Find the average rate of change of of f(x)=x^{3}-3 x^{2}+2
NOVEMBER 30, 2023
Problem:
Find the average rate of change of from to . 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 from to , we will follow a systematic approach.
Answer
The average rate of change of from to is ,which is already a simplified form.
Method: Hints
To find the average rate of change, we will:
- Evaluate the function at the starting point .
- Evaluate the function at the ending point .
- Subtract the function value at the starting point from the function value at the ending point.
- Subtract the starting point -value from the ending point -value.
- Divide the difference in function values by the difference in -values.
Step-by-Step Calculations
Evaluate :
Evaluate :
Find the difference in function values:
Find the difference in -values:
Divide the difference in function values by the difference in -values:
Verification
Double-checking the calculations:
- is correct.
- is correct.
- The difference in function values is .
- The difference in -values is .
- The division yields .
All steps are correct, and the average rate of change is indeed .
Related Knowledge Points
- The average rate of change is analogous to the slope of the secant line that passes through the points and 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 is a cubic polynomial, and its average rate of change from to tells us how quickly the function values are increasing or decreasing on average over this interval. The result of indicates that, on average, for each unit increase in , the function value increases by units over the interval from to .