Math Problem Statement
The table gives the average rates of change of a function over different intervals. On which of the intervals does the function increase the most?
Interval 1: 0≤x≤1 and Average Rate of Change=10 Interval 2: 1≤x≤4 and Average Rate of Change=-5 Interval 3: 4≤x≤8 and Average Rate of Change=2 Interval 4: 8≤x≤10 and Average Rate of Change=6
Solution
The average rate of change (AROC) of a function over an interval gives information about how the function behaves (increasing or decreasing) over that interval.
- If the AROC is positive, the function is increasing.
- If the AROC is negative, the function is decreasing.
- The larger the positive value of the AROC, the greater the increase over that interval.
Examining the intervals:
- Interval 1 (0≤x≤1): AROC = 10 (positive, function increases)
- Interval 2 (1≤x≤4): AROC = -5 (negative, function decreases)
- Interval 3 (4≤x≤8): AROC = 2 (positive, function increases)
- Interval 4 (8≤x≤10): AROC = 6 (positive, function increases)
Among the intervals where the function increases (Interval 1, 3, and 4), Interval 1 has the highest rate of change (AROC = 10). Therefore, the function increases the most on Interval 1 (0≤x≤1).
Would you like further details or have any questions? Here are some related questions for practice:
- What does the average rate of change represent geometrically on a graph?
- How can you determine whether a function is increasing or decreasing from its derivative?
- What would it mean if the AROC were zero in an interval?
- How do you find the instantaneous rate of change at a specific point?
- What is the relationship between average rate of change and slope of a secant line?
Tip: A positive average rate of change indicates an increasing function over the interval, while a negative rate suggests a decrease.
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Math Problem Analysis
Mathematical Concepts
Average Rate of Change
Function Behavior
Intervals
Formulas
AROC = (f(b) - f(a)) / (b - a)
Theorems
Rate of Change Theorem
Suitable Grade Level
Grades 9-11
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