Business Calculus - University of Central Arkansas
Business Calculus Recall: Profit = Revenue – Cost
4-7 Revenue = x(price demand)
Marginal Cost, Revenue, & Profit
• Instantaneous rate of change relative to production at a given production level
• Derivative of Cost, Revenue, or Profit functions
• Interpretation: If [pic]= 100, then we could interpret it as, “At a production level of 3 units, the cost is increasing at a rate of $100 per unit.”
• The interpretation can be translated directly to Revenue and Profit
Average Cost, Revenue, & Profit
• The total divided by the number of units (x).
• Average cost: [pic], Average revenue: [pic], Average profit: [pic]
• Interpretation #1: If[pic], then “The average cost for producing 3 units is $100 per unit.”
• Interpretation #2: : If[pic], then “At a production level of 3 units, the average cost per unit is $100.”
Marginal Average Cost, Revenue, & Profit: [pic], [pic], [pic]
• The derivative of the Average Cost, Revenue, & Profit functions
• First find[pic], and then take the derivative.
• Interpretation: If[pic], then “At a production level of 3 units, the average cost is increasing at a rate of $100 per unit.”
The Radio Company has determined from data that it has collected that the price-demand function is given by[pic], 0 < x < 10,000. The cost given by the function[pic], where x is the number of radios produced and sold , p(x) is the price per radio in dollars, and C(x) is the total cost for producing x radios.
a) Find the Revenue Function.
b) Find the Marginal Revenue Function.
c) Find the Profit Function.
d) Find the Marginal Profit Function.
e) Find the Average Profit Function.
f) Find the Marginal Average Profit Function.
g) Find the marginal cost function.
h) Evaluate the marginal cost function at x = 100 and interpret the results.
i) Find the EXACT cost of producing the 100th radio. Compare this to your answer in part F.
j) Find the Average Cost Function.
k) Find the Marginal Average Cost Function.
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