The problem asks to find the average rate of change of the function $k(x) = 19x^2$ over the interval $[0, 2]$.

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

$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$

Here, $f(x) = 19x^2$, $a = 0$, and $b = 2$.

Let's compute it step-by-step:

1. Evaluate $f(2)$: $f(2) = 19(2^2) = 19 \times 4 = 76$

2. Evaluate $f(0)$: $f(0) = 19(0^2) = 19 \times 0 = 0$

3. Apply the formula: $\text{Average rate of change} = \frac{f(2) - f(0)}{2 - 0} = \frac{76 - 0}{2 - 0} = \frac{76}{2} = 38$

Thus, the average rate of change is 38, which corresponds to the third option in the list.

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the geometric interpretation of the average rate of change?
2. How is the average rate of change related to the derivative?
3. What is the rate of change of the function $f(x) = 19x^2$ at $x = 1$?
4. How would the average rate of change change if the interval was $[1, 3]$?
5. How can we calculate the instantaneous rate of change for this function?

Tip: The average rate of change over an interval is the slope of the secant line connecting two points on the function over that interval.