The demand function for the Luminar desk lamp is given by
p = f(x) = −0.1x^2 − 0.7x + 20
where x is the quantity demanded in thousands and p is the unit price in dollars.
(a) Find f '(x).
(b) What is the rate of change of the unit price (in dollars per 1,000 lamps) when the quantity demanded is 3,000 units (x = 3)?
(a) differentiating ... f'(x) = -0.2 x - 0.7
(b) to find dp/dx , plug x=3 into f'(x)
(a) To find f'(x), we need to take the derivative of the demand function.
Using the power rule, which states that if f(x) = x^n, then f'(x) = nx^(n-1), we can find the derivative of each term in the demand function.
Taking the derivative of the first term, -0.1x^2, we get:
d/dx (-0.1x^2) = -0.1 * 2x^(2-1) = -0.2x
Taking the derivative of the second term, -0.7x, we get:
d/dx (-0.7x) = -0.7
Taking the derivative of the constant term, 20, we get:
d/dx (20) = 0
Combining the derivatives of each term, we get:
f'(x) = -0.2x - 0.7
Therefore, the derivative of the demand function is f'(x) = -0.2x - 0.7.
(b) To find the rate of change of the unit price when the quantity demanded is 3,000 units (x = 3), we substitute x = 3 into the derivative we found in part (a).
f'(3) = -0.2(3) - 0.7 = -0.6 - 0.7 = -1.3
Therefore, the rate of change of the unit price when the quantity demanded is 3,000 units is -1.3 dollars per 1,000 lamps.
To find the derivative of the demand function f(x), we can use the power rule for differentiation. The power rule states that for any term in the form ax^n, the derivative is given by nx^(n-1).
(a) So, for the given demand function f(x) = -0.1x^2 - 0.7x + 20, we can find the derivative by differentiating each term separately:
f'(x) = d/dx (-0.1x^2) + d/dx (-0.7x) + d/dx (20)
= -0.1 * 2x^(2-1) - 0.7 * 1x^(1-1) + 0
= -0.2x - 0.7
Therefore, f '(x) = -0.2x - 0.7.
(b) To find the rate of change of the unit price (in dollars per 1,000 lamps) when the quantity demanded is 3,000 units (x = 3), we can evaluate f '(x) at x = 3:
f '(3) = -0.2 * 3 - 0.7
= -0.6 - 0.7
= -1.3
Therefore, the rate of change of the unit price when the quantity demanded is 3,000 units is -1.3 dollars per 1,000 lamps.