Math Problem Statement
Solution
We are given a problem about population change modeled by the function . The task is to determine which of the given statements are always true. Let's examine each statement.
Statement (i):
This is the definition of the derivative of at time , using the limit form. To confirm this, we need to find .
Given , we apply the general rule for differentiating exponential functions: Thus, the derivative of is . Therefore, the given limit expression is consistent with this derivative formula. So, (i) is true.
Statement (ii):
First, we compute . Using the formula from statement (i): Now, the average rate of change of between and is: Since and , the average rate of change is: Now, we compare with the average rate of change, .
The value of is approximately , so: Since is greater than , statement (ii) is false because is not less than the average rate of change.
Statement (iii):
We already calculated the average rate of change as , so (iii) is false.
Conclusion:
From the analysis above, only statement (i) is true. Therefore, the correct answer is:
Would you like any further clarification or details on these steps?
Related Questions:
- How is the derivative of an exponential function with a base other than calculated?
- Why is the average rate of change of a function between two points not the same as the instantaneous rate of change?
- Can you explain how the natural logarithm plays a role in differentiating exponential functions?
- What is the significance of the limit definition of the derivative in calculus?
- How can we interpret the derivative of a population growth function in real-world terms?
Tip:
For any exponential function , the derivative is , where is the natural logarithm of the base.
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Math Problem Analysis
Mathematical Concepts
Calculus
Exponential Functions
Rates of Change
Derivatives
Formulas
P'(t) = lim(h→0) (5.4^h(5.4^t − 1))/h
P'(t) = 5.4^t * ln(5.4)
Average rate of change = (P(1) − P(0))/(1 − 0)
Theorems
Derivative of exponential functions
Limit definition of the derivative
Average rate of change
Suitable Grade Level
College Calculus
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