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# Derivatives, Revenue Function, Maximizing Profit - Lemonade Stand

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Data:
regression equation: y= -100x + 250
regression coefficient: r= -1

X Y Predicted value
0.25 225 225
0.5 200 200
0.75 175 175
1 150 150
1.25 125 125
1.5 100 100
1.75 75 75
2 50 50
2.25 25 25
2.5 0 0

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:

1. The prices from the data above

2. The revenue you will make at each price

3. The value of the derivative at each price

D. At what price is your revenue maximized

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?