Math Problem Statement
Solution
Let's analyze the problem based on the image provided.
The problem describes a population change according to the formula . The goal is to determine which of the following statements are always true:
- (i.e., the derivative of ).
- is less than the average rate of change of between 0 and 1.
- The average rate of change between 0 and 1 is .
Step-by-Step Breakdown:
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Derivative Calculation:
- The derivative is the instantaneous rate of change, which can be simplified as follows:
- This result is derived using the chain rule and the fact that the derivative of an exponential function is .
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Evaluating :
- Substituting into the derivative:
- The derivative at depends on , and if , the derivative will be positive.
-
Average Rate of Change Between 0 and 1:
- The average rate of change between and is given by:
- From the formula :
- Therefore, the average rate of change is:
Verifying the Statements:
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Statement (i): This is simply the derivative expression given. Since we've confirmed the derivative of , statement (i) is true.
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Statement (ii): We compare with the average rate of change .
- Whether is less than the average rate of change depends on the value of . If , the value of is positive, so could be greater or less, depending on the magnitude. Therefore, statement (ii) is not always true.
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Statement (iii): The average rate of change between 0 and 1 is . This is a correct statement if the formula is interpreted as . Hence, statement (iii) is true.
Conclusion:
- (i) and (iii) are always true.
- The correct answer is (C).
Would you like more details on any part of this solution? Here are 5 follow-up questions you might find helpful:
- What is the significance of the derivative in population models?
- How does the average rate of change differ from the instantaneous rate of change?
- What would happen if ? How would that affect the rates?
- Can you explain the difference between exponential and linear population growth models?
- How do we apply limits when calculating derivatives for exponential functions?
Tip: When comparing instantaneous rates with average rates, it's helpful to calculate both and interpret them based on the context of the function's behavior.
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Math Problem Analysis
Mathematical Concepts
Calculus
Exponential Functions
Rates of Change
Formulas
P(t) = 5A^t
Derivative of exponential function A^t
Average rate of change formula
Theorems
Chain Rule for Differentiation
Exponential Growth Models
Suitable Grade Level
Undergraduate or advanced high school calculus
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