Math Problem Statement
Solution
Let's solve this problem step by step.
Given equation:
(a) Average rate of change of over the intervals
The average rate of change of a function over an interval is given by:
Interval 2, 3:
Interval 2, 2.5:
Interval 2, 2.1:
(b) Instantaneous rate of change at
The instantaneous rate of change is the derivative of with respect to . Given , the derivative is:
At :
Final answers:
- (a) Average rates of change:
- (b) Instantaneous rate of change at : 4
Would you like more details or have any questions? Here are five related questions for practice:
- How do you compute the average rate of change for a cubic function over an interval?
- What is the difference between average and instantaneous rates of change?
- How would the instantaneous rate of change change if the function were ?
- What is the meaning of the instantaneous rate of change in real-world scenarios?
- How can you approximate the instantaneous rate of change using average rates?
Tip: The instantaneous rate of change at a point is essentially the slope of the tangent line to the function at that point.
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Math Problem Analysis
Mathematical Concepts
Algebra
Calculus
Rate of Change
Formulas
Average rate of change: (f(x2) - f(x1)) / (x2 - x1)
Instantaneous rate of change: Derivative of the function
Theorems
Derivative of a quadratic function
Suitable Grade Level
Grades 11-12, Early College
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