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Let's analyze the problem:

The population is given by the function \( P(t) = 5.4^t \).

### Part (i)
\( P'(t) \) is the derivative of \( P(t) \), which represents the instantaneous rate of change of the population at time \( t \).

We calculate \( P'(t) \) using the chain rule for exponentials:

\[
P'(t) = \frac{d}{dt} \left( 5.4^t \right) = 5.4^t \ln(5.4)
\]

So, \( P'(t) = 5.4^t \ln(5.4) \).

### Part (ii)
This statement claims that \( P'(1) \) is less than the average rate of change of \( P(t) \) between \( t = 0 \) and \( t = 1 \).

The average rate of change between \( t = 0 \) and \( t = 1 \) is calculated as:

\[
\text{Average rate of change} = \frac{P(1) - P(0)}{1 - 0} = P(1) - P(0)
\]

Since \( P(1) = 5.4^1 = 5.4 \) and \( P(0) = 5.4^0 = 1 \):

\[
\text{Average rate of change} = 5.4 - 1 = 4.4
\]

Now, calculate \( P'(1) \):

\[
P'(1) = 5.4^1 \ln(5.4) = 5.4 \ln(5.4)
\]

The natural logarithm \( \ln(5.4) \) is approximately 1.6864, so:

\[
P'(1) \approx 5.4 \times 1.6864 \approx 9.111
\]

Since \( 9.111 > 4.4 \), the statement in (ii) is **false**.

### Part (iii)
The average rate of change between \( t = 0 \) and \( t = 1 \) is 4.4, not 5.4, so statement (iii) is also **false**.

### Conclusion
- (i) is true.
- (ii) is false.
- (iii) is false.

Thus, the correct answer is **(A) (i) only**.

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Do you have any questions or would you like more details? Here are 5 related questions to expand on this:

1. How do you compute the derivative of other exponential functions like \( a^t \)?
2. What is the significance of the average rate of change compared to the instantaneous rate of change?
3. Can we apply similar reasoning to functions with different bases, like \( 2^t \) or \( e^t \)?
4. How would the problem change if the base of the exponential was smaller, like 3 or 2?
5. What is the interpretation of the logarithm \( \ln(5.4) \) in the context of exponential growth?

**Tip**: For exponential growth, the derivative always involves the natural logarithm of the base, which scales the rate of growth.

A population changes according to the formula P(t) = 5.4^t. Which of the following statements are always true? (i) P'(t) = lim h→0 (5.4^h (5.4^t − 1) / h), (ii) P'(1) is less than the average rate of change of P between 0 and 1, (iii) The average rate of change between 0 and 1 is 5.4.

Learn how to analyze population growth using the exponential function P(t) = 5.4^t. This problem covers concepts like calculating the derivative using the chain rule, understanding the average rate of change, and comparing instantaneous and average rates of change. Suitable for high school students in Grades 11-12.

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