Let's go step by step to understand and solve this problem.

Part 1 (A) — Finding the derivative of demand with respect to price:

We are given that:

$\frac{dx}{dp} = \frac{-300}{(0.3p + 1)^2}$

This represents the rate of change of demand ($x$) with respect to price ($p$).

Part 2 (B) — Find the demand and the instantaneous rate of change of demand with respect to price when the price is $330:

• Demand when the price is $330: The demand $x$ is given as 100 when the price $p$ is $330. This is directly stated in the problem.

• Instantaneous rate of change of demand with respect to price: We can calculate the rate of change of demand using the formula we already have:

$\frac{dx}{dp} = \frac{-300}{(0.3p + 1)^2}$

Substitute $p = 330$:

$\frac{dx}{dp} = \frac{-300}{(0.3(330) + 1)^2}$

Simplify:

$\frac{dx}{dp} = \frac{-300}{(99 + 1)^2} = \frac{-300}{100^2} = \frac{-300}{10000} = -0.03$

So, the instantaneous rate of change of demand with respect to price when the price is $330 is $-0.03$. This means the demand is decreasing by 0.03 units per dollar when the price is $330.

Part 3 (Interpretation of Results):

At a price of $330, the demand is 100 units (DVD players per week), and the demand is decreasing at a rate of 3 units per $1 (negative 0.03), which means that for every additional dollar increase in price, the demand decreases by 3 units.

Part 4 (C) — Estimate the demand when the price is increased to $331:

To estimate the demand when the price increases to $331, we can use the linear approximation based on the rate of change.

We know that:

• The demand when the price is $330 is 100 units.
• The rate of change of demand when the price is $330 is $-3$ units per dollar.

The approximate change in demand $\Delta x$ for a small increase in price ($\Delta p = 1$) is given by:

$\Delta x \approx \frac{dx}{dp} \cdot \Delta p$

Substitute the values:

$\Delta x \approx -3 \cdot 1 = -3$

Thus, the demand at $p = 331$ is approximately:

$x(331) = x(330) + \Delta x = 100 - 3 = 97$

So, the estimated demand when the price is increased to $331 is 97 units.

Final Summary:

• Demand at $330 = 100 units
• Rate of change of demand at $330 = $-3$ units per dollar
• Estimated demand at $331 = 97 units

Would you like any further details on any of these steps? Let me know if you need more clarification!

Further Questions for Exploration:

1. How do you interpret the relationship between demand and price based on this model?
2. How does the derivative $\frac{dx}{dp}$ affect the demand curve for a product?
3. What would happen if the rate of change of demand were positive instead of negative?
4. How would the estimation of demand change if the price increase were larger than 1 dollar?
5. How do changes in the rate of change $\frac{dx}{dp}$ influence the long-term trend in demand?

Tip: When estimating demand changes, using the linear approximation formula gives a quick estimate but is most accurate for small changes in price!