Thanks for the clarification! Given that the function for the number of people infected is:

$N(t) = 2 \times 3^t + 30$

where $N(t)$ represents the number of people infected and $t$ represents the time in days.

Problem 1a: Average Rate of Change from Day 1 to Day 2

The average rate of change of a function $N(t)$ from $t = 1$ to $t = 2$ is given by:

$\text{Average Rate of Change} = \frac{N(2) - N(1)}{2 - 1}$

Let's calculate $N(1)$ and $N(2)$ first:

1. Calculate $N(1)$: $N(1) = 2 \times 3^1 + 30 = 2 \times 3 + 30 = 6 + 30 = 36$

2. Calculate $N(2)$: $N(2) = 2 \times 3^2 + 30 = 2 \times 9 + 30 = 18 + 30 = 48$

Now, calculate the average rate of change:

$\text{Average Rate of Change} = \frac{N(2) - N(1)}{1} = \frac{48 - 36}{1} = 12$

So, the average rate of change between Day 1 and Day 2 is 12 people per day.

Would you like further details on this, or help with the next parts?

Follow-up Questions:

1. How does the constant 30 in the function affect the rate of change?
2. If the base of the exponential function were different, how would the average rate of change be impacted?
3. What does the average rate of change tell us about the virus's spread?
4. How would the results differ if the time interval were from Day 2 to Day 3?
5. Can you derive the instantaneous rate of change at Day 2 from this function?

Tip:

The average rate of change gives a useful measure of how a quantity is increasing or decreasing over a specific interval, which can be insightful for understanding trends in data, such as the spread of a virus over time.