- Identify the coordinates \((2, 1.26)\) and \((7, 1.91)\) from the provided table for the interval \([2, 7]\).
- Apply the average rate of change formula: \(\frac{f(x_2) - f(x_1)}{x_2 - x_1}\).
- Calculate the difference in y-values (\(0.65\)) and x-values (\(5\)).
- Divide the differences to find the average rate of change, which is \(\boxed{0.13}\).
Explanation
- Problem Analysis and Data Extraction
To determine the average rate of change of the function \(y = \sqrt[3]{x}\) over the interval \([2, 7]\), we first extract the necessary values from the provided table. For \(x_1 = 2\), the corresponding function value is \(y_1 = 1.26\). For \(x_2 = 7\), the corresponding function value is \(y_2 = 1.91\). These points define the endpoints of the interval for which we want to find the rate of change.
- Objective and Formula Application
The average rate of change of a function \(f(x)\) over an interval \([x_1, x_2]\) is defined as the slope of the secant line connecting the points \((x_1, f(x_1))\) and \((x_2, f(x_2))\). The formula is given by:
\[\text{Average Rate of Change} = \frac{\Delta y}{\Delta x} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}\]
We will use this formula with our extracted values: \(x_1 = 2, y_1 = 1.26\) and \(x_2 = 7, y_2 = 1.91\).
- Calculation of the Average Rate of Change
Substitute the identified values into the average rate of change formula:
\[\frac{\Delta y}{\Delta x} = \frac{1.91 - 1.26}{7 - 2}\]
First, calculate the change in the output values (the numerator):
\[1.91 - 1.26 = 0.65\]
Next, calculate the change in the input values (the denominator):
\[7 - 2 = 5\]
Now, divide the two results:
\[\frac{0.65}{5} = 0.13\]
The average rate of change over the interval \([2, 7]\) is \(0.13\).
- Final Answer Selection
Comparing the calculated value of \(0.13\) with the given options:
A\(0.35\)
B\(0.13\)
C\(0.65\)
D\(1.62\)
The correct option is b. The final answer for the average rate of change is \(\boxed{0.13}\).
Examples
The concept of the average rate of change is essential in many fields, such as economics, where it is used to measure the average change in the price of a commodity or demand over a specific period. For instance, if a company wants to understand how its sales grew over the last five months, it can use the average rate of change to quantify the growth per month, allowing for better forecasting and strategic planning. By observing the values at the beginning and the end of the interval, the company can effectively determine the average trend despite fluctuations during the intermediate months.