)[/tex] and [tex]\( f(4) \)[/tex] into the difference quotient formula:
[tex]\[
\frac{f(4 + h) - f(4)}{h} = \frac{(h^2 + 6h + 9) - 9}{h}
\][/tex]
Simplify the expression in the numerator:
[tex]\[
\frac{h^2 + 6h + 9 - 9}{h} = \frac{h^2 + 6h}{h}
\][/tex]
Factor out [tex]\( h \)[/tex] in the numerator:
[tex]\[
\frac{h(h + 6)}{h}
\][/tex]
Since [tex]\( h \neq 0 \)[/tex], we can cancel [tex]\( h \)[/tex] in the numerator and the denominator:
[tex]\[
h + 6
\][/tex]
So, the simplified form of the difference quotient is:
[tex]\[
\frac{f(4 + h) - f(4)}{h} = h + 6
\][/tex]
Conclusively, the simplified form of the difference quotient for the function [tex]\( f(x) = x^2 - 2x + 1 \)[/tex] at [tex]\( x = 4 \)[/tex] is:
[tex]\[
h + 6
\][/tex]
