Math Problem Statement
Find the average rate of change of the function over the given intervals. f(x)equals7 x cubed plus 7; a) [6,8], b) [negative 4,4] Question content area bottom Part 1 a) The average rate of change of the function f(x)equals7 x cubed plus 7 over the interval [6,8] is enter your response here. (Simplify your answer.)
Solution
To find the average rate of change of the function over an interval , we use the formula for the average rate of change:
Part a) Interval :
- Step 1: Evaluate and
- Step 2: Apply the formula for the average rate of change
Thus, the average rate of change over the interval is 1036.
Would you like the details for part b or more questions on related topics? Here are some you might consider:
- What is the geometric meaning of the average rate of change of a function?
- How does the average rate of change relate to the derivative of a function?
- Can the average rate of change be negative? If so, under what conditions?
- How does the average rate of change vary for quadratic or linear functions?
- What is the average rate of change of a constant function?
Tip: The average rate of change over an interval is essentially the slope of the secant line between two points on the curve.
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Math Problem Analysis
Mathematical Concepts
Calculus
Average Rate of Change
Polynomial Functions
Formulas
Average rate of change formula: (f(b) - f(a)) / (b - a)
Theorems
Secant Line Slope
Polynomial Function Properties
Suitable Grade Level
Grades 10-12
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