Find the average rate of change of the given function on the interval
[ 0 , 3 ] . f ( x )

The average rate of change of a function $f(x)$ on the interval $[a, b]$ is given by the formula:

$\frac{f(b) - f(a)}{b - a}.$

Given:

• Function: $f(x) = x^3 - x^2 - 2x - 4$
• Interval: $[0, 3]$

We need to find $f(0)$ and $f(3)$ to apply the formula.

1. Calculate $f(0)$:

$f(0) = 0^3 - 0^2 - 2 \times 0 - 4 = -4.$

2. Calculate $f(3)$:

$f(3) = 3^3 - 3^2 - 2 \times 3 - 4 = 27 - 9 - 6 - 4 = 8.$

3. Apply the average rate of change formula:

$\frac{f(3) - f(0)}{3 - 0} = \frac{8 - (-4)}{3 - 0} = \frac{8 + 4}{3} = \frac{12}{3} = 4.$

So, the average rate of change of the function $f(x) = x^3 - x^2 - 2x - 4$ on the interval $[0, 3]$ is 4.

Would you like more details or have any questions?

Here are 5 related questions to consider:

1. What is the physical interpretation of the average rate of change in a real-world context?
2. How does the average rate of change differ from the instantaneous rate of change?
3. How can the average rate of change be used to estimate function behavior over an interval?
4. What would the average rate of change be for different intervals of the function $f(x) = x^3 - x^2 - 2x - 4$?
5. How can you use the derivative to find the instantaneous rate of change at a point?

Tip: The average rate of change of a function over an interval is the slope of the secant line connecting the points on the graph corresponding to the interval's endpoints.