To find the average rate of change of the function $k(x) = \frac{-19}{x}$ over the interval $[-18, -14]$, we use the formula:

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

Where $a = -18$ and $b = -14$.

1. First, calculate $k(a) = k(-18)$:

$k(-18) = \frac{-19}{-18} = \frac{19}{18}$

2. Then, calculate $k(b) = k(-14)$:

$k(-14) = \frac{-19}{-14} = \frac{19}{14}$

3. Now, apply the average rate of change formula:

$\text{Average Rate of Change} = \frac{\frac{19}{14} - \frac{19}{18}}{-14 - (-18)}$

4. Simplify the expression:

$\text{Average Rate of Change} = \frac{\frac{19}{14} - \frac{19}{18}}{4}$

5. Find a common denominator to subtract the fractions:

$\frac{19}{14} - \frac{19}{18} = 19 \left(\frac{18 - 14}{14 \times 18}\right) = 19 \left(\frac{4}{252}\right) = \frac{76}{252} = \frac{19}{63}$

6. So, the average rate of change is:

$\text{Average Rate of Change} = \frac{\frac{19}{63}}{4} = \frac{19}{252}$

Thus, the average rate of change of $k(x)$ over the interval $[-18, -14]$ is $\frac{19}{252}$.

Would you like further details or have any questions?

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Here are some related questions you might find interesting:

1. How would you interpret the average rate of change geometrically?
2. What is the significance of the average rate of change in real-world applications?
3. How does the average rate of change relate to the derivative of the function?
4. Can the average rate of change be negative, and what does that imply?
5. How would the average rate of change differ over a different interval for the same function?

Tip: The average rate of change can be thought of as the slope of the secant line connecting two points on the curve of the function.