To find the difference quotient for the function [tex]\( f(x) = 5x^2 - 2x + 2 \)[/tex], follow these steps:
Step 1: Compute [tex]\( f(x + h) \)[/tex]
First, substitute [tex]\( x + h \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(x + h) = 5(x + h)^2 - 2(x + h) + 2 \][/tex]
Step 2: Expand [tex]\( f(x + h) \)[/tex]
Expand the squared term and simplify:
[tex]\[
(x + h)^2 = x^2 + 2xh + h^2
\][/tex]
Therefore,
[tex]\[
f(x + h) = 5(x^2 + 2xh + h^2) - 2(x + h) + 2
\][/tex]
[tex]\[
= 5x^2 + 10xh + 5h^2 - 2x - 2h + 2
\][/tex]
Step 3: Compute [tex]\( f(x + h) - f(x) \)[/tex]
Now, subtract [tex]\( f(x) \)[/tex] from [tex]\( f(x + h) \)[/tex]:
[tex]\[
f(x + h) - f(x) = (5x^2 + 10xh + 5h^2 - 2x - 2h + 2) - (5x^2 - 2x + 2)
\][/tex]
When we distribute the subtraction, we get:
[tex]\[
f(x + h) - f(x) = 5x^2 + 10xh + 5h^2 - 2x - 2h + 2 - 5x^2 + 2x - 2
\][/tex]
Combining like terms, the [tex]\( 5x^2 \)[/tex], [tex]\( -2x \)[/tex], and [tex]\( +2 \)[/tex] cancel out, leaving us with:
[tex]\[
f(x + h) - f(x) = 10xh + 5h^2 - 2h
\][/tex]
Step 4: Form the difference quotient
Divide the result by [tex]\( h \)[/tex]:
[tex]\[
\frac{f(x + h) - f(x)}{h} = \frac{10xh + 5h^2 - 2h}{h}
\][/tex]
Step 5: Simplify the difference quotient
Factor out [tex]\( h \)[/tex] from the numerator:
[tex]\[
\frac{10xh + 5h^2 - 2h}{h} = \frac{h(10x + 5h - 2)}{h}
\][/tex]
Since [tex]\( h \neq 0 \)[/tex], we can cancel the [tex]\( h \)[/tex]:
[tex]\[
10x + 5h - 2
\][/tex]
Therefore, the simplified difference quotient is:
[tex]\[
\frac{f(x+h)-f(x)}{h} = 10x + 5h - 2
\][/tex]
