- Define the average rate of change formula: \(\frac{f(x_2) - f(x_1)}{x_2 - x_1}\).
- Identify the function values from the table: \(f(18) = 40\) and \(f(24) = 39\).
- Substitute the values into the formula: \(\frac{39 - 40}{24 - 18}\).
- Calculate the result to find the average rate of change, which is \(\boxed{-\frac{1}{6}}\).
Explanation
- Understanding Average Rate of Change
The average rate of change of a function \(f(x)\) over an interval \([x_1, x_2]\) measures the average amount the function's value changes per unit of \(x\) as \(x\) increases from \(x_1\) to \(x_2\). This is represented by the slope of the secant line passing through the points \((x_1, f(x_1))\) and \((x_2, f(x_2))\) on the graph of the function. The formula to calculate this is:
\[\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}\]
In this problem, we are looking for the average rate of change over the interval \([18, 24]\).
- Identifying Data Points
Based on the provided table, we identify the values of the function at the endpoints of the interval \([18, 24]\):
- For \(x_1 = 18\), the value is \(f(18) = 40\).
- For \(x_2 = 24\), the value is \(f(24) = 39\).
These two points, \((18, 40)\) and \((24, 39)\), represent the endpoints on the function's curve for the given interval.
- Calculating the Average Rate of Change
Now we substitute these values into the average rate of change formula:
\[\text{Average Rate of Change} = \frac{f(24) - f(18)}{24 - 18}\]
Substituting the known values:
\[\text{Average Rate of Change} = \frac{39 - 40}{24 - 18}\]
Performing the subtractions in the numerator and the denominator:
\[\text{Numerator} = 39 - 40 = -1\]
\[\text{Denominator} = 24 - 18 = 6\]
Thus, the average rate of change is:
\[\frac{-1}{6}\]
- Final Answer
The average rate of change of the function \(f(x)\) over the interval \([18, 24]\) is \(\boxed{-\frac{1}{6}}\).
Examples
Average rate of change is essential in many fields. For example, if you are tracking the speed of a car, the average rate of change of distance with respect to time represents the car's average velocity over a specific period. By understanding this, you can determine how quickly an object moves between two points, which is a fundamental concept in both physics and daily navigation.