Math Problem Statement
Solution
To find the average rate of change of the function on the interval , we use the formula for the average rate of change:
where and .
Step 1: Determine and
From the graph:
- (this is the y-value corresponding to ),
- (this is the y-value corresponding to ).
Step 2: Apply the formula
Now, substitute these values into the formula:
Conclusion:
The average rate of change of on the interval is .
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Relative Questions:
- How do you interpret the average rate of change of a function geometrically?
- What does a positive average rate of change imply about the function over the given interval?
- How is the average rate of change related to the slope of a secant line?
- Can the average rate of change be negative? Under what conditions?
- How does the concept of the derivative relate to the average rate of change?
Tip:
The average rate of change between two points on a function is essentially the slope of the line connecting those two points (the secant line).
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Math Problem Analysis
Mathematical Concepts
Calculus
Average Rate of Change
Functions
Formulas
Average Rate of Change = (f(b) - f(a)) / (b - a)
Theorems
Secant Line Slope Theorem
Suitable Grade Level
Grades 10-12
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