To solve this, we need to find and simplify the difference quotient for the given function $f(x) = 8x^2$. The difference quotient formula is:

$\frac{f(x+h) - f(x)}{h}, \quad h \neq 0$

Step-by-step solution:

1. Substitute $f(x) = 8x^2$ into $f(x + h)$: $f(x + h) = 8(x + h)^2$ Expand $(x + h)^2$: $(x + h)^2 = x^2 + 2xh + h^2$ So: $f(x + h) = 8(x^2 + 2xh + h^2) = 8x^2 + 16xh + 8h^2$

2. Find $f(x + h) - f(x)$: $f(x + h) - f(x) = \big(8x^2 + 16xh + 8h^2\big) - 8x^2$ Simplify: $f(x + h) - f(x) = 16xh + 8h^2$

3. Divide by $h$: $\frac{f(x + h) - f(x)}{h} = \frac{16xh + 8h^2}{h}$ Simplify the fraction: $\frac{16xh + 8h^2}{h} = 16x + 8h$

Final Answer:

$\frac{f(x + h) - f(x)}{h} = 16x + 8h, \quad h \neq 0$

Let me know if you'd like further clarification or additional details!

Questions to Explore:

1. How is the difference quotient related to derivatives in calculus?
2. What happens as $h \to 0$ in the difference quotient?
3. How does this problem change if $f(x)$ were a cubic function instead of quadratic?
4. What is the geometric interpretation of the difference quotient?
5. Can the formula for the difference quotient be applied to functions with fractions?

Tip:

The difference quotient is the foundation of derivative computation. Practice simplifying it step-by-step to build a strong understanding of limits and calculus!