Math Problem Statement
Given the function g, of, x, equals, minus, x, squared, plus, 3, x, plus, 5g(x)=−x 2 +3x+5, determine the average rate of change of the function over the interval minus, 4, is less than or equal to, x, is less than or equal to, 6−4≤x≤6.
Solution
The average rate of change of a function over an interval is given by the formula:
In this case, the function is and the interval is .
Step 1: Calculate
Step 2: Calculate
Step 3: Apply the formula for the average rate of change
So, the average rate of change of the function over the interval is 1.
Would you like further details on any part of this? Here are some related questions:
- How does the average rate of change relate to the slope of a secant line?
- How would the average rate of change differ over a smaller interval?
- Can the rate of change be negative, and in what cases?
- What does the sign of the average rate of change tell us about the function?
- How would we compute instantaneous rate of change at a point?
Tip: The average rate of change formula is similar to the slope formula for a straight line, so it gives an approximation of how the function behaves between two points.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Functions
Rate of Change
Formulas
Average Rate of Change = (g(b) - g(a)) / (b - a)
Quadratic function: g(x) = ax^2 + bx + c
Theorems
Secant Line Slope Theorem
Suitable Grade Level
Grades 9-10
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