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

Problem:

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