Find and simplify.
f(x) = 8 − x2
(a)
f(x + h) − f(x)

(b)
f(x + h) − f(x)
h

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 \( f(x + h) \):**
   \[
   f(x + h) = 8 - (x + h)^2
   = 8 - (x^2 + 2xh + h^2)
   = 8 - x^2 - 2xh - h^2
   \]

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:
   \[
   f(x + h) - f(x) = 8 - x^2 - 2xh - h^2 - 8 + x^2
   = -2xh - h^2
   \]

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

---

### (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:
\[
\frac{f(x + h) - f(x)}{h} = -2x - h
\]

---

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.

Find and simplify. f(x) = 8 − x^2 (a) f(x + h) − f(x) (b) f(x + h) − f(x) / h

This problem demonstrates how to simplify the difference quotient for the function f(x) = 8 − x^2. It covers algebraic concepts like polynomials and the difference quotient, providing a step-by-step solution suitable for students in Grades 10-12.

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