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Let's go through the problem.

### Given:
The population function is \( P(t) = 5.4^t \), and we are asked which of the following statements are always true:

#### (i) \( P'(t) = \lim_{h \to 0} \frac{5.4^h(5.4^t - 1)}{h} \)
This expression is trying to represent the derivative of \( P(t) \). We'll need to calculate the derivative of \( P(t) = 5.4^t \) to see if this limit holds.

#### (ii) \( P'(1) \) is less than the average rate of change of \( P \) between 0 and 1.
We need to compute \( P'(1) \) and compare it to the average rate of change of \( P \) between \( t = 0 \) and \( t = 1 \).

#### (iii) The average rate of change between \( 0 \) and \( 1 \) is 5.4.
We'll compute the average rate of change between \( t = 0 \) and \( t = 1 \) and see if it equals 5.4.

### Step-by-step process:

1. **Find the derivative \( P'(t) \):**
   Using the chain rule, the derivative of \( P(t) = 5.4^t \) is:
   \[
   P'(t) = 5.4^t \ln(5.4)
   \]
   So, we need to compare this to the limit in (i).

2. **Compare the given limit in (i):**
   The limit expression in (i) seems to be an approximation of the derivative using first principles. We'll check if it simplifies to \( 5.4^t \ln(5.4) \), the correct derivative.

3. **Calculate \( P'(1) \):**
   Substituting \( t = 1 \) into the derivative:
   \[
   P'(1) = 5.4 \ln(5.4)
   \]

4. **Average rate of change between \( 0 \) and \( 1 \):**
   The average rate of change between \( t = 0 \) and \( t = 1 \) is:
   \[
   \frac{P(1) - P(0)}{1 - 0} = \frac{5.4^1 - 5.4^0}{1} = 5.4 - 1 = 4.4
   \]

5. **Compare \( P'(1) \) with the average rate of change:**
   We need to check whether \( P'(1) \) is less than the average rate of change, which is 4.4.

Now, let's compute these values to finalize the analysis.Here's what we find:

1. **For statement (i):** The derivative \( P'(t) = 5.4^t \ln(5.4) \) holds true, and the given limit in (i) matches this definition, so this statement is true.
   
2. **For statement (ii):** The derivative at \( t = 1 \), \( P'(1) = 9.11 \), is **greater** than the average rate of change (which is \( 4.4 \)). Therefore, statement (ii) is false.

3. **For statement (iii):** The average rate of change between \( 0 \) and \( 1 \) is 4.4, not 5.4, so statement (iii) is false.

Thus, only **statement (i)** is true. Therefore, the correct answer is:

\[
\boxed{A}
\]

Would you like more details on any part of this solution or have further questions?

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Here are five related questions to extend your understanding:

1. How would the derivative of a more complex exponential function, such as \( P(t) = 7.3^t \), be calculated?
2. Why does the natural logarithm \( \ln(5.4) \) appear in the derivative of an exponential function with base 5.4?
3. Can the average rate of change ever be equal to the derivative at a specific point?
4. What does the difference between the average rate of change and the instantaneous rate of change (derivative) tell us about a function's behavior?
5. How would the derivative calculation change if the base of the exponent were Euler's number \( e \)?

**Tip:** The derivative of an exponential function \( a^t \) is given by \( a^t \ln(a) \), which arises from the chain rule and the properties of logarithms.

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.

This problem involves the population function P(t) = 5.4^t. We compute the derivative P'(t) and determine if the given statements about the derivative and the average rate of change are true. Step-by-step solutions are provided to evaluate the limits, calculate the rates, and compare values.

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