The question in the image involves a population $P(t)$ that changes according to the formula $P(t) = 5.4^t$, and the task is to determine which of the following statements are always true:

1. $P'(t) = \lim\limits_{h \to 0} \frac{5.4^h(5.4^t - 1)}{h}$
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 5.4.

Let's break these down:

1. Understanding $P'(t)$:

• $P(t) = 5.4^t$ represents an exponential function, and its derivative $P'(t)$ will involve the natural logarithm of the base, 5.4.
• The derivative of $P(t) = a^t$ is given by $P'(t) = a^t \ln(a)$, where $a = 5.4$.
  • Thus, $P'(t) = 5.4^t \ln(5.4)$.

2. Evaluating $P'(1)$:

• $P'(1) = 5.4^1 \ln(5.4) = 5.4 \ln(5.4)$.
  • Using an approximation, $\ln(5.4) \approx 1.686$, so $P'(1) \approx 5.4 \times 1.686 \approx 9.11$.
  • We need to compare this with the average rate of change between $t = 0$ and $t = 1$.

3. Average rate of change:

• The average rate of change between $t = 0$ and $t = 1$ is given by: $\text{Average rate of change} = \frac{P(1) - P(0)}{1 - 0} = P(1) - P(0)$
  • Since $P(1) = 5.4^1 = 5.4$ and $P(0) = 5.4^0 = 1$, the average rate of change is: $5.4 - 1 = 4.4.$

Conclusion:

• $P'(1) \approx 9.11$, and the average rate of change is 4.4.
• Thus, $P'(1)$ is greater than the average rate of change between 0 and 1, so statement (ii) is false.
• Statement (iii) is true, as the average rate of change is indeed 4.4.
• Statement (i) involves the derivative, which is true as it correctly expresses the limit definition of $P'(t)$.

Final Answer:

The correct option is (A), which includes only statement (i).

Would you like more details or have any questions?

Here are some related questions for further understanding:

1. What is the general rule for finding the derivative of exponential functions?
2. How do you calculate the average rate of change for any function?
3. What is the significance of the natural logarithm in exponential derivatives?
4. How does the average rate of change compare to the instantaneous rate of change?
5. What are the applications of exponential growth in real-world scenarios?

Tip: Always calculate the derivative carefully, especially when comparing it with the average rate of change over an interval!