The question is asking to construct and simplify the difference quotient for the function $f(x) = x^2 - 2$. The difference quotient is given by:

$\frac{f(x+h) - f(x)}{h}$

Step-by-step solution:

1. Substitute $f(x) = x^2 - 2$ into the difference quotient:

$f(x+h) = (x+h)^2 - 2$ Simplify $f(x+h)$: $(x+h)^2 - 2 = x^2 + 2xh + h^2 - 2$

2. Now, compute $f(x+h) - f(x)$: $f(x+h) - f(x) = (x^2 + 2xh + h^2 - 2) - (x^2 - 2)$ Simplify: $f(x+h) - f(x) = x^2 + 2xh + h^2 - 2 - x^2 + 2 = 2xh + h^2$

3. Divide by $h$: $\frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2}{h}$ Simplify: $\frac{f(x+h) - f(x)}{h} = 2x + h$

Final Answer:

The simplified difference quotient is:

$2x + h$

Would you like further details or explanations on any part of this process?

Here are some related questions for practice:

1. What is the difference quotient for $f(x) = 3x^2 - 4x + 1$?
2. How does the difference quotient relate to the derivative of a function?
3. What happens to the difference quotient as $h$ approaches 0?
4. Can the difference quotient be used to find the slope of a secant line? Why?
5. What are some other applications of the difference quotient in calculus?

Tip: The difference quotient helps estimate the slope of the tangent line to a curve at any point and is crucial in finding derivatives.