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The question involves the demand function $ N(p) = 70 - 4p^2 $ (where $ 1 \leq p \leq 4 $), which describes the number of kilograms of tomatoes bought per day as a function of the price per kilogram, $ p $.

We are tasked with answering three subparts:

### a. **Average Rate of Change**
The average rate of change of demand from $ p = 2 $ to $ p = 3 $ is calculated by:
\[
\text{Average Rate of Change} = \frac{N(3) - N(2)}{3 - 2}
\]
First, we need to compute $ N(2) $ and $ N(3) $.

- $ N(2) = 70 - 4(2)^2 = 70 - 16 = 54 $
- $ N(3) = 70 - 4(3)^2 = 70 - 36 = 34 $

Now, we compute the average rate of change:
\[
\frac{N(3) - N(2)}{3 - 2} = \frac{34 - 54}{1} = -20
\]

So, the average rate of change is **$-20$ kilograms per dollar**.

### b. **Instantaneous Rate of Change at $ p = 2 $**
The instantaneous rate of change of demand is found using the derivative of $ N(p) $. The derivative of $ N(p) = 70 - 4p^2 $ is:
\[
\frac{dN}{dp} = -8p
\]

At $ p = 2 $:
\[
\frac{dN}{dp} \bigg|_{p=2} = -8(2) = -16
\]
So, the instantaneous rate of change of demand at $ p = 2 $ is **$-16$ kilograms per dollar**.

### c. **Instantaneous Rate of Change at $ p = 3 $**
At $ p = 3 $:
\[
\frac{dN}{dp} \bigg|_{p=3} = -8(3) = -24
\]
So, the instantaneous rate of change of demand at $ p = 3 $ is **$-24$ kilograms per dollar**.

Let me know if you need further details or explanations!

Here are five related questions you could explore:
1. How does the demand function behave for prices outside the given range $ 1 \leq p \leq 4 $?
2. What would happen to the average rate of change if the price were increased by smaller intervals?
3. How is the derivative of the demand function related to its concavity?
4. How would a shift in the demand function (e.g., adding a constant term) affect the rates of change?
5. How can this rate of change be applied in a real business setting to optimize pricing?

**Tip**: The instantaneous rate of change gives the best estimate of how demand responds to small changes in price at a specific point, while the average rate of change provides a broader perspective over an interval.

Restaurant is interested in buying N(p) kilograms of tomatoes a day at p dollars per kilogram, as described by the function: N(p) = 70 - 4p^2, 1 <= p <= 4. a) Calculate the average rate of change of demand for a change in price from $2 to $3 per kilogram. b) Calculate the instantaneous rate of change of demand when the price is $2 per kilogram. c) Calculate the instantaneous rate of change of demand when the price is $3 per kilogram.

Learn how to calculate the average and instantaneous rate of change for a demand function N(p) = 70 - 4p^2, which models the demand for tomatoes based on price. The problem involves finding the average rate of change from $2 to $3 per kilogram and the instantaneous rates of change at specific prices. Suitable for students in Grades 10-12.

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