The common economic functions can be summarized as follows:
C(x) is the total cost of producing x items.
C(x)/x is the average cost per item if x items are made.
p(x) is the price per item when x items are sold.
R(x) = x*p(x) is the revenue collected when x items are sold.
P(x) = R(x)-C(x) is the profit made when x items are sold.
Of course, the domain of x must be [0,+inf) for a real economic problem. And if we want to stay in business, we need to have P(x) > 0.
Given C(x) = 2x + 12 and p(x) = 10 - x.
1. Plot C(x) on a piece of graph paper. Let the x-axis represent the number of items, and the y-axis represent the number of dollars.
2. For what values of x is C(x) increasing?
3. Explain why this makes economic sense.
4. What is the value of C(0)?
5. In economic terms, what does C(0) represent?
6. Plot p(x) on the same graph.
7. For what values of x is p(x) decreasing?
8. Explain why this makes economic sense.
9. What is the value of p(0)?
10. In economic terms, what does p(0) represent?
11. Plot R(x) = x(10-x) on the same graph.
12. Shade in the portion of the graph where P(x) > 0.
13. In economic terms, what does the shaded area represent?
14. What are the values of x for the break-even points?
15. What is the number of items that need to be sold in order to make the maximum profit?
16. What is the average cost per item at the maximum profit point?