Let's break down the problem presented in the image. The problem asks about a population $P(t)$ that changes according to the formula:

$P(t) = 5 \cdot 4^t$

You are asked to evaluate which of the following statements are always true:

1. $P'(t) = \lim_{h \to 0} \frac{5 \cdot 4^{t+h} - 5 \cdot 4^t}{h}$
2. $P'(1)$ is less than the average rate of change of $P$ between $t = 0$ and $t = 1$.
3. The average rate of change between $t = 0$ and $t = 1$ is 5.4.

Step 1: Differentiate $P(t) = 5 \cdot 4^t$

To find the derivative $P'(t)$, we need to apply the chain rule:

[ P'(t) = \frac{d}{dt} [5 \cdot 4^t] = 5 \cdot \frac{d}{dt} [4^t] = 5 \cdot 4^t \cdot \ln(4) ]

Thus:

$P'(t) = 5 \cdot 4^t \cdot \ln(4)$

Step 2: Evaluate the statements

Statement (i):

$P'(t) = \lim_{h \to 0} \frac{5 \cdot 4^{t+h} - 5 \cdot 4^t}{h}$ This is the definition of the derivative for $P(t) = 5 \cdot 4^t$, so this statement is true.

Statement (ii):

To evaluate whether $P'(1)$ is less than the average rate of change between $t = 0$ and $t = 1$, we first calculate both.

1. $P'(1) = 5 \cdot 4^1 \cdot \ln(4) = 5 \cdot 4 \cdot \ln(4) = 20 \cdot \ln(4) \approx 27.73$.
2. The average rate of change between $t = 0$ and $t = 1$ is: $\text{Average rate of change} = \frac{P(1) - P(0)}{1 - 0} = \frac{5 \cdot 4^1 - 5 \cdot 4^0}{1} = 5 \cdot 4 - 5 = 20 - 5 = 15$

Thus, $P'(1) \approx 27.73$ is greater than the average rate of change, which is 15, so statement (ii) is false.

Statement (iii):

The average rate of change between $t = 0$ and $t = 1$ was already calculated as 15, not 5.4. Hence, statement (iii) is false.

Conclusion:

Only statement (i) is true. The correct answer is (B) i only.

Do you want further details or have any questions?

Related questions:

1. How is the derivative of exponential functions calculated in general?
2. Can the chain rule be applied to logarithmic and trigonometric functions?
3. What is the difference between average and instantaneous rates of change?
4. How do you find the derivative of functions with exponential bases other than $e$?
5. How would the problem change if the base of the exponential function were different?

Tip:

When finding the derivative of an exponential function like $a^t$, remember to multiply by the natural logarithm of the base $a$.