Certainly! Let's solve this step-by-step.
Given the function [tex]\( f(x) = x^3 - 2x \)[/tex], we are asked to find the value of the difference quotient [tex]\(\frac{f(h+2) - f(2)}{h}\)[/tex].
First, we need to compute the value of [tex]\( f(h+2) \)[/tex] and [tex]\( f(2) \)[/tex].
### Step 1: Calculate [tex]\( f(h+2) \)[/tex]
To find [tex]\( f(h+2) \)[/tex], we substitute [tex]\( x = h+2 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(h+2) = (h+2)^3 - 2(h+2)
\][/tex]
We expand and simplify the expression:
[tex]\[
(h+2)^3 = (h+2)(h+2)(h+2)
\][/tex]
Expanding the product:
[tex]\[
(h+2)(h+2) = h^2 + 4h + 4
\][/tex]
[tex]\[
(h^2 + 4h + 4)(h+2) = h^3 + 2h^2 + 4h^2 + 8h + 4h + 8 = h^3 + 6h^2 + 12h + 8
\][/tex]
Thus:
[tex]\[
(h+2)^3 = h^3 + 6h^2 + 12h + 8
\][/tex]
Next, compute the linear term:
[tex]\[
- 2(h + 2) = -2h - 4
\][/tex]
Combining these:
[tex]\[
f(h+2) = (h^3 + 6h^2 + 12h + 8) - 2(h + 2)
\][/tex]
[tex]\[
f(h+2) = h^3 + 6h^2 + 12h + 8 - 2h - 4
\][/tex]
Simplifying:
[tex]\[
f(h+2) = h^3 + 6h^2 + 10h + 4
\][/tex]
### Step 2: Calculate [tex]\( f(2) \)[/tex]
To find [tex]\( f(2) \)[/tex], we substitute [tex]\( x = 2 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(2) = 2^3 - 2 \cdot 2
\][/tex]
[tex]\[
f(2) = 8 - 4
\][/tex]
[tex]\[
f(2) = 4
\][/tex]
### Step 3: Calculate the difference quotient
We have:
[tex]\[
f(h+2) = h^3 + 6h^2 + 10h + 4
\][/tex]
[tex]\[
f(2) = 4
\][/tex]
Now, we calculate the difference quotient:
[tex]\[
\frac{f(h+2) - f(2)}{h} = \frac{(h^3 + 6h^2 + 10h + 4) - 4}{h}
\][/tex]
Simplify the numerator:
[tex]\[
(h^3 + 6h^2 + 10h + 4) - 4 = h^3 + 6h^2 + 10h
\][/tex]
Thus:
[tex]\[
\frac{f(h+2) - f(2)}{h} = \frac{h^3 + 6h^2 + 10h}{h}
\][/tex]
Factor out [tex]\( h \)[/tex] in the numerator:
[tex]\[
\frac{h(h^2 + 6h + 10)}{h}
\][/tex]
Cancel [tex]\( h \)[/tex]:
[tex]\[
h^2 + 6h + 10
\][/tex]
Therefore, the value of [tex]\(\frac{f(h+2) - f(2)}{h}\)[/tex] is:
[tex]\[
\boxed{h^2 + 6h + 10}
\][/tex]
