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Let's take a closer look at the problem from your image. The problem states:

A population changes according to the formula \( P(t) = 5A^t \). You are asked to determine which of the following statements are always true:

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

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Let’s break it down:

1. The first option is a definition of the derivative \( P'(t) \), using the limit definition. The formula looks correct for differentiating an exponential function, but we’ll evaluate the correctness fully.
   
2. The second option refers to the behavior of the derivative, comparing it to the average rate of change.
   
3. The third option provides the value of the average rate of change, claiming it’s \( 5A \), which can be verified.

I'll walk through each statement to confirm whether it's true or not.

### Step 1: Deriving \( P'(t) \)

Given \( P(t) = 5A^t \), the derivative \( P'(t) \) using the chain rule is:

\[
P'(t) = 5A^t \ln(A)
\]

### Step 2: Average Rate of Change between \( t = 0 \) and \( t = 1 \)

The average rate of change of \( P(t) \) from 0 to 1 is calculated as:

\[
\text{Average Rate} = \frac{P(1) - P(0)}{1 - 0} = \frac{5A^1 - 5A^0}{1} = 5(A - 1)
\]

### Step 3: Verifying Statements

1. The limit definition of the derivative in the first option matches the derivative rule for exponential functions, so **Statement (i) is true**.

2. For \( P'(t) \), we found \( P'(t) = 5A^t \ln(A) \). To determine whether \( P'(t) \) is less than the average rate of change, we need to compare \( 5A^t \ln(A) \) and \( 5(A - 1) \). This requires knowing the value of \( A \), but typically, for \( A > 1 \), the derivative will be less than the average rate over time. Therefore, **Statement (ii) can be true in typical cases**.

3. The average rate of change was calculated as \( 5(A - 1) \), not \( 5A \), so **Statement (iii) is false**.

### Conclusion:
- Statements (i) and (ii) are true.
- Statement (iii) is false.

The correct option is likely **(B) (i) and (ii) only**.

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Would you like further details or explanations? Here are 5 related questions you could consider:

1. How do you compute the derivative of an exponential function like \( 5A^t \)?
2. What is the significance of the average rate of change in calculus?
3. How do we compare the instantaneous rate of change and the average rate of change?
4. What role does the base \( A \) play in the growth of the function \( P(t) = 5A^t \)?
5. How can we interpret the derivative in the context of population growth models?

### Tip:
Always carefully compare the limit definition of the derivative with the actual derivative computation to ensure correctness!

A population changes according to the formula P(t) = 5A^t. Which of the following statements are always true?

(i) P'(t) = lim (h → 0) [5A^(t+h) - 5A^t] / h
(ii) P'(t) 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 5A.
Options: (A) (i) only, (B) (i) and (ii) only, (C) (i) and (iii) only, (D) none of them, (E) (ii) and (iii) only, (F) all of them, (G) (iii) only, (H) (ii) only

Learn how to calculate the derivative of an exponential function and compare it with the average rate of change over a given interval. This problem focuses on exponential growth and its calculus applications, suitable for Calculus I students.

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