Question

all parts have been answered.
the total profit (in dollars) from the sale of x espresso machines is
p(x) = 90x - 0.6x² - 280.
evaluate the rate of change of the profit at the following values:
(a) p(200) =
(b) p(350) =

Explanation:

Step1: Find the derivative of \( P(x) \)

The profit function is \( P(x) = 90x - 0.6x^2 - 280 \). Using the power rule for differentiation, the derivative \( P'(x) \) is:
\( P'(x) = \frac{d}{dx}(90x) - \frac{d}{dx}(0.6x^2) - \frac{d}{dx}(280) \)
\( P'(x) = 90 - 1.2x \) (since the derivative of \( ax^n \) is \( anx^{n - 1} \), and the derivative of a constant is 0)

Step2: Evaluate \( P'(200) \)

Substitute \( x = 200 \) into \( P'(x) \):
\( P'(200) = 90 - 1.2(200) \)
\( P'(200) = 90 - 240 \)
\( P'(200) = -150 \)

Step3: Evaluate \( P'(350) \)

Substitute \( x = 350 \) into \( P'(x) \):
\( P'(350) = 90 - 1.2(350) \)
\( P'(350) = 90 - 420 \)
\( P'(350) = -330 \)

Answer:

(A) \( P'(200) = -150 \)
(B) \( P'(350) = -330 \)