The demand function for the Luminar desk lamp is given by the following function where x is the quantity demanded in thousands and p is the unit price in dollars.
p = f(x) = -0.1x2 - 0.3x + 39
(a) Find f '(x).
f '(x) =
(b) What is the rate of change of the unit price when the quantity demanded is 3000 units (x = 3)?
$ per 1000 lamps
What is the unit price at that level of demand?
$
To find f '(x), the derivative of the given demand function f(x), we need to use the power rule for differentiation.
(a) Finding f '(x):
Using the power rule, we differentiate each term with respect to x.
The power rule states:
d/dx (x^n) = nx^(n-1)
Given: f(x) = -0.1x^2 - 0.3x + 39
Differentiating each term:
f '(x) = d/dx (-0.1x^2) + d/dx (-0.3x) + d/dx (39)
Using the power rule:
f '(x) = -0.1 * 2x^(2-1) - 0.3 * x^(1-1) + 0
Simplifying:
f '(x) = -0.2x - 0.3
So, f '(x) = -0.2x - 0.3.
(b) To find the rate of change of the unit price when the quantity demanded is 3000 units (x = 3), we substitute x = 3 into the derivative of f.
So, we substitute x = 3 into f '(x) = -0.2x - 0.3.
Substituting x = 3:
f '(3) = -0.2(3) - 0.3
= -0.6 - 0.3
= -0.9
The rate of change of the unit price when the quantity demanded is 3000 units is -0.9 dollars per 1000 lamps.
To find the unit price at that level of demand, we substitute x = 3 into the demand function f(x).
Given: f(x) = -0.1x^2 - 0.3x + 39
Substituting x = 3:
f(3) = -0.1(3^2) - 0.3(3) + 39
= -0.1(9) - 0.3(3) + 39
= -0.9 - 0.9 + 39
= 37.2
The unit price at a quantity demanded of 3000 units (x = 3) is $37.2.
(a) To find f'(x), we need to take the derivative of f(x) with respect to x.
f(x) = -0.1x^2 - 0.3x + 39
Taking the derivative of each term, we get:
f'(x) = -0.1 * 2x - 0.3 * 1 + 0
Simplifying, we have:
f'(x) = -0.2x - 0.3
Therefore, f'(x) = -0.2x - 0.3.
(b) To find the rate of change of the unit price when the quantity demanded is 3000 units (x = 3), we substitute x = 3 into the derivative f'(x) that we found in part (a).
f'(x) = -0.2x - 0.3
Substituting x = 3, we have:
f'(3) = -0.2(3) - 0.3
Simplifying, we get:
f'(3) = -0.6 - 0.3
f'(3) = -0.9
Therefore, the rate of change of the unit price when the quantity demanded is 3000 units is -0.9 dollars per 1000 lamps.
To find the unit price at that level of demand, we substitute x = 3 into the demand function f(x).
p = f(x) = -0.1x^2 - 0.3x + 39
Substituting x = 3, we have:
p = -0.1(3)^2 - 0.3(3) + 39
Simplifying, we get:
p = -0.1(9) - 0.9 + 39
p = -0.9 - 0.9 + 39
p = 37.2
Therefore, the unit price at a quantity demanded of 3000 units is $37.2.