Copyright © Cengage Learning. All rights reserved. 3 Applications of the Derivative

Copyright © Cengage Learning. All rights reserved. Business and Economics Applications 3.5

33  Solve business and economics optimization problems.  Find the price elasticity of demand for demand functions.  Recognize basic business terms and formulas. Objectives

44 Optimization in Business and Economics

55 Example 1 – Finding the Maximum Revenue A company has determined that its total revenue (in dollars) for a product can be modeled by where x is the number of units produced (and sold). What production level will yield a maximum revenue?

66 1. A sketch of the revenue function is shown in Figure Figure 3.37 Maximum revenue occurs when dR/dx = 0. Example 1 – Solution

77 2. The primary equation is the given revenue function. 3. Because R is already given as a function of one variable, you do not need a secondary equation. 4. The feasible domain of the primary equation is This is determined by finding the x-intercepts of the revenue function, as shown in Figure cont’d

88 5. To maximize the revenue, find the critical numbers. The only critical number in the feasible domain is x = 350. From the graph of the function, you can see that the production level of 350 units corresponds to a maximum revenue. Example 1 – Solution cont’d

99 Optimization in Business and Economics To study the effects of production levels on cost, one method economists use is the average cost function which is defined as where C = f (x) is the total cost function and x is the number of units produced.

10 Example 2 – Finding the Minimum Average Cost A company estimates that the cost (in dollars) of producing x units of a product can be modeled by Find the production level that minimizes the average cost per unit.

11 1. C represents the total cost, x represents the number of units produced, and represents the average cost per unit. 2. The primary equation is Example 2 – Solution

12 3. Substituting the given equation for C produces 4. The feasible domain of this function is because the company cannot produce a negative number of units. Example 2 – Solution cont’d

13 5. You can find the critical numbers as shown. Example 2 – Solution cont’d

14 By choosing the positive value of x and sketching the graph of as shown in Figure 3.38, you can see that a production level of x = 2000 minimizes the average cost per unit. Minimum average cost occurs when Figure 3.38 Example 2 – Solution cont’d

15 Price Elasticity of Demand

16 One way in which economists measure the responsiveness of consumers to a change in the price of a product is with price elasticity of demand. For example, a drop in the price of fresh tomatoes might result in a much greater demand for fresh tomatoes; such a demand is called elastic. On the other hand, the demand for items such as coffee and gasoline is relatively unresponsive to changes in price; the demand for such items is called inelastic. Price Elasticity of Demand

17 More formally, the elasticity of demand is the percent change of a quantity demanded x, divided by the percent change in its price p. You can develop a formula for price elasticity of demand using the approximation which is based on the definition of the derivative. Using this approximation, you can write Price Elasticity of Demand

18 Price Elasticity of Demand

19 Price Elasticity of Demand Price elasticity of demand is related to the total revenue function, as indicated in Figure 3.42 and the list below. 1. If the demand is elastic, then a decrease in price is accompanied by an increase in unit sales sufficient to increase the total revenue. Figure 3.42 Revenue Curve

20 Price Elasticity of Demand 2. If the demand is inelastic, then a decrease in price is not accompanied by an increase in unit sales sufficient to increase the total revenue.

21 Example 5 – Comparing Elasticity and Revenue The demand function for a product is modeled by where p is the price per unit (in dollars) and x is the number of units. (See Figure 3.43.) a. Determine when the demand is elastic, inelastic, and of unit elasticity. b. Use the result of part (a) to describe the behavior of the revenue function. Figure 3.43

22 Example 5(a) – Solution The price elasticity of demand is given by

23 Example 5(a) – Solution The demand is of unit elasticity when In the interval [0, 144], the only solution of the equation is x = 64. cont’d

24 So, the demand is of unit elasticity when x = 64. For x-values in the interval (0, 64), which implies that the demand is elastic when 0 < x < 64. For x-values in the interval (64, 144), which implies that the demand is inelastic when 64 < x <144. Example 5(a) – Solution cont’d

25 From part (a), you can conclude that the revenue function R is increasing on the open interval (0, 64), is decreasing on the open interval (64, 144) and is a maximum when x = 64, as indicated in Figure Example 5(b) – Solution cont’d Figure 3.44

26 Business Terms and Formulas

27 Business Terms and Formulas This section concludes with a summary of the basic business terms and formulas used in this section.

28 Business Terms and Formulas A summary of the graphs of the demand, revenue, cost, and profit functions is shown in Figure Demand Function Quantity demanded increases as price decreases. Revenue Function The low prices required to sell more units eventually result in a decreasing revenue. Figure 3.45

29 Business Terms and Formulas Cost Function The total cost to produce x units includes the fixed cost. Profit Function The break-even point occurs when R = C. Figure 3.45 cont’d