Let's break down the problem based on the image.

The demand function for the Luminar desk lamp is given by:

$p = f(x) = -0.1x^2 - 0.4x + 38$

Where:

• $x$ is the quantity demanded in thousands, and
• $p$ is the unit price in dollars.

(a) Find $f'(x)$

From the image, the derivative $f'(x)$ has already been computed as:

$f'(x) = -0.2x - 0.4$

This gives us the rate of change of the unit price with respect to the quantity demanded.

(b) What is the rate of change of the unit price when the quantity demanded is 3,000 units?

Here, $x = 3$ (since the quantity is in thousands, 3,000 units corresponds to $x = 3$).

Substitute $x = 3$ into $f'(x)$:

$f'(3) = -0.2(3) - 0.4 = -0.6 - 0.4 = -1$

So, the rate of change of the unit price when the quantity demanded is 3,000 units is $-1 per 1,000 lamps. This means that for every additional 1,000 lamps sold, the price decreases by $1.

(c) What is the unit price at that level of demand?

To find the unit price when $x = 3$, we substitute $x = 3$ into the original demand function $p = f(x)$:

$p = f(3) = -0.1(3)^2 - 0.4(3) + 38 = -0.1(9) - 1.2 + 38 = -0.9 - 1.2 + 38 = 35.9$

Thus, the unit price when 3,000 units are demanded is $35.90.

Would you like further details or explanations on any part? Here are some related questions for further practice:

1. What does the derivative $f'(x)$ tell us about the relationship between price and quantity?
2. How would you interpret a positive value for $f'(x)$?
3. If the quantity demanded increases to 4,000 units, what would be the rate of change in unit price?
4. Calculate the unit price when the quantity demanded is 5,000 units.
5. How does the quadratic nature of the demand function affect the pricing as quantity increases?

Tip: The derivative of a demand function helps in understanding how sensitive the price is to changes in demand quantity.