Let's analyze the problem based on the image provided.

The problem describes a population change according to the formula $P(t) = 5A^t$. The goal is to determine which of the following statements are always true:

1. $P'(t) = \lim_{h \to 0} \frac{5A^{t+h}(5A^h - 1)}{h}$ (i.e., the derivative of $P(t)$).
2. $P'(1)$ is less than the average rate of change of $P$ between 0 and 1.
3. The average rate of change between 0 and 1 is $5A$.

Step-by-Step Breakdown:

1. Derivative Calculation:

   • The derivative $P'(t)$ is the instantaneous rate of change, which can be simplified as follows: $P'(t) = \frac{d}{dt} (5A^t) = 5A^t \ln(A).$
   • This result is derived using the chain rule and the fact that the derivative of an exponential function $A^t$ is $A^t \ln(A)$.

2. Evaluating $P'(1)$:

   • Substituting $t = 1$ into the derivative: $P'(1) = 5A^1 \ln(A) = 5A \ln(A).$
   • The derivative at $t = 1$ depends on $A$, and if $A > 1$, the derivative will be positive.

3. Average Rate of Change Between 0 and 1:

   • The average rate of change between $t = 0$ and $t = 1$ is given by: $\frac{P(1) - P(0)}{1 - 0} = P(1) - P(0).$
   • From the formula $P(t) = 5A^t$: $P(1) = 5A^1 = 5A \quad \text{and} \quad P(0) = 5A^0 = 5.$
   • Therefore, the average rate of change is: $5A - 5 = 5(A - 1).$

Verifying the Statements:

1. Statement (i): This is simply the derivative expression given. Since we've confirmed the derivative of $P(t)$, statement (i) is true.

2. Statement (ii): We compare $P'(1) = 5A \ln(A)$ with the average rate of change $5(A - 1)$.

   • Whether $P'(1)$ is less than the average rate of change depends on the value of $A$. If $A > 1$, the value of $\ln(A)$ is positive, so $P'(1)$ could be greater or less, depending on the magnitude. Therefore, statement (ii) is not always true.

3. Statement (iii): The average rate of change between 0 and 1 is $5(A - 1)$. This is a correct statement if the formula is interpreted as $P(1) - P(0)$. 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:

1. What is the significance of the derivative $P'(t)$ in population models?
2. How does the average rate of change differ from the instantaneous rate of change?
3. What would happen if $A < 1$? How would that affect the rates?
4. Can you explain the difference between exponential and linear population growth models?
5. 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.