To find the average rate of change of demand over a price interval $[p_1, p_2]$, we use the formula for the average rate of change:

$\text{Average Rate of Change} = \frac{D(p_2) - D(p_1)}{p_2 - p_1}$

Where $D(p)$ is the demand function.

Given:

The demand function is: $D(p) = \frac{1000}{p} - 1$

We will calculate the average rate of change for each interval.

(a) From $p = 1$ to $p = 16$:

1. Find $D(1)$: $D(1) = \frac{1000}{1} - 1 = 1000 - 1 = 999$

2. Find $D(16)$: $D(16) = \frac{1000}{16} - 1 = 62.5 - 1 = 61.5$

3. Calculate the average rate of change: $\text{Average Rate of Change} = \frac{D(16) - D(1)}{16 - 1} = \frac{61.5 - 999}{16 - 1} = \frac{-937.5}{15} = -62.5$

(b) From $p = 16$ to $p = 100$:

1. Find $D(100)$: $D(100) = \frac{1000}{100} - 1 = 10 - 1 = 9$

2. Calculate the average rate of change: $\text{Average Rate of Change} = \frac{D(100) - D(16)}{100 - 16} = \frac{9 - 61.5}{100 - 16} = \frac{-52.5}{84} = -\frac{52.5}{84} = -\frac{35}{56}$

Final Answers:

• (a) The average rate of change from $p = 1$ to $p = 16$ is $-62.5$.
• (b) The average rate of change from $p = 16$ to $p = 100$ is $-\frac{35}{56}$.

Would you like further details or have any questions?

Here are 5 related questions you might consider:

1. What does the average rate of change represent in terms of demand?
2. How does increasing the price affect demand based on this function?
3. Can the demand become negative for any price?
4. How would the rate of change differ for a smaller interval, like $p = 1$ to $p = 2$?
5. How does the behavior of $D(p)$ change as $p$ approaches zero?

Tip: The average rate of change gives an idea of how the demand is shifting over a certain price interval.