Part (a): Find the derivative of $f(x)$

Given the function:
$f(x) = -0.1x^2 - 0.3x + 30$

To find $f'(x)$, we will differentiate each term of $f(x)$ with respect to $x$:

• The derivative of $-0.1x^2$ is $-0.2x$ (using the power rule: $\frac{d}{dx}(ax^n) = n \cdot ax^{n-1}$).
• The derivative of $-0.3x$ is $-0.3$ (since the derivative of $x$ is 1).
• The derivative of the constant 30 is 0 (since the derivative of a constant is always 0).

Thus, the derivative is: $f'(x) = -0.2x - 0.3$

Part (b): Rate of change when $x = 2$

To find the rate of change when the quantity demanded is 2,000 units (which corresponds to $x = 2$ because the units are in thousands), we will substitute $x = 2$ into the derivative $f'(x)$.

Substituting $x = 2$ into $f'(x)$: $f'(2) = -0.2(2) - 0.3 = -0.4 - 0.3 = -0.7$

Thus, the rate of change of the unit price when the quantity demanded is 2,000 units is $-0.7$ dollars per 1,000 lamps. This means the price decreases by $0.70 for each additional 1,000 lamps demanded.

Unit price at $x = 2$

To find the unit price at $x = 2$, we substitute $x = 2$ into the original price function $f(x)$:

$f(2) = -0.1(2)^2 - 0.3(2) + 30$ $f(2) = -0.1(4) - 0.6 + 30 = -0.4 - 0.6 + 30 = 29$

Thus, the unit price when the quantity demanded is 2,000 units is $29.

Summary of Answers:

• (a) $f'(x) = -0.2x - 0.3$
• (b) The rate of change of the unit price at $x = 2$ is $-0.7$ dollars per 1,000 lamps.
• The unit price at $x = 2$ (2,000 units) is $29.

Let me know if you'd like more details or have further questions!

Related Questions:

1. What happens to the rate of change of price if $x$ increases further (e.g., at $x = 3$)?
2. How does the rate of change of price impact a firm's pricing strategy in real-life scenarios?
3. What would be the price when the demand is at 1,000 units ($x = 1$)?
4. How can the second derivative $f''(x)$ help analyze the concavity of the price function?
5. If the price increases by $0.70 per 1,000 units, what effect would that have on total revenue?

Tip: When interpreting the rate of change in a function like this, think of it as a slope that tells you how much the price will increase or decrease as demand changes.