§2.4 Optimization.

1. §2.4 Optimization. The student will learn how to optimization of a function.

2. Review of Business Functions Total Cost Function C (x) = a + b x (Where a is the fixed cost and b x is the variable cost.) Price Function p (x) (Which gives the price p at which consumers will buy exactly x units of the product.) Revenue Function R = x p (where p is the unit price and x is the quantity sold.) Profit Function P = R – C

3. Average Cost Function Marginal Average Cost Function Review of Business Functions Total Cost Function C (x) = a + b x Marginal Cost Function C ' (x) The same is true for revenue, price and profit. Consider going to my website and downloading the handout on business functions.

4. Review of Business Functions We are going to see how to maximize or minimize these business functions. This is why you are in this course! 4

5. (5000, 500000) from calculator R (x) Example 1 A company manufactures and sells x television sets per month. The monthly cost and price-demand equations are: C (x) = 60,000 + 60x and p = 200 – x/50 for 0 ≤ x ≤ 8,000 a. Find the maximum revenue. R (x) = xp = How do we maximize? Explain! R ‘ (x) = Maximum at R ‘ (x) = 0 or x = 5,000, and 0  x  8,000 0  y  600,000. R(5000) = $500,000 Explain!

6. P (x) (3500, 185000) From calculator. Example 1 continued C (x) = 60,000 + 60x and R = 200x – x 2/50 for 0 ≤ x ≤ 8,000 b. Find the maximum profit and the production level that will realize the maximum profit. P (x) = WOW! How do we maximize? P ‘ (x) = Solving P ‘ (x) = 0 gives a value at x = 3500. P (3500) = $185,000 0  x  8,000 0  y  600,000. Production level Maximum profit

7. R C P Example 1 continued C (x) = 60,000 + 60x and R = 200x – x 2/50 for 0 ≤ x ≤ 8,000 c. Graph the cost and the revenue and the profit functions on the same graph. P (x) = Note the break even points (profit is 0). Note that max profit and max revenue do not occur at the same x value. Max R at x = 5000. 0  x  8,000 0  y  600,000. Max P at x = 3500.

8. R C P Example 1 continued C (x) = 60,000 + 60x and R = 200x – x 2/50 for 0 ≤ x ≤ 8,000 This brings us to a classic economic criteria for maximum profit C ’ = R ’ 60 = 200 – x/25 x/25 = 140 x = 3500 as before.

9. Example 1 concluded C (x) = 60,000 + 60x and p = 200 – x/50 for 0 ≤ x ≤ 8,000 R = 200x – x 2/50 and P(x) = Maximum revenue of $500,000 occurred at a sales level of 5,000. Maximum profit of $185,000 occurred at a sales level of 3,500. What price should you charge to maximize your profit????? How many do you want to sell? x = 3500 as before. p (3500) = 200 – 3500/50 = 200 – 70 = $130 powerful Now that is some mathematics!

10. Maximizing Tax Revenue If a tax on imports is too high, fewer will be sold and tax revenues will go down. To maximize revenue we need to find the tax rate that will produce the appropriate sales so that revenue is maximized. Suppose that the relationship between the tax rate t on an item and its total sales S is S (t) = 4 – 6 At a tax rate of t = 0, sales will be S (0) = 4 – 6 Sales = 4 million dollars. What is the tax revenue? $ 0 At a tax rate of t = .08, sales will be S (0.08) = 4 – 6 Sales = 1.41 million dollars. What is the revenue? $ 0.11 million Continued

11. 0 ≤ x ≤ 0.2 0 ≤ y ≤ 0.15 Tax Revenue Continued If a tax on imports is too high, fewer will be sold and tax revenues will go down. To maximize revenue we need to find the tax rate that will produce the appropriate sales so that revenue is maximized. S (t) = 4 – 6 The previous has shown the relationship between tax rate and sales. But we want the total revenue. R (t) = rate · sales = R (t) = rate · sales = t · S (t) = t (4 – 6 ) = 4t – 6 t 4/3 Maximize R with your calculator! t = 0.125 or 12.5%

12. Optimization Although all of the optimizations we did today involved maximums it is also possible to optimize using minimums. For instance one might want to minimize the amount of materials or labor used in making a product. To do this follow the same methods we used today but when graphing use the minimum option under the appropriate menu on your calculator.

13. Summary. • We reviewed basic business functions. • We learned how to optimize a function using our calculators and a derivative. • We learned how to optimization a tax revenue function.

14. ASSIGNMENT §2.4 on my website. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.