Questions: Determine the average rate of change of the following function between the given values of the variable: f(x)=x^4-x ; x=-2, x=3

Transcript text: Determine the average rate of change of the following function between the given values of the variable: \[ f(x)=x^{4}-x ; \quad x=-2, x=3 \]

Solution

Solution Steps

Step 1: Define the Function

We are given the function \( f(x) = x^4 - x \).

Step 2: Identify the Interval

We need to determine the average rate of change of the function between the points \( x = -2 \) and \( x = 3 \).

Step 3: Calculate the Average Rate of Change

The average rate of change of a function \( f \) over the interval \([x_1, x_2]\) is calculated using the formula: \[ \text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \] Substituting \( x_1 = -2 \) and \( x_2 = 3 \), we find: \[ \text{Average Rate of Change} = \frac{f(3) - f(-2)}{3 - (-2)} \]

Step 4: Evaluate the Function at the Endpoints

Calculating \( f(3) \): \[ f(3) = 3^4 - 3 = 81 - 3 = 78 \] Calculating \( f(-2) \): \[ f(-2) = (-2)^4 - (-2) = 16 + 2 = 18 \]

Step 5: Substitute Values into the Formula

Now substituting the values into the average rate of change formula: \[ \text{Average Rate of Change} = \frac{78 - 18}{3 - (-2)} = \frac{60}{5} = 12.0 \]