1 Lesson 10: Applications of Derivatives and Differentiation

Transcription

1 1 Lesson 10: Applications of Derivatives and Differentiation Chapter 5 Material: pages in the textbook: As in Lesson 9, this lesson covers a few applications that illustrate the use of derivatives in mathematical modeling of business applications. Again, the idea is to present applications that use the various tools developed in Chapter 4 that are related to the derivatives. The examples in this lesson visit different applications that require some of the same calculus techniques along with a few alternate ways to look at the mathematical models that arise in these types of applications. The applications are listed below. Elasticity of Demand: Section 5.4 presents a very standard concept in a university level business curriculum. Elasticity of demand refers to how elastic the demand for a product or service will remain when small changes are made to the inputs. Small changes usually are modeled via derivatives. Productivity Models: One of the most common applications in business calculus courses involves the mathematical model for productivity. The content in Section 5.5 presents the simple Cobb-Douglas productivity function along with some extensions. Several techniques from Chapter 4 in the textbook are used to analyze these models. The Law of Diminishing Returns: This concept appears in many applications including business and economics. The law can also be used in logistic models of populations to describe certain processes. The basic idea is that even though a profit function may be increasing, the rate of increase will be decreasing. The point at which the rate of increase of the profit is no longer increasing is called the point of diminishing returns. There are a number of applications where this is important. For example, how much additional capital should be invested in continuing a project or additional advertising should be used to sell a product. If the cost is not worth the additional gain, the process has likely crossed a point of diminishing returns. The last section in Chapter 5 spends a bit of time applying related rates to productivity functions like the Cobb-Douglas model. The applications in this lesson and Chapter 5 in general have been chosen to illustrate the application of various calculus techniques related to the derivative. It is very important to emphasize that these examples are simple versions of more complicated models and that the calculus techniques illustrated in the mathematical models of these applications can be useful in the analysis of more complicated models. Prerequisite Content for Lesson 9 Students should understand all of the content in Chapter 4. The techniques developed in Chapter 4 are used in the specific mathematical models to illustrate the calculus techniques. In this lesson students must be able to compute derivatives of elementary functions, understand the algebraic properties of differentiable functions, and apply the chain rule and differentials. The algebraic properties are used in models of most of the examples in this lesson. Formulas like η = p x dx dp = p x dp dx are used to provide point elasticity estimates predictions of profit. Students should understand that the derivatives x (p) and p (x) are reciprocals of each other. In addition, students must know how to apply implicit differentiation to equations where there are a number of terms. The result may include applications of derivatives to sums and differences of terms, applications of the product rule (e.g; the Cobb-Douglas model), or any other basic algebraic rule of differentiation. Students must know how to apply the chain rule in determining solutions of related rates problems and as these methods apply to productivity models. The second derivative is used in the law of diminishing returns. Students must be able able to apply an interval analysis on the second derivative to do this. Finally, students must be able to apply differentials in the elasticity of demand application in this lesson. Goals and Objectives for Lesson 10 Students who complete this lesson should be able to: 1

2 understand the application of algebraic properties of derivatives in the analysis of real world mathematical models, use the chain rule effectively in related rates and implicit differentiation techniques, use differentials as illustrated in the section of force of interest, and use the second derivative to find the point of diminishing returns in applications. The goals and objectives of this lesson are less mathematical and more practical than other parts of the the course. 2

3 1.1 Lecture Notes for Lesson 10 - Day 1 This lesson works through a number of additional mathematical models of business applications. The first of the applications in this lesson involves the analysis of models of elasticity of demand. The basic definitions of terms used in the modeling process are presented as the text progresses in Section 5.4. Elasticity of Demand In the first part of the lesson, it is very important to spend time on the difference between the discrete and continuous mathematical models of elasticity of demand. Note that there are a lot of applications that use the term elasticity and students will need to focus only on those definitions included in Section 5.4. These definitions have been taken from a number of sources in the business literature. The initial definition of Price Elasticity of Demand in Definition 60 is the starting point for the mathematical models in the section. Make sure students understand this particular definition. On page 224, a discrete version of the price elasticity of demand is given. There are actually a number of different discrete models of this concept that could be used. The definition oc Arc Elasticity is one possible version of a model. The point price demand of elasticity is the instantaneous rate of change or the derivative at the current production level. This means that the particular definition is not unique. Point Price Demand of Elasticity One algebraic property of the derivative is the reciprocal relationship on page 225 between p (x) and x (p). In Chapter 4 a great deal of time was taken to determine the independent and dependent variable relationship in models. This application is one place where this is important. In some cases, x (p) may be easier to compute and in other cases, p (x) may be easier to compute. Since the these are reciprocals of each other, it is easy to compute one from the other. Example 171 illustrates the computation of the elasticity parameter η. Emphasize that if η > 1 the demand will be elastic and if η < 1 the demand will be inelastic. Differentials and Elasticity Applications Section presents the same concept of elasticity in terms of differentials. You should emphasize that the relationship written in terms of differentials displays how small changes in the price, dp, and small changes in demand, dx, effect the relationship. Example 172 illustrates this process. 3

4 1.2 Lecture Notes for Lesson 10 - Day 2 In most business calculus textbooks, the Cobb-Douglas model is used often to illustrate various properties of differentiable functions. The Cobb-Douglas model is a relatively simple mathematical model of a productivity function. The Cobb-Douglas model is usually restricted to two inputs. In many examples, the two inputs are captial investment and labor investment. In more general models of productivity, any number of inputs can be included. For a first course in calculus these models are too complicated. In order to study calculus applied to functions of a single real variable, the number of inputs needs to be restricted to make any progress. Productivity Functions Definition 63 provides an illustration that general productivity functions are very complicated. If this definition is causing confusion amongst your students it is ok to skip this part and move on to the simpler Cobb-Douglas model in the on page 228. Students should understand that the Cobb-Douglas model is a idealized model of productivty. In general, the real world application of these types of models is much more complicated. Students should also understand that the use of simpler models makes it easier to illustrate application of calculus techniques to these models. Once students understand the application of the models to simple models, more complicated versions of the process can be analyzed. Productivity Functions: The Cobb-Douglas Model The Cobb-Douglas model is defined on page 228. Students need to understand Definition 64 in order to be successful in this lesson. Examples 173 and 174 should be used to illustrate how to use the mathematical model. Page 230 includes three cases to consider in analyzing the Cobb-Douglas model. In most textbooks, the first case is the only case used. There is a brief example after each of the definitions to illustrat the differences. You should work through each of the definitions on page 230 along with the brief calculation following each of the definitions. Implicit Differentiation Section is where calculus techniques start being used on these models. Students will need to use implicit differentiation to work through Example 175 and be able to understand how to compute the rate of change associated with each of the inputs. Make sure that students understand the reciprocal relationship that appears on page

5 1.3 Lecture Notes for Lesson 10 - Day 3 Two applications are covered in the last part of this lesson. The first is the law of diminishing returns and the other is an application of the concept of related rates to the simple Cobb-Douglas model discussed earlier. The Law of Diminishing Returns The law of diminishing returns is articulated in Definition 68. The definition needs to be translated into mathematical terms. To do this, Definition 69 states that the point of diminishing returns involves an inflection point in a function. In particular, the focus of Definition 69 is when there is a point of diminishing returns in a profit function. Inflection points are locations where a point of diminishing returns will be found. Example 176 illustrates the computations to find the point of diminishing returns. This is a good application of interval analysis on the second derivative. Emphasize that an inflection point requires two properties. These are (1) the second derivative is zero, and (2) the sign of the second derivative changes. Both mathematical properties are necessary to find an inflection point and in terms of this particular application, both properties are needed to define a point where a point of diminishing returns occurs in the mathematical model. Related Rates Applied to the Cobb-Douglas Model Section 5.7 revisits the Cobb-Douglas model in terms of related rates. The Cobb-Douglas model is modified to include the effect of a time variable. To explain the more complicated model you can tell students that if the two inputs are capital investment and increasing a labor force for a company, both inputs may vary as a function of time. The rate of change of productivity as a function of time requires the rate of change of time with respect to each of the inputs. Example 177 illustrates this. In the second edition of the book, I will include more examples and more on related rates. 5

6 1.4 Homework Price Elasticity of Demand Problem 1: In each of the problems compute the price elasticity of demand using the same approach as in Example 171 on page 226. a. R(x) = 100 x 0.03 x x 3, x = 1200, p = 28 b. R(x) = 210 x x x 3, x = 2300, p = 15 6

7 c. R(x) = 210 x x ln(x) x 2 e x, x = 2300, p = 15 7

8 Elasticity for a Profit Model Problem 2: In this problem, mathematical models of profit obtained from the production and sales of an item are analyzed in terms of the point price elasticity of demand. Suppose that two mathematical models of a profit function are given as follows. and P 1 (x) = 85x (0.003) x 2 ln(x) 1400 P 2 (x) = 85x (0.003) x Use the model that profit is equal to revenue minus cost with C(x) = 15x to write the revenue in terms of a price multiplied by the demand level, x, and then find the elasticity parameter from this relationship (see Example 171). Compare the two models. 8

9 Differentials and Price Elasticity of Demand Problem 3: Using the formula η x dp = p dx relating differentials in the elasticity application do the following problems. a. With η = 1/2, p = 20, x = 675, and the differential in demand dx = compute the differential in price, dp. b. With η = 1/2, p = 20, x = 675, and the differential in demand dx = compute the differential in price, dp. 9

10 c. With η = 1/2, p = 20, x = 675, and the differential in demand dp = compute the differential in price, dx. d. Compare the results obtained in the first three parts of this problem. 10

11 Productivity: Cobb-Douglas Models Problem 4: Given the Cobb-Douglas model P (x, y) = A x a y b with A = 195, a = 0.35, and b = a. Compute P (50, 60). b. Suppose that P (x, y) = 450, compute y (x). 11

12 c. Suppose that P (x, y) = 450, compute x (y). d. Suppose that x = 50, compute P (y). e. Suppose that y = 60, compute P (x). 12

13 Comparison of Cases: Cobb-Douglas Model Problem 5: Using P (x, y) = A x a y b with A = 195, x = 50, and y = 60. a. Given a = 0.25 and a + b = 1. compute P (x, y). b. Given a = 0.25 and a + b = compute P (x, y). 13

14 c. Given a = 0.25 and a + b = compute P (x, y). d. Compare the results obtained in the other parts of the problem. 14

15 Reciprocal Relation for Derivatives Problem 6: For the Cobb-Douglas model P (x, y) = 295 x 0.15 y 0.85 with P (x, y) = 699 compute x (y) using implicit differentiation. Use the reciprocal relationship for derivatives to compute y (x). 15

16 Interval Analysis of the Second Derivative Problem 7: For each of the following functions, compute intervals on which the second derivative is positive or negative, and points where the second derivative is zero. Determine which of these point(s) are inflection points. This analysis is an interval analysis of the second derivative. a. f(x) = x e 2x2 b. f(x) = 3 x 2 2 x 3 16

17 c. g(x) = e x 1 + e x 17

18 Point of Diminishing Returns Problem 8: Find the point of diminishing returns for each of the following functions. a. P (x) = 10 x 0.07 x x 3 b. g(x) = e x 1 + e x 18

19 Related Rates: Cobb-Douglas Revisited Problem 9: In each of the following problem, compute the desired rate given the rest of the indicated information. a. Compute P (t) given P (t) = 145 (x(t)) 0.27 (y(t)) 0.73 and x (t) = 3.7 and y (t) = 1.6. b. Compute y (t) given P (t) = 145 (x(t)) 0.27 (y(t)) 0.73 and x (t) = 3.7 and P (t) =