The problem in the image asks to calculate the average rate of change of the function $f(x) = 4x^2 + 8$ over the interval $[-4, 6]$.

Formula for Average Rate of Change

The average rate of change of a function $f(x)$ over an interval $[a, b]$ is given by:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

For this problem:

• $a = -4$
• $b = 6$
• $f(x) = 4x^2 + 8$

Step 1: Compute $f(a)$ and $f(b)$

• $f(-4) = 4(-4)^2 + 8 = 4(16) + 8 = 64 + 8 = 72$
• $f(6) = 4(6)^2 + 8 = 4(36) + 8 = 144 + 8 = 152$

Step 2: Apply the formula

$\text{Average Rate of Change} = \frac{f(6) - f(-4)}{6 - (-4)} = \frac{152 - 72}{6 + 4} = \frac{80}{10} = 8$

So, the average rate of change is 8.

Would you like further details or have any questions?

Related Questions:

1. What is the interpretation of the average rate of change in a real-world context?
2. How do you find the instantaneous rate of change of a function?
3. Can the average rate of change be negative? Under what conditions?
4. How would the calculation change if the function were linear instead of quadratic?
5. What is the difference between the slope of a tangent line and the average rate of change?

Tip:

The average rate of change is conceptually similar to the slope of a line connecting two points on the curve of the function.