1. Business and Economic Applications

2. Summary of Business Terms and Formulas  x is the number of units produced (or sold)  p is the price per unit  R is the total revenue from selling x units R = xp R = xp  C is the total cost of producing x units

3. Summary of Business Terms and Formulas  P is the total profit from selling x units P = R – C P = R – C  The break-even point is the number of units for which R = C.

4. Marginals

5. Using Marginals as Approximations  A manufacturer determines that the profit derived from selling x units of a certain item is given by P = 0.0002x 3 + 10x.  a. Find the marginal profit for a production level of 50 units  b. Compare this with the actual gain in profit obtained by increasing the production from 50 to 51 units.

6. Demand Function  The number of units x that consumers are willing to purchase at a given price p is defined as the demand function  p = f(x)

7. Finding the Demand Function  A business sells 2000 items per month at a price of $10 each. It is predicted that monthly sales will increase by 250 items for each $0.25 reduction in price. Find the demand function corresponding to this prediction.

8. Steps  First find the number of units produced:

9. Steps  Now solve this equation for p

10. Finding Marginal Revenue  A fast-food restaurant has determine that the monthly demand for its hamburgers is  Find the increase in revenue per hamburger (marginal revenue) for monthly sales of 20,000 hamburgers.

11. Finding Marginal Revenue  Because the total revenue is given by  R = xp, you have  and the marginal revenue is

12. Finding Marginal Revenue  When x = 20,000 the marginal revenue  Now lets look at the graph. Notice that as the price decreases, more hamburgers are sold. (Make sense?)

13. Finding Marginal Profit  Suppose that the cost of producing those same x hamburgers is  C = 5000 + 0.56x  (Fixed costs = $5000; variable costs are 56¢ per hamburger)

14. Finding Marginal Profit  Find the total profit and the marginal profit for 20,000, for 24,400, and for 30,000 units.  Solution: Because P = R – C, you can use the revenue function to obtain

15. Finding Marginal Profit

16.  Now do the derivative to find a marginal Demand20,00024,40030,000Profit$23,800$24,768$23,200 Marginal profit $0.44$0.00 - $0.56

17. Finding Maximum Profit  In marketing a certain item, a business has discovered that the demand for the item is  The cost of producing x items is given by C = 0.5x + 500. Find the price per unit that yields a maximum profit.

18. Finding Maximum Profit  From the given cost function, you obtain  P = R – C = xp – (0.5x + 500).  Substituting for p (from the demand function) produces

19. Finding Maximum Profit  To find maximums using calculus, you now do a derivative of the profit equation and then set it equal to zero.

20. Finding Maximum Profit  This gives us the number of units needed to be produced in order to get the maximum profit. How do we find the price?

21. Minimizing the Average Cost  A company estimates that the cost (in dollars) of producing x units of a certain product is given by  C = 800 + 0.4x + 0.0002x 2.  Find the production level that minimizes the average cost per unit.

22. Minimizing Average Cost  Substituting from the given equation for C produces

23. Practice  Your turn  Homework (From CD Appendix G)  p. G5 problems 1 – 33 odd