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 ?
To find the revenue function, R(x), we can use the formula R(x) = x * p(x), where p(x) is the unit selling price at a demand level of x units.
Given that p(x) = 100e^(-0.0001x), we can substitute this equation into the revenue function formula.
R(x) = x * p(x) = x * 100e^(-0.0001x)
Therefore, the revenue function is R(x) = 100xe^(-0.0001x).
To find the marginal revenue function, R', we need to differentiate the revenue function R(x) with respect to x.
R'(x) = d/dx [100xe^(-0.0001x)]
Let's apply the product rule of differentiation, which states that if we have two functions f(x) and g(x), their derivative is given by the formula:
d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
In this case, f(x) = 100x and g(x) = e^(-0.0001x).
Differentiating f(x) and g(x):
f'(x) = 100
g'(x) = (-0.0001)e^(-0.0001x)
Applying the product rule:
R'(x) = (100 * e^(-0.0001x)) + (100x * (-0.0001)e^(-0.0001x))
= 100e^(-0.0001x) - 0.0001xe^(-0.0001x)
Therefore, the marginal revenue function is R'(x) = 100e^(-0.0001x) - 0.0001xe^(-0.0001x).
To find the marginal revenue when x = 10, we can substitute x = 10 into the marginal revenue function R'(x):
R'(10) = 100e^(-0.0001(10)) - 0.0001(10)e^(-0.0001(10))
Calculating the values:
R'(10) = 100e^(-0.001) - 0.001e^(-0.001)
Using a calculator or software, evaluate the expression to find the numerical value of R'(10).