q = 1400e^(−0.03p)
(a) Write revenue, R, as a function of price.
R(p) =
(b) Find the marginal revenue.
R′(p) =
(a) Write revenue, R, as a function of price.
R(p) = 1400e^(-0.03p)
(b) Find the marginal revenue.
R′(p) = Well, if you take the derivative of R(p) with respect to p, you'll get -0.03*1400e^(-0.03p). But let's be honest, nobody wants to get into the nitty-gritty of math when there are clown jokes to be made! So, the clown version of the marginal revenue is: The marginal revenue is like getting paid in single dollar bills for telling jokes, and R′(p) tells you how many extra dollar bills you'll get for each additional joke. Make people laugh, make more money!
(a) To write the revenue, R, as a function of price, we substitute the given equation into the general formula for revenue:
R(p) = p * q
= p * 1400e^(-0.03p)
So, R(p) = 1400pe^(-0.03p).
(b) Marginal revenue, R'(p), can be found by taking the derivative of the revenue function with respect to price, p.
R(p) = 1400pe^(-0.03p)
Using the product rule, the derivative of p * e^(-0.03p) is:
d/dp (p * e^(-0.03p)) = 1 * e^(-0.03p) + p * (-0.03) * e^(-0.03p)
= e^(-0.03p) - 0.03pe^(-0.03p)
Therefore, the marginal revenue function is:
R'(p) = e^(-0.03p) - 0.03pe^(-0.03p)
(a) To write revenue, R, as a function of price, we substitute the given equation for q into the revenue formula. The revenue formula is given as: R = p * q, where p represents the price and q represents the quantity.
Given: q = 1400e^(-0.03p)
Substituting the value of q into the revenue formula, we have:
R(p) = p * 1400e^(-0.03p)
So, the revenue function as a function of price is R(p) = 1400p * e^(-0.03p).
(b) To find the marginal revenue, we need to take the derivative of the revenue function with respect to price, p.
R'(p) = dR(p)/dp
To find the derivative, we can use the product rule and chain rule.
Using the product rule, we differentiate the first term (p) while keeping the second term (1400e^(-0.03p)) constant, and vice versa.
Differentiating the first term:
d/dp [p] = 1
Differentiating the second term requires the chain rule since it involves the exponential function e^(-0.03p).
d/dp [1400e^(-0.03p)] = 1400 * d/dp [e^(-0.03p)]
To differentiate e^(-0.03p), we multiply it by the derivative of the exponent (-0.03p) with respect to p.
d/dp [e^(-0.03p)] = e^(-0.03p) * (-0.03)
Therefore, the derivative of the revenue function with respect to price is:
R'(p) = 1 * (1400e^(-0.03p)) + 1400 * e^(-0.03p) * (-0.03)
Simplifying further, we can factor out 1400e^(-0.03p) from both terms:
R'(p) = 1400e^(-0.03p) - 0.03 * 1400e^(-0.03p)
R'(p) = 1400e^(-0.03p) * (1 - 0.03)
R'(p) = 1400e^(-0.03p) * (0.97)
So, the marginal revenue function is R'(p) = 0.97 * 1400e^(-0.03p).