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# Derivatives, Second Derivatives and Profit Function

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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 profit function, and create another table with the price, profit, and value of the first derivative at the prices below. Can you tell what your profit maximizing price is from this table?

Price x Profit the derivative
0.25 -56.25 250
0.5 0 200
0.75 43.75 150
1 75 100
1.25 93.75 50
1.5 100 0
1.75 93.75 -50
2 75 -100
2.25 43.75 -150
2.5 0 -200

C. Calculate the second derivative, and also use the first derivative to find the profit maximizing price. What is the price, and what does the second derivative tell you?

##### Solution Summary

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

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###### Education
• BSc , Wuhan Univ. China
• MA, Shandong Univ.
###### Recent Feedback
• "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
• "excellent work"
• "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
• "Thank you"
• "Thank you very much for your valuable time and assistance!"

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