The problem asks for the average rate of change of the function $f(x) = 3x$ from $x_1 = 0$ to $x_2 = 4$.

Solution

The average rate of change of a function $f(x)$ over an interval $[x_1, x_2]$ is given by the formula:

$\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

Step 1: Compute $f(x_1)$ and $f(x_2)$

• Given $f(x) = 3x$, find $f(0)$ and $f(4)$:

  $f(0) = 3(0) = 0$

  $f(4) = 3(4) = 12$

Step 2: Apply the Formula

$\text{Average Rate of Change} = \frac{f(4) - f(0)}{4 - 0} = \frac{12 - 0}{4} = \frac{12}{4} = 3$

Thus, the average rate of change is $\mathbf{3}$.

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Related Questions:

1. What is the geometric interpretation of the average rate of change?
2. How does the average rate of change relate to the derivative?
3. What happens if the function was quadratic instead of linear?
4. How can we interpret a negative average rate of change?
5. What are some real-world applications of the average rate of change?

Tip:

For a linear function like $f(x) = 3x$, the average rate of change over any interval is always the same as the slope of the function.