A. Create a function to determine how much revenue you will make at each price you charge. This is done by multiplying the price times your function. For example if your function is Cups Sold = 1000 - 100*Price, your revenue function would be Price*(1000 - 100*Price). For simplicity sake, you can write Price as "P".

B. What is the derivative of your revenue function?

C. Create a table (you can use Excel) with each of the columns:

E. What is the value of the derivative when you are charging more than the revenue maximizing price? How about when you are charging less? Based on this, how would you use the derivative to help you decide how much to charge for a cup of lemonade?

Solution Summary

Derivatives, Revenue Function, Maximizing Profit are investigated for a lemonade stand. The solution is detailed and well presented.

A. Write a function for your profits for each price you charge. This is done by multiplying (P-.5) times your function (y= -100x + 250). I.e. if your function is Cups Sold = 1000 - 100P, your profit function would be (P - .5)*(1000 - 100P).
B. Calculate the first derivative of your profitfunction, and create another table

A. Your function, and specify what the slope and intercept of your function is.
B. What this function tells you about the relationship between the price and the number of cups sold?
C. How you plan on using this function to help you maximize the profits of your lemonadestand?
data:
regression equation: y= -100x + 250

With the total-revenue schedule of problem 1 (revised derivatives) and the total-cost schedule of problem 7 (cost function), show the profit-maximizing level of output?
Problem history:
1. Derive the total-revenue, average-revenue, and marginal-revenue schedules from Q = 0 to Q = 4 by 1s
Average revenue (AR) = total re

A monopolist's demand function is given by
P = 80-3Q
(with MR = 80-6Q).
Its total cost function is
TC = 20Q + 200
(with MC = 20).
(i) Using algebra determine the profitmaximizing output, price and optimal profit for the firm.
(ii) Suppose that instead of maximizingprofit, the firm wants to maximize total revenue

Suppose that during a football game, lemonade sells for $15 per gallon but only costs $4 per gallon to make. If they run out of lemonade during the game, it will be impossible to get more. On the other hand, leftover lemonade has a negligible value. Assume that you believe the fans would buy 10 gallons with probability 3/10, 11

Assume a monopolist with the following:
Qd =100-10p
TC = 1 + 2Q
Find the following:
a) Price at profitmaximizing output
b) Profitmaximizing output
c) Total Revenue at profitmaximizing output
d) Total Cost at profitmaximizing output
e)Profit

Suppose that a monopoly faces an inverse market demand function:
P = 100-2Q
and its marginal cost function is:
MC = 40 - 2Q.
a. What should be the monopoly's profit-maximizing output?
b. What is the monopoly's price?

(i) A competitive firmÃ¢??s total cost function is given by
TC = .25Q2 + 25
(with MC = .5Q).
The firm faces a market price of $15. Algebraically calculate the profitmaximizing output and the level of optimal profit for the firm.
(ii) Suppose that fixed costs increase by $50 but the prevailing market price rema