Find the average rate of change of of f(x)=x^{3}-3 x^{2}+2

Problem:

Find the average rate of change of $f(x)=x^{3}-3 x^{2}+2 x$ from $x=-3$ to $x=2$. Simplify your answer as much as possible.

Text explaination：

The average rate of change of a function over an interval is a measure of how much the function's value changes on average between two points. To find the average rate of change of the function $f(x)=x^{3}-3 x^{2}+2 x$ from $x=-3$ to $x=2$, we will follow a systematic approach.

Answer

The average rate of change of $f(x)$ from $x=-3$ to $x=2$ is $12$，which is already a simplified form.

Method: Hints

To find the average rate of change, we will:

1. Evaluate the function at the starting point $x=-3$.
2. Evaluate the function at the ending point $x=2$.
3. Subtract the function value at the starting point from the function value at the ending point.
4. Subtract the starting point $x$-value from the ending point $x$-value.
5. Divide the difference in function values by the difference in $x$-values.

Step-by-Step Calculations

1. Evaluate $f(-3)$: $ f(-3) = (-3)^3 - 3(-3)^2 + 2(-3) = -27 - 27 - 6 = -60. $

2. Evaluate $f(2)$: $ f(2) = (2)^3 - 3(2)^2 + 2(2) = 8 - 12 + 4 = 0. $

3. Find the difference in function values: $ f(2) - f(-3) = 0 - (-60) = 60. $

4. Find the difference in $x$-values: $ 2 - (-3) = 5. $

5. Divide the difference in function values by the difference in $x$-values: $ \frac{f(2) - f(-3)}{2 - (-3)} = \frac{60}{5} = 12. $

Verification

Double-checking the calculations:

• $f(-3) = -60$ is correct.
• $f(2) = 0$ is correct.
• The difference in function values is $60$.
• The difference in $x$-values is $5$.
• The division yields $12$.

All steps are correct, and the average rate of change is indeed $12$.

Related Knowledge Points

• The average rate of change is analogous to the slope of the secant line that passes through the points $(a, f(a))$ and $(b, f(b))$ on the graph of the function.
• This concept is a precursor to the derivative, which measures the instantaneous rate of change at a single point.

Detailed Explanation

The average rate of change of a function between two points gives us an overall idea of the function's behavior over that interval. It is a useful measure when we want to understand the general trend of a function without looking at its specific instantaneous changes. In this case, the function $f(x)$ is a cubic polynomial, and its average rate of change from $x=-3$ to $x=2$ tells us how quickly the function values are increasing or decreasing on average over this interval. The result of $12$ indicates that, on average, for each unit increase in $x$, the function value increases by $12$ units over the interval from $x=-3$ to $x=2$.