The unit selling price p (in dollars) and the quantity demanded x (in pairs) of a certain brand of women’s shoes are given by the demand equation
p(x) = 100e^-0.0001x f or 0 _< x _< 20,000
a. Find the revenue function,R. (Hint: R(x)= x(p(x)), since the
revenue function is the unit selling price at a demand level of x units
times the number of units demanded.)
b. Find the marginal revenue function, R'.
c. What is the marginal revenue when, x = 10 ?
a. To find the revenue function R(x), we need to multiply the unit selling price p(x) by the quantity demanded x:
R(x) = x * p(x)
Substituting the given demand equation:
R(x) = x * (100e^(-0.0001x))
b. To find the marginal revenue function R', we need to differentiate the revenue function with respect to x:
R'(x) = d(R(x))/dx
Using the product rule for differentiation:
R'(x) = 100e^(-0.0001x) + (-0.0001x)(100e^(-0.0001x))
Simplifying:
R'(x) = 100e^(-0.0001x) - 0.01xe^(-0.0001x)
c. To find the marginal revenue when x = 10, substitute x = 10 into the marginal revenue function R':
R'(10) = 100e^(-0.0001(10)) - 0.01(10)e^(-0.0001(10))
Simplifying:
R'(10) = 100e^(-0.001) - 0.1e^(-0.001)
To find the revenue function, R(x), we need to multiply the unit selling price, p(x), by the quantity demanded, x. Using the given demand equation:
p(x) = 100e^(-0.0001x)
We can substitute this expression into the revenue function equation:
R(x) = x * p(x) = x * (100e^(-0.0001x))
Now, let's move on to finding the marginal revenue function, R'.
The marginal revenue function, R', represents the rate at which the revenue is changing with respect to the quantity demanded. To find R', we need to differentiate the revenue function, R(x), with respect to x.
Using the product rule of differentiation, we have:
R'(x) = x * (d/dx[100e^(-0.0001x)]) + (d/dx[x]) * (100e^(-0.0001x))
To find the derivative of the exponential function, we need to use the chain rule. The derivative of e^(-0.0001x) is (-0.0001)e^(-0.0001x).
Differentiating further:
R'(x) = x * (-0.0001) * e^(-0.0001x) + 100e^(-0.0001x)
Now, let's move on to finding the marginal revenue when x = 10.
To do this, substitute x = 10 into the derived marginal revenue function, R'(x):
R'(10) = 10 * (-0.0001) * e^(-0.0001 * 10) + 100e^(-0.0001 * 10)
Simplifying this expression will give you the numerical value of the marginal revenue when x = 10.