1. Copyright © Cengage Learning. All rights reserved. 2 Differentiation

2. Copyright © Cengage Learning. All rights reserved. Rates of Change: Velocity and Marginals 2.3

3. 33  Find the average rates of change of functions over intervals.  Find the instantaneous rates of change of functions at points.  Find the marginal revenues, marginal costs, and marginal profits for products. Objectives

4. 44 Average Rate of Change

5. 55 You have studied the two primary applications of derivatives. 1. Slope The derivative of f is a function that gives the slope of f the graph of at a point (x, f (x)). 2. Rate of Change The derivative of f is a function that gives the rate of change of f (x) with respect to x at the point (x, f (x)). Average Rate of Change

6. 66 In this section, you will see that there are many real-life applications of rates of change. A few are velocity, acceleration, population growth rates. Although rates of change often involve change with respect to time, you can investigate the rate of change of one variable with respect to any other related variable. When determining the rate of change of one variable with respect to another, you must be careful to distinguish between average and instantaneous rates of change. Average Rate of Change

7. 77 The distinction between these two rates of change is comparable to the distinction between the slope of the secant line through two points on a graph and the slope of the tangent line at one point on the graph. Average Rate of Change

8. 88 Figure 2.18

9. 99 A common application of an average rate of change is to find the average velocity of an object that is moving in a straight line. That is, This formula is demonstrated in Example 2. Average Rate of Change

10. 10 Example 2 – Finding an Average Velocity A free-falling object is dropped from a height of 100 feet. Neglecting air resistance, the height h (in feet) of the object at time t (in seconds) is given by Some falling objects have considerable air resistance. Other falling objects have negligible air resistance. When modeling a falling-body problem, you must decide whether to account for air resistance or neglect it. Figure 2.20

11. 11 Example 2 – Finding an Average Velocity Find the average velocity of the object over each interval. a. [1, 2] b. [1, 1.5] c. [1, 1.1] cont’d

12. 12 Example 2 – Solution You can use the position equation h = –16t 2 + 100 to determine the heights at t = 1, 1.1, 1.5, and 2 as shown in the table. a. For the interval [1, 2], the object falls from a height of 84 feet to a height of 36 feet. So, the average velocity is

13. 13 Example 2 – Solution b. For the interval [1, 1.5] the average velocity is c. For the interval [1, 1.1] the average velocity is cont’d

14. 14 Instantaneous Rate of Change and Velocity

15. 15 Suppose in Example 2 you wanted to find the rate of change of h at the instant t = 1 second. Such a rate is called an instantaneous rate of change. You can approximate the instantaneous rate of change at t = 1 by calculating the average rate of change over smaller and smaller intervals of the form [1, 1 +  t ] as shown in the table. Instantaneous Rate of Change and Velocity

16. 16 From the table, it seems reasonable to conclude that the instantaneous rate of change of the height at t = 1 is –32 feet per second. Instantaneous Rate of Change and Velocity

17. 17 The general position function for a free-falling object, neglecting air resistance, is where h is the height (in feet), t is the time (in seconds), v 0 is the initial velocity (in feet per second), and h 0 is the initial height (in feet). Remember that the model assumes that positive velocities indicate upward motion and negative velocities indicate downward motion. Instantaneous Rate of Change and Velocity

18. 18 The derivative is the velocity function. The absolute value of the velocity is the speed of the object. Instantaneous Rate of Change and Velocity

19. 19 Example 4 – Finding the Velocity of a Diver At time t = 0, a diver jumps from a diving board that is 32 feet high, as shown in Figure 2.21. Because the diver’s initial velocity is 16 feet per second, the position of the diver is given by a. When does the diver hit the water? b. What is the diver’s velocity at impact? Figure 2.21

20. 20 Example 4(a) – Solution To find the time at which the diver hits the water, let h = 0 and solve for t. The solution t = –1 does not make sense in the problem because it would mean that the diver hits the water 1 second before jumping. So, you can conclude that the diver hits the water at t = 2 seconds.

21. 21 Example 4(b) – Solution The velocity at time is given by the derivative The velocity at time t = 2 is cont’d

22. 22 In Example 4, note that the diver’s initial velocity is v 0 = 16 feet per second (upward) and the diver’s initial height is h 0 = 32 feet. Instantaneous Rate of Change and Velocity

23. 23 Rates of Change in Economics: Marginals

24. 24 Another important use of rates of change is in the field of economics. Economists refer to marginal profit, marginal revenue, and marginal cost as the rates of change of the profit, revenue, and cost with respect to x, the number of units produced or sold. An equation that relates these three quantities is where P, R, and C represent the following quantities. P = total profit, R = total revenue, and C = total cost Rates of Change in Economics: Marginals

25. 25 The derivatives of these quantities are called the marginal profit, marginal revenue, and marginal cost, respectively. Rates of Change in Economics: Marginals

26. 26 In many business and economics problems, the number of units produced or sold is restricted to nonnegative integer values, as indicated in Figure 2.22. (Of course, it could happen that a sale involves half or quarter units, but it is hard to conceive of a sale involving units.) The variable that denotes such units is called a discrete variable. Rates of Change in Economics: Marginals Figure 2.22 Function of a Discrete Variable

27. 27 To analyze a function of a discrete variable x, you can temporarily assume that x is a continuous variable and is able to take on any real value in a given interval, as indicated in Figure 2.23. Rates of Change in Economics: Marginals Figure 2.23 Function of a Continuous Variable

28. 28 Then, you can use the methods of calculus to find the x-value that corresponds to the marginal revenue, maximum profit, minimum cost, or whatever is called for. Finally, you should round the solution to the nearest sensible x-value—cents, dollars, units, or days, depending on the context of the problem. Rates of Change in Economics: Marginals

29. 29 Example 5 – Finding the Marginal Profit The profit derived from selling x units of an alarm clock is given by a. Find the marginal profit for a production level of 50 units. b. Compare the marginal profit with the actual gain in profit obtained by increasing the production level from 50 to 51 units. Solution: a. The profit is The marginal profit is given by the derivative

30. 30 Example 5 – Solution When x = 50, the marginal profit is cont’d

31. 31 Example 5 – Solution b. For x = 50, the actual profit is and for x = 51, the actual profit is cont’d

32. 32 Example 5 – Solution So, the additional profit obtained by increasing production from 50 to 51 units is cont’d

33. 33 Example 5 – Solution Note that the actual profit increase of $11.53 (when x increases from 50 to 51 units) can be approximated by the marginal profit of $11.50 per unit (when x = 50), as shown in Figure 2.24. cont’d Figure 2.24

34. 34 The profit function in Example 5 is unusual in that the profit continues to increase as long as the number of units sold increases. In practice, it is more common to encounter situations in which sales can be increased only by lowering the price per item. Such reductions in price will ultimately cause the profit to decline. Rates of Change in Economics: Marginals

35. 35 The number of units x that consumers are willing to purchase at a given price per unit p is given by the demand function The total revenue R is then related to the price per unit and the quantity demanded (or sold) by the equation Rates of Change in Economics: Marginals

36. 36 Example 7 – Finding the Marginal Revenue A fast-food restaurant has determined that the monthly demand for its hamburgers is given by Figure 2.26 shows that as the price decreases, the quantity demanded increases. Figure 2.26 As the price decreases, more hamburgers are sold.

37. 37 Example 7 – Finding the Marginal Revenue The table shows the demands for hamburgers at various prices. Find the increase in revenue per hamburger for monthly sales of 20,000 hamburgers. In other words, find the marginal revenue when x = 20,000. cont’d

38. 38 Example 7 – Solution Because the demand is given by and the revenue is given by R = xp, you have

39. 39 Example 7 – Solution By differentiating, you can find the marginal revenue to be So, at x = 20,000, the marginal revenue is cont’d

40. 40 Example 7 – Solution So, for monthly sales of 20,000 hamburgers, you can conclude that the increase in revenue per hamburger is $1. cont’d

41. 41 Example 8 – Finding the Marginal Profit For the fast-food restaurant in Example 7, the cost of producing hamburgers is Find the profit and the marginal profit for each production level. a. x = 20,000b. x = 24,400 c. x = 30,000

42. 42 Example 8 – Solution From Example 7, you know that the total revenue from selling x hamburgers is Because the total profit is given by P = R – C, you have

43. 43 Example 8 – Solution So, the marginal profit is cont’d Figure 2.27

44. 44 Example 8 – Solution Using these formulas, you can compute the profit and marginal profit. cont’d