Math Problem Statement
Solution
To solve this problem, we need to find the value of where the instantaneous rate of change of at is equal to the average rate of change of over the interval .
Step 1: Find the Average Rate of Change of over
The average rate of change of over is given by: For this problem, and .
Calculate and :
Now, the average rate of change is:
Step 2: Find the Instantaneous Rate of Change of at
The instantaneous rate of change of is the derivative .
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Differentiate :
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Set equal to the average rate of change and solve for :
Answer
The value of is approximately , which corresponds to option C.
Would you like more details on any of the steps, or do you have any questions?
Here are 5 related questions for practice:
- How would you find the instantaneous rate of change at any given point?
- What does the derivative of represent in this context?
- How would the process change if we were given a different function, such as ?
- What if the interval was instead of ? How would that affect the result?
- How would you interpret the result of the average rate of change in real-world terms?
Tip: Always make sure to carefully evaluate logarithmic functions to avoid approximation errors.
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Math Problem Analysis
Mathematical Concepts
Calculus
Rates of Change
Differentiation
Logarithmic Functions
Formulas
Average rate of change: (f(b) - f(a)) / (b - a)
Instantaneous rate of change (derivative): f'(x)
Theorems
Mean Value Theorem
Suitable Grade Level
Grades 11-12 or College Calculus
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