Let's solve the given expressions step by step for $f(x) = 8 - x^2$.

(a) $f(x + h) - f(x)$

We start by calculating $f(x + h)$ and subtracting $f(x)$.

1. Find $f(x + h)$: = 8 - (x^2 + 2xh + h^2) = 8 - x^2 - 2xh - h^2$$
2. Now, subtract $f(x)$ from $f(x + h)$: $f(x + h) - f(x) = \left(8 - x^2 - 2xh - h^2\right) - \left(8 - x^2\right)$ Simplifying: = -2xh - h^2$$

Thus, the simplified expression for part (a) is: $f(x + h) - f(x) = -2xh - h^2$

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

Using the result from part (a):

$\frac{f(x + h) - f(x)}{h} = \frac{-2xh - h^2}{h}$

Simplify the fraction: $\frac{-2xh - h^2}{h} = -2x - h$

Thus, the simplified expression for part (b) is: $\frac{f(x + h) - f(x)}{h} = -2x - h$

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Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. What is $\frac{f(x+h) - f(x)}{h}$ as $h \to 0$?
2. How do you interpret the result of part (b) in terms of the derivative?
3. Find the derivative of $f(x) = 8 - x^2$ using first principles.
4. If $f(x) = x^3$, what is $\frac{f(x+h) - f(x)}{h}$?
5. How does the difference quotient relate to the slope of a tangent line?

Tip: The expression $\frac{f(x + h) - f(x)}{h}$ is called a difference quotient, and it is fundamental for understanding the derivative of a function.