1. Copyright © 2016, 2012 Pearson Education, Inc. 2.6 - 1

2. 2.6 Copyright © 2016, 2012 Pearson Education, Inc. 2.6 - 2 Marginals and Differentials OBJECTIVE Find marginal cost, revenue, and profit. Find ∆y and dy. Use differentials for approximations.

3. Copyright © 2016, 2012 Pearson Education, Inc. 2.6 - 3 DEFINITIONS: Let C(x), R(x), and P(x) represent, respectively, the total cost, revenue, and profit from the production and sale of x items. The marginal cost at x, given by C (x), is the approximate cost of the (x + 1) th item: C (x) ≈ C(x + 1) – C(x), or C(x + 1) ≈ C(x) + C (x). 2.6 Marginals and Differentials

4. Copyright © 2016, 2012 Pearson Education, Inc. 2.6 - 4 DEFINITIONS (concluded): The marginal revenue at x, given by R (x), is the approximate revenue from the (x + 1) th item: R (x) ≈ R(x + 1) – R(x), or R(x + 1) ≈ R(x) + R (x). The marginal profit at x, given by P (x), is the approximate profit from the (x + 1) th item: P (x) ≈ P(x + 1) – P(x), or P(x + 1) ≈ P(x) + P (x). 2.6 Marginals and Differentials

5. Copyright © 2016, 2012 Pearson Education, Inc. 2.6 - 5 Example 1: Given find each of the following: a) Total profit, P(x). b) Total cost, revenue, and profit from the production and sale of 50 units of the product. c) The marginal cost, revenue, and profit when 50 units are produced and sold. 2.6 Marginals and Differentials

6. Copyright © 2016, 2012 Pearson Education, Inc. 2.6 - 6 Example 1 (continued): a) b) 2.6 Marginals and Differentials

7. Copyright © 2016, 2012 Pearson Education, Inc. 2.6 - 7 Example 1 (concluded): c) So, when 50 units have been made, the approximate cost of the 51 st unit will be $6200, and the approximate revenue from the sale of the 51 st unit will be $6340 for an approximate profit on the 51 st unit of $140. 2.6 Marginals and Differentials

8. Copyright © 2016, 2012 Pearson Education, Inc. 2.6 - 8 Example 2: For x = 4, and ∆x = 0.1, find ∆y. 2.6 Marginals and Differentials

9. Copyright © 2016, 2012 Pearson Education, Inc. 2.6 - 9 2.6 Marginals and Differentials Quick Check 1 For, and, find.

10. Copyright © 2016, 2012 Pearson Education, Inc. 2.6 - 10 For f a continuous, differentiable function, and small ∆x. 2.6 Marginals and Differentials

11. Copyright © 2016, 2012 Pearson Education, Inc. 2.6 - 11 Example 3: Approximate using Let and x equal a number close to 27 and A number for which the square root is easy to compute. So, here let x = 25 with ∆x = 2. Then, 2.6 Marginals and Differentials

12. Copyright © 2016, 2012 Pearson Education, Inc. 2.6 - 12 Example 3 (concluded): Now we can approximate 2.6 Marginals and Differentials

13. Copyright © 2016, 2012 Pearson Education, Inc. 2.6 - 13 2.6 Marginals and Differentials Quick Check 2 Approximate using. To five decimal places,. How close is your approximation? Let and equal a number close to 98 and a number for which the square root is easy to compute. So here let with. Then, and

14. Copyright © 2016, 2012 Pearson Education, Inc. 2.6 - 14 2.6 Marginals and Differentials Quick Check 2 Concluded Now we can approximate : This is within 0.001 of the actual value of

15. Copyright © 2016, 2012 Pearson Education, Inc. 2.6 - 15 DEFINITION: For y = f (x), we define dx, called the differential of x, by dx = ∆x and dy, called the differential of y, by dy = f   (x)dx. 2.6 Marginals and Differentials

16. Copyright © 2016, 2012 Pearson Education, Inc. 2.6 - 16 Example 4: For a) Find dy. b) Find dy when x = 5 and dx = 0.2. a) 2.6 Marginals and Differentials

17. Copyright © 2016, 2012 Pearson Education, Inc. 2.6 - 17 Example 4 (concluded): b) 2.6 Marginals and Differentials

18. Copyright © 2016, 2012 Pearson Education, Inc. 2.6 - 18 2.6 Marginals and Differentials Section Summary If represents the cost for producing items, then marginal cost is its derivative, and. Thus, the cost to produce the can be approximated by If represents the revenue from selling items, then marginal revenue is its derivative, and. Thus, the revenue from the item can be approximated by

19. Copyright © 2016, 2012 Pearson Education, Inc. 2.6 - 19 2.6 Marginals and Differentials Section Summary Continued If represents profit from selling items, then marginal profit a is its derivative, and. Thus, the profit from the item can be approximated by In general, profit = revenue – cost, or In delta notation,, and. For small values of, we have which is equivalent to The differential of is. Since, we have. In general,, and the approximation can be very close for sufficiently small.