Math Problem Statement
Solution
Let's work through the problem you've uploaded.
Given that:
Part (a) - Finding the average rate of change
The average rate of change of a function over an interval is calculated as:
1. From to :
We need to compute and :
The average rate of change is:
2. From to :
We compute :
The average rate of change is:
3. From to :
We compute :
The average rate of change is:
Part (b) - Instantaneous rate of change at
The instantaneous rate of change is given by the derivative .
First, we compute the derivative of :
At , we get:
Final Answers:
- (a) Average rates of change:
- From to :
- From to :
- From to :
- (b) Instantaneous rate of change at :
Would you like further clarification or details on any part?
Here are 5 related questions to expand on:
- How does the average rate of change differ from the instantaneous rate of change?
- Can the derivative give us information about concavity of the function?
- How would the average rate of change change for other intervals around ?
- What happens to the rate of change as approaches 2 from either side?
- How can we visualize the derivative graphically?
Tip: The average rate of change between two points on a curve is the slope of the secant line connecting them. The instantaneous rate of change at a point is the slope of the tangent line at that point.
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Math Problem Analysis
Mathematical Concepts
Calculus
Average Rate of Change
Instantaneous Rate of Change
Derivatives
Formulas
f(x) = x^2 - 2x
Average rate of change formula: (f(b) - f(a)) / (b - a)
Derivative of f(x): f'(x) = 2x - 2
Theorems
Derivative as Instantaneous Rate of Change
Suitable Grade Level
Grades 10-12
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