- Identify the interval \([3, 6]\) and the corresponding function values from the table: \(f(3) = 43\) and \(f(6) = 13\).
- Apply the average rate of change formula: \(\frac{f(6) - f(3)}{6 - 3}\).
- Calculate the difference in values: \(\frac{13 - 43}{3} = \frac{-30}{3}\).
- Simplify to obtain the final result: \(\boxed{-10}\).
Explanation
- Problem Analysis and Identification
The average rate of change of a function \(f(x)\) over an interval \([a, b]\) is defined as the change in the function values divided by the change in the input values, given by the formula \(\frac{f(b) - f(a)}{b - a}\). To determine the average rate of change on the interval \(3 \le x \le 6\), we identify the corresponding values from the provided table: \(a = 3\), \(b = 6\), \(f(3) = 43\), and \(f(6) = 13\).
- Applying the Formula
We substitute the identified values into the average rate of change formula: \(\frac{f(6) - f(3)}{6 - 3}\). Substituting the function values gives \(\frac{13 - 43}{6 - 3}\).
- Calculation and Final Result
Now, we perform the arithmetic calculations:
\[\frac{13 - 43}{6 - 3} = \frac{-30}{3} = -10\]
The average rate of change of the function over the given interval is \(-10\).
Examples
The concept of average rate of change is vital in real-world applications, such as tracking the speed of a vehicle. If a car travels 43 miles in 3 hours and 13 miles in 6 hours from a starting point, calculating the average rate of change \(\frac{13-43}{6-3} = -10\) miles per hour indicates that the vehicle's position is changing at an average of -10 miles per hour over that duration. This helps in understanding trends and performance over specific time intervals.