A Calculus Approach to Mathematical Modeling. Marcel B. Finan Arkansas Tech University All Rights Reserved

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1 A Calculus Approach to Mathematical Modeling Marcel B. Finan Arkansas Tech University All Rights Reserved

2 2 PREFACE Mathematical modeling is a fastly developing science which is vitally important to students majoring in the disciplince of mathematics. Not only should they be exposed to its utility, but they should also gain some basic experience and skill in using mathematics to provide quantified solutions to an increasing number of society s technical problems. The aim of this course is to teach undergraduate students how to become not only competent mathematicians but also skilled users of mathematics in the solution of problems arising in the real world. This will be achieved by requiring students to work on problems which give them some experience of applying their mathematical knowledge to the sort of problems that arise in industry and commerce. The only prerequisite for this course is a course in differential and integral calculus of one single variable. Marcel B. Finan Russellville, Arkansas May 2002.

3 Contents 1 The Modeling Process Introductory Remarks A Mathematical Modeling Approach An Example of a Mathematical Model Models in Management and Economics Mathematics of Finance Short-Term Financing: Simple Interest Long-Term Financing: Compound Interest Loans or Investments Problems: Annuity Amortization and Sinking Funds Linear Programming and the Simplex Method Miscellaneous Models Models in Physics and Engineering Sciences Models Based on 1st Order Differential Equations Models Based on 2nd Order Linear Differential Equations Models Based on 1st Order Linear Difference Equations Models Based on Second Order Difference Equations Miscellaneous Models Models in Computer Sciences Cryptography Basic Cryptographic Algorithms Asymmetric Algorithms: RSA Algorithm Finite-State Automaton Introduction to the Analysis of Algorithms Time Complexity and O-Notation Logarithmic and Exponential Complexities Θ- and Ω-Notations

4 4 CONTENTS

5 Chapter 1 The Modeling Process 1.1 Introductory Remarks The process of representing a phenomenon mathematically, i.e. by means of a function or an equation, is referred to as mathematical modeling. In this chapter we will introduce the reader to the basic elements of mathematical modeling. We will discuss how mathematical models are formulated, solved and applied, and a concise description of the mathematical techniques used in the process. The models in this course have been taken mainly from management, economics, physics, engineering, life sciences, and computer sciences. 1.2 A Mathematical Modeling Approach The mathematical modeling approach to problem solving that we will adopt in this course consists of the following five steps: 1. Ask a question. 2. Set up a model. 3. Formulate the mathematical model. 4. Solve the mathematical model. 5. Answer the question. We can summarise these steps of modeling into three stages:formulation, solution, and application. The formulation stage consists of steps 1 through 3. The solution stage consists of step 4, and the application stage consists of step 5. These stages are important in modeling however not all modeling will follow this exact pattern. This is just a guide to what modeling is about. 5

6 6 CHAPTER 1. THE MODELING PROCESS 1.3 An Example of a Mathematical Model In this section we discuss a simple example to illustrate the modeling process discussed in the previous section: Ms. Brown has $1, 000 to invest. If invested with the NBA bank, interest will be compounded annually at 7%, but if invested with the NFL bank, interest is paid at the nominal annual rate of 6.9% but compounded monthly. Ms. Brown s problem is to know which account will, at the end of the year, have the larger value. Step 1. We would like to know which choice is the best for Ms. Brown. Would it be investing with NBA bank or the NFL bank? Step 2. We introduce the variables that we need in our model. Let P be the present value or the money to be invested, i.e. P = $1, 000; S be the future value or the amount to which P would have grown if deposited in an interest bearing bank account; r be the annual nominal rate (APR) compounded n times a year; and t be the period of investment. Step 3. In this step we will find the relationship between P, A, r, n, and t. At the end of the first period, the balance is P + r n P = P (1 + r n ). At the end of the second period the balance in the account is P (1 + r n ) + r n P (1 + r n ) = P (1 + r n )2. Continuing in this fashion, we find that the balance at the end of the third period is P (1 + r n )3 and at the end of the first year the balance in the account will be P (1 + r n )n. Repeating this, we find that the balance at the end of the second year of investment is P (1 + r n )2n and if the investment is over t years then the amount of money in the account will be A = P (1 + r n )nt. Step 4. Next we use the data given in the problem. We will see which deal is better by computing the effective annual rate. In this case, t = 1. If money is invested in the NBA bank, then r =.07 and n = 1 so that A = 1, 000(1 +.07) = $1, 070 and therefore the effective annual rate is 7% which is the nominal interest rate. If money is rather invested in the NFL bank then r =.069 and n = 12 so that A = 1, 000( )12 $ and in this case the effective annual rate is 7.1%. Step 5. As a conclusion to our solution, Ms. Brown would be advised to invest with NFL bank.

7 1.3. AN EXAMPLE OF A MATHEMATICAL MODEL 7 Class Activity 1.1: Let P be the amount of deposit and r the interest per unit time period for an account which pays simple interest, that is, at the end of each time period, the interest payable is rp. Let A be the total amount in the account after n time periods. Find the relationship among A, P, r, and n. Assuming simple interest at 7% per year, what is the present value of receiving $1, 000 in 4 years time?

8 8 CHAPTER 1. THE MODELING PROCESS Class Activity 1.2 Two years ago your parents purchased a home by financing $80, 000 for 20 years, paying monthly payments of $ with a monthly interest of 1%. They have made 24 payments and wish to know how much they owe on the mortgage, which they are considering paying off with an inheritance they received. Hint: Let b n be the amount owed after n months. Find a relationship between b n+1 and b n. The important mathematical concept introduced in this problem is the concept of a sequence. A sequence is a function f with domain the set of nonnegative integers. We write f(n) = a n or simply {a n } n=0. We call a n the nth term.

9 1.3. AN EXAMPLE OF A MATHEMATICAL MODEL 9 Class Activity 1.3 Find a formula for the nth term of the sequence. (a) {1, 4, 16, 64, 256, }. (b) {2, 0, 2, 0, 2, 0, }.

10 10 CHAPTER 1. THE MODELING PROCESS Project I You wish to buy a new car and narrow your choices to a Saturn, Cavalier, and Hyundai. Each company offers its best deal: Saturn $13, 990 $1, 000 down 3.5% interest for up to 60 months Cavalier $13, 550 $1, 500 down 4.5% interest f or up to 60 months Hyundai $12, 400 $500 down 6.5% interest f or up to 48 months You are able to spend at most $475 a month on a car payment. which car to buy. Determine

11 Chapter 2 Models in Management and Economics 2.1 Mathematics of Finance In this chapter we will develop the formulas to solve various sorts of present-value or future-value problems such as loans, savings accounts and other investments, mortgages, and annuities. As you ll notice, though there are lots of names for these problems they re really all the same thing looked at from different angles Short-Term Financing: Simple Interest Interest is the rent or fee charged for the use of money. In other words interest is what you pay for the use of somebody else s money. Or it is what somebody else pays for the use of your money. The principal is the amount of money borrowed or loaned. The rate of interest is the percentage of the principal that will be charged to the borrower or paid to the lender over a specific period of time usually one year. Most people are both borrowers and lenders. They owe money on a home mortgage or on a car loan. However, when you make a deposit into your savings account you are lending the money to the bank. Simple interest is only charged on the original principal and is paid at the end of the term of the loan: Interest = P rincipal Rate T ime or I = P rt 11

12 12 CHAPTER 2. MODELS IN MANAGEMENT AND ECONOMICS where I is the amount of simple interest, P is the principal amount invested, r is the annual interest rate, and t is the time period in years. Note that if any three of the four variables I, P, r, or t are known the formula shown above may be easily manipulated to solve for the unknown variable. Class Activity 2.1 What amount of interest will be charged if $5, 000 is borrowed for 8 months at a simple rate of interest of 8.5% per year? What principal will earn $75 in interest if it is invested at 7.5% simple interest for 6 months? Determine the annual rate of simple interest required for $900 to earn $90 in interest in 16 months? How long will it take a principal of $1, 200 to earn $80 in interest at 9.5% simple interest?

13 2.1. MATHEMATICS OF FINANCE 13 The maturity value or future value S of an investment is defined to be as the sum of the principal and the interest. That is, S = P + I = P (1 + rt). Class Activity 2.2 Find the maturity value of $2, 000 invested for 2 years at 7.5% simple interest. Find the principal that will amount to $3, in 18 months if money is worth 6.5% simple interest. A debt can be paid off by making payments of $1, 000 two years from now and $2, 500 five years from now. Determine the single equivalent payment made now that would settle the debt if money is worth 8.5% simple interest.

14 14 CHAPTER 2. MODELS IN MANAGEMENT AND ECONOMICS Long-Term Financing: Compound Interest The term compound interest refers to a procedure for computing interest whereby the interest for a specified interest period is added to the original principal. The resulting sum becomes a new principal for the next interest period. The interest earned in the earlier interest periods earn interest in the future interest periods. The future value is the total amount due on the maturity date of the loan or the investment. The maturity value is calculated by using the compound amount formula which is given by A = P (1 + r n )nt where P is the present value or principal, r is the nominal annual interest rate, n is the number of periods in a year, and t is the time in years. Possible values of n are: 1 (annually), 2 (semiannually), 4 (quarterly), 12 (monthly), and 365 (daily). When interest is compounded more frequently than once a year, the account effectively earns more than the nominal rate. Thus, we distinguish between nominal rate and effective rate. The effective annual rate tells how much interest the investment actually earns. The quantity (1 + r n )n 1 is known as the effective interest rate. Class Activity 2.3 Translating a value to the future is referred to as compounding. What will be the maturity value of an investment of $15, 000 invested for four years at 9.5% compounded semi-annually? Translating a value to the present is referred to as discounting. We call (1 + r n ) nt the discount factor. What principal invested today will amount to $8, 000 in 4 years if it is invested at 8% compounded quarterly? What is the effective rate of interest corresponding to a nominal interest rate of 5% compounded quarterly?

15 2.1. MATHEMATICS OF FINANCE Loans or Investments Problems: Annuity An annuity is an account to which regular payments are made. We start by looking at a loan type problems. Let A be the original amount of a loan; P be the amount of each payment of a loan or periodic payment; r is interest rate per unit period; and B n the loan balance after n payments have been made also known as payoff amount. We start first by finding a formula for B n. Class Activity 2.4: What is the value of B 0? What are the values of B 1, B 2, B 3? Find a compact form for the sum S n = 1 + (1 + r) + (1 + r) (1 + r) n 1. Find a formula for B n in terms of A, n, P, and r. You have a $18, 000 car loan at 14.25% for 36 months. You have just made your 24th payment of $ and you would like to know the payoff payment.

16 16 CHAPTER 2. MODELS IN MANAGEMENT AND ECONOMICS Class Activity 2.5: What will the payment amount P be for a certain number of payments? You are buying a $250, 000 house, with 10% down, on a 30-year mortgage at a fixed rate of 7.8%. What is the monthly payment? What is B 12? What conclusion can you make from that?

17 2.1. MATHEMATICS OF FINANCE 17 Class Activity 2.6: Suppose now that you know your periodic payment P and you want to figure out a formula for the total number of payments n. Find that formula. Mr. X offers to lend you $3, 500 at 6% for that new stereo you want. If you pay him back $100 a month, how long will it take? What will be the final payment?

18 18 CHAPTER 2. MODELS IN MANAGEMENT AND ECONOMICS Class Activity 2.7: Suppose you know what monthly payment you can afford, and about what interest rate you d be paying. From that, can you figure how big a purchase you can afford? So suppose you know the payment amount P, the interest rate r, and the number of payments n. How much are you actually borrowing (i.e. what is A?) You re looking to buy furniture for your living room. You can afford to pay about $60 a month over the next three years, and your credit card charges 16.9% interest. How much furniture can you buy?

19 2.1. MATHEMATICS OF FINANCE 19 Now, solving a problem of an investment type or annuity type is just a variation of the loan type discussed above. For example, a savings account can be regarded as a loan from you to the bank. The difference is that payments can be made into the account or withdrawn from the account: In the first case we call them deposits while in the second they are referred to as withdrawals. In the formula for B n, We count a withdrawal as positive P and a deposit as a negative P (since deposit must increase your balance whereas withdrawals must decrease your balance.) Class Activity 2.8: At the end of every month, you put $100 into a mutual fund that pays 6%. How much will you have at the end of five years?

20 20 CHAPTER 2. MODELS IN MANAGEMENT AND ECONOMICS An annuity is a contract, usually with an insurance company, for you to receive a fixed amount of money at stated intervals, usually monthly. This is also the same as a loan, except that the payments move only one way. Whole life insurance works this way once you cash it in: you can take the cash value of the insurance or use it to buy an annuity. You can also purchase the annuity with a lump sum. (An insurance annuity is typically more complicated, because it factors in your life expectancy. The payments are lower than they would otherwise be, because the company guarantees to pay you until you die, or to pay your heirs for the stated period if you die early. Here we re just concerned with a straight annuity that pays for a definite period.) Class Activity 2.9: You want to purchase a 20-year annuity that will pay $500 a month. If the guaranteed interest rate is 4%, how much will the annuity cost?

21 2.1. MATHEMATICS OF FINANCE 21 Class Activity 2.10: Suppose you have a goal, and you need to come out with a plan for how to reach it. In other words, you know a future value F that you want to reach, by making n periodic payments P that earn interest r. Find an explicit formula for the periodic payment P in terms of F, n, and r. You re saving up for a down payment on a house. You expect to buy in about five years, and you ll be looking in the $250, 000 range. You need to make at least a 10% down payment, plus $2, 500 for closing costs. If your money fund pays 5.5%, how much a month do you need to deposit?

22 22 CHAPTER 2. MODELS IN MANAGEMENT AND ECONOMICS Class Activity 2.11: Suppose you need to meet an investment goal. You know how much a month you can save, and what interest you ll earn. How long it will take to reach your goal? On the same day every year, you put $2, 000 into stocks. If the market rises 8% a year, how many years will it take you to accumulate $40, 000?

23 2.1. MATHEMATICS OF FINANCE Amortization and Sinking Funds Some loans are made so that only interest payments are made on a periodic basis with the principal due in whole at some future dates. In order to accumulate money to pay off the principal one might set up a seperate fund, called a sinking fund. The interest earned on the sinking fund will in general be different from that on the loan. Now, the question of finding P in Class Activity 2.5 is referred to as the amortization problem. Class Activity 2.12 A car loan for $20, 000 can be made for 5 years at 12% with interest only at the end of each year, the balance is due at the end of 5 years. A sinking fund can be set up at 9% to accumulate the balance at the end of 5 years. (a) What are the total annual payments needed to cover the interest on the car loan and the payments to the sinking fund? (b) What interest rate r on an amortized loan result in the same payments?

24 24 CHAPTER 2. MODELS IN MANAGEMENT AND ECONOMICS Project II Consider the value of a savings certificate initially worth $1000 that accumulates interest paid each month at 1% per month. Let a n be the value of the certificate after n months. Find a formula for a n in terms of n.

25 2.2. LINEAR PROGRAMMING AND THE SIMPLEX METHOD Linear Programming and the Simplex Method In this section we discuss a powerful method in optimizing a function subject to certain constraints. Mathematical programming is a field of mathematics that deals with the maximization or minimization of objective functions that are subject to constraints. A word of caution of what is meant by the word programmig. In this context, programming stands for planning. By an objective function we mean a function in one or more variables that one is interested in either maximizing or minimizing. The function could represent the cost or profit of some manufacturing process. Constraints are equalities or inequalities that describe restrictions involved with the minimization or maximization of the objective function. Linear Programming (LP) is a subcategory of mathematical programming. As the name suggests, both the objective function and the constraints are linear, i.e. of the form a 1 x 1 + a 2 x a n x n + b. A standard form LP has the following characteristics: the objective function must be maximized, all variables in the problem are nonnegative, all constraints are in the form a 1 x 1 + a 2 x a n x n nonnegative number Example Because of new federal regulations on pollution, a chemical plant introduced a new, more expensive process to supplement or replace an older process used in the production of a particular chemical. The older process emitted 15 grams of sulfur dioxide and 40 grams of particulate matter into the atmosphere for each gallon of chemical produced. The new process emits 5 grams of sulfur dioxide and 20 grams of particulate matter for each gallon of chemical produced. The company makes a profit of 30 cents per gallon and 20 cents per gallon on the old and new processes respectively. If the government allows the plant to emit no more than 10,500 grams of sulfur dioxide and no more than 30,000 grams of particulate matter daily, how many gallons of the chemical should be produced by each process to maximize daily profit? What is the maximum profit? Let x 1 be the number of gallons produced by the old process and x 2 be the number of gallons produced by the new process. Then the problem is an LP problem. Maximize: P =.3x 1 +.2x 2 Subject to: 15x 1 + 5x 2 10, x x 2 30, 000 x 1, x 2, 0

26 26 CHAPTER 2. MODELS IN MANAGEMENT AND ECONOMICS By graphing the constraints we find the following region known as the feasible region. Feasible Region in Red Thus, the feasible region is the set of points that satisfy all of the constraints of a linear program. A point in the feasible region is called a feasible solution. Note that a feasible region can be either bounded or unbounded. The graphs of the constraints when considered as equalities are called the edges of the feasible region. The intersection of two edges is called a vertex. Linear programming problems are usually solved by the Simplex Method which we start by discussing its steps. We will use the previous example as a reference. Step 1. All inequality constraints must be converted to equalities by using slack variables. 15x 1 + 5x 2 + s 1 = x x 2 + s 2 = 30.3x 1.2x 2 + P = 0 x 1, x 2, s 1, s 2, 0 Step 2. Set up the initial tableau. Label the rows. The bottom row always has the label P. The other rows are labeled by the variable that heads the column with a 1 in that row and zeros as all other entries in that column obtaining x 1 x 2 s 1 s 2 P s s P At this stage, s 1, s 2 are referred to as basic variables and x 1, x 2 as nonbasic variables.

27 2.2. LINEAR PROGRAMMING AND THE SIMPLEX METHOD 27 Step 3. Decide which non-basic variable to make basic, i.e. which variable should enter the set of basic variables. This is the pivot column which is the column containing the negative number of largest magnitude in the bottom row (other than the last column.) In this case, the entering variable is x 1. Note that if there are more than one candidates then you can choose either one. Now, if none of the numbers is negative then stop. x 1 x 2 s 1 s 2 P s s P Step 4. Decide which basic variable to make nonbasic. This is the variable exiting the set of basic variables. Its row is called the pivot row. It is found as follows. In each row except the one for the objective variable (bottom row), calculate the ratio entry in rightmost column entry in the pivot column. The pivot row is the row for which this ratio is the smallest non-negative number. Since = 0.70 and 40 = 0.75 then the exiting variable is s 1. Note that if all entries in the pivot column are nonpositive (i.e. 0) then the feasible region is unbounded and there is no solution. Note also that 0 1 is considered positive and 0 ( 1) is considered negative even though both are numerically zero. Step 5. The pivot element is the intersection of the pivot column and the pivot row. In this case, it is 15. Now, divide the pivot row by 15 to make the pivot element 1. Thus, obtaining x 1 x 2 s 1 s 2 P x s P Step 6. Get a 0 in every entry of the pivot column except the pivot element obtaining a column of a basic variable. In our problem we perform the following elementary row operations: Replace the second row R 2 by R 2 40R 1 and R 3 by R 3 +.3R 1 to obtain the new tableau

28 28 CHAPTER 2. MODELS IN MANAGEMENT AND ECONOMICS x 1 x 2 s 1 s 2 P x s P a vertex of fea- This tableau corresponds to the feasible basic solution (i.e. sible region) (.7, 0). Now we repeat the above algorithm. Step 3. The entering variable is x 2 so that the column of x 2 is the pivot column. Step 4. Since and 2.3 then the exiting variable is s so that the row of s 2 is the pivot row. Step 5. Divide the pivot row by the pivot element obtaining x 1 x 2 s 1 s 2 P x x P Step 6. To get zeros in the x 2 column (except in the second row) we perform the following operations: Replace R 1 by R R 2 and replace R 3 by R 3 +.1R 2 obtaining x 1 x 2 s 1 s 2 P x x P This tableau corresponds to the basic feasible solution (.6,.3) Step 3. The entering variable is s 1 so that the column of s 1 is the pivot column. Step 4. Since = 3 and.4 < 0 (ignore) then the exiting variable is x 1 so that the row of x 1 is the pivot row. Step 5. Divide the pivot row by the pivot element.2 obtaining x 1 x 2 s 1 s 2 P s x P Step 6. To get zeros in the s 1 column (except in the first row) we perform the follow-

29 2.2. LINEAR PROGRAMMING AND THE SIMPLEX METHOD 29 ing operations: Replace R 2 by R 2.4R 1 and replace R 3 by R 3.2R 1 obtaining x 1 x 2 s 1 s 2 P s x P Thus, the bottom row has all nonnegative entries so that the optimal solution accurs at (0, 1.5) and the maximum profit is $300.

30 30 CHAPTER 2. MODELS IN MANAGEMENT AND ECONOMICS Class Activity 2.13 Maximize: P = 10x x x 3 Subject to: 3x 1 + 3x 2 + 3x x 1 2x 2 + 4x x 1 + 6x 2 + 9x x 1, x 2, x 3, 0

31 2.2. LINEAR PROGRAMMING AND THE SIMPLEX METHOD 31 Class Activity 2.14 The Precision Tool Company is a manufacturer of precision screws. It has two main lines, wood screws and metal screws, which it sells for $20 and $25 respectively per box. The material costs for each box are $10 and $8 respectively, and overhead costs are $5, 000 per week. All the screws have to pass through a slotting and a threading machine. A box of wood screws requires 3 minutes in the slotting machine and 2 minutes on the threading machine, whereas a box of metal screws requires 2 minutes on the slotting machine and 8 minutes on the threading machine. In a week, each machine is available for 60 hours. The company wishes to maximize its weekly earnings.

32 32 CHAPTER 2. MODELS IN MANAGEMENT AND ECONOMICS Class Activity 2.15 A finance company has a total of $20 million between home loans and car loans. Home loans return 10% and car loans return 12% per year. However, the amount in home loans must be at least four times the amount in car loans. How much on each type of loan in order to maximize the return?

33 2.2. LINEAR PROGRAMMING AND THE SIMPLEX METHOD 33 Project III A carpenter makes tables and bookcases. He is trying to determine how many of each type of furniture he should make each week. The carpenter wishes to determine a weekly production for tables and bookcases that maximize his profits. It costs $5 and $7 to produce tables and bookcases respectively. The carpenter realizes a net unit profit of $25 per table and $30 per bookcase. He has up to 600 board feet of lumber to devote weekly and up to 40 hours of labor. He estimates that it requires 20 board feet of lumber and 5 hours of labor to complete a table and 30 board feet of lumber and 4 hours of labor for a bookcase. Moreover, he has signed a contract to deliver four tables and two bookcases every week. The carpenter wishes to determine a weekly production schedule for tables and bookcases that maximizes his profits. (Remark: When using slack variables, a constraint in the form ax + by c is replaced by ax + by s = c, where s 0.)

34 34 CHAPTER 2. MODELS IN MANAGEMENT AND ECONOMICS 2.3 Miscellaneous Models We conclude this chapter by discussing various models that arise in the field of Economics. Background: A polynomial of degree n is an expression of the form P (x) = a n x n + a n 1 x n a 1 x + a 0. A rational function is a function that can be written as the ratio of two polynomials. If in the long run the function approaches a value a then we call the line y = a a horizontal asymptote. Geometrically, this says that as x or x the graph of the function is close to the value a. Be aware that a graph can cross a horizontal asymptote in the short run behavior but not in the long run behavior. Another important fact to remember here is the limit lim x ± c x n = 0. Model Problem 1. The Bakewell Organization has spent an increasing percentage of gross income from its sale of baked beans on media advertising both in the press and on television. There has been no doubt that this effort has been successful in achieving widespread awareness of Bakewell s product. The issue that the Chairman has posed to his marketing director is how much should they spend on overall advertising. While their experience so far suggests that advertising increases sales, there is some doubt whether this effect will be maintained indefinitely. Fitting a curve to the graph of company profits y the total advertising expenditure x has produced the formula y = 22x + 11 x + 2 linking profits and advertising, where x and y are measured in $100, 000s. Before presenting these results to his chairman, the marketing director wishes to determine: (i) Is the model realistic in that profit increases with advertising expenditure? (ii) Do profits increase indefinitely with advertising expenditure, or is there some fixed value which they can never exceed?

35 2.3. MISCELLANEOUS MODELS 35 Background: Recall that if q is the quantity demanded of a product at a unit price p then the revenue function R is the product of p and q. If C is the cost function and R is the revenue function then the profit function P is the difference R C. Model Problem 2. An old establishing engineering firm, Wonder Works, has designed a stabilized children s bicycle of revolutionary design which is about to be put into production, but the board has met to discuss the recommended selling price of the Wonderbike, as the marketing division believes that this will strongly influence the number sold and thus the production ordered. The marketing director estimates that if the selling price is fixed at $50 then 5500 bicycles will be sold in the first month, whereas if the price is doubled to $100 only 1000 will be sold. He believes that the relation between selling price p and the quantity demanded q is linear and therefore given by the demand equation q = 10, p The marketing director suggests that the selling price be fixed at the value p for which revenue achieves its maximum. The production manager is quick to point out that this need not maximize profits. The Chairman then puts the question What should the selling price p of each Wonderbike be in order to give the largest possible company profit? Assume that the cost function is linear and is given by C(p) = 704, p.

36 36 CHAPTER 2. MODELS IN MANAGEMENT AND ECONOMICS Background: Piecewise Defined Functions. A piecewise defined function is a function which is defined explicitly by more than one expression. For example, the absolute value function f(x) = x is a piecewise defined function since { x, if x 0 f(x) = x, if x < 0 Model Problem 3. A bus charter company offers a travel club the following arrangements: If no more than 100 people go on a certain tour, the cost will be $500 per person, but the cost per person will be reduced by $4 for each person in excess of 100 who takes the tour. a. Express the total revenue R obtained by the charter company as a function of the number of people who go on the tour. b. Sketch the graph R. Estimate the number of people that results in the greatest total revenue for the charter company.

37 2.3. MISCELLANEOUS MODELS 37 Background: This model requires the use of the concept of piecewise defined functions and the process of finding the coordinates of the point of intersection of two curves. Recall that in order to find the point of intersection of two curves, express y in terms of x in the first equation and plug in this expression in the second equation to obtain an equation in the variable x. Solve this new equation for x and then use either of the original equations to find the value of y. Model Problem 4. The famous author John Uptight must decide between two publishers who are vying for the rights to his new book, Zen and the Art of Taxidermy. Publisher A offers royalties of 1% of net proceeds on the first 30, 000 copies sold and 3.5% on all copies in excess of that figure, and expects to net $2 on each copy sold. Publisher B will pay no royalties on the first 4, 000 copies sold but will pay 2% on the net proceeds of all copies sold in excess of 4,000 copies, and expects to net $3 on each copy sold. a. Express the revenue P A John should expect if he signs with Publisher A as a function of books sold, x. Likewise, find the revenue P B associated with Publisher B. b. Sketch the graphs of P A (x) and P B (x) on the same coordinates axes. c. For what value(s) of x are the two offers equivalent? d. With whom should he sign if he expects to sell 5,000 copies? 100,000 copies? e. State a simple criterion for determining which publisher he should choose if he expects to sell N copies.

38 38 CHAPTER 2. MODELS IN MANAGEMENT AND ECONOMICS Background: The sensitivity of demand to changes in price varies with the product. For example, a change in the price of light bulbs may not affect the demand for light bulbs much, whereas a change in the price of a particular make of a car have a significant effect on the demand for that car. An important quantity in economic analysis is the elasticity of demand, defined by E = p dq q dp where q is the number of units of a commodity demanded when the price is p dollars per unit. Changing the price of an item by 1% causes a change of E % in the quantity of goods sold. If E > 1, a one percent increase in price causes demand to drop by more than one percent. In this case we say that demand is elastic. If 0 E < 1, a one percent increase in price causes demand to drop by less than one percent, and we say that the demand is inelastic. In general, if the elasticity is large, then a change in price will cause a large change in the number of sales. Model Problem 5. (a) Show that dr dq = R q [ 1 ± 1 ]. E That is the marginal revenue is (1 ± 1 E ) times the average revenue. (b) Raising the price of hotel rooms from $ 75 to $ 80 per night reduces weekly sales from 100 rooms to 90 rooms. What is the elasticity of demand for rooms at a price of $ 75? Should the owner raise the price?

39 2.3. MISCELLANEOUS MODELS 39 Background: In economics and business, the terms marginal cost, marginal revenue, and marginal profit are used for the rate of change of cost, revenue, and profit. The term marginal is used to highlight the rate of change as an indicator of how the cost, revenue, or profit changes in response to a one unit (i.e. marginal) change in the independent variable. The net earnings of an industrial process for the production range a x b are given by the definite integral b a P (x)dx where P (x) = R(x) C(x). Geometrically, the net earnings is the area between the curves R (x) and C (x). Model Problem 6. For a certain industrial process, the marginal cost and marginal revenue (in thousands of dollars) associated with producing x units are C (x) = 0.1x 2 + 4x + 10 and R (x) = 70 x respectively. What are the net earnings of the process as x ranges from x = 0 to x = x m, where x m is the level of production at which marginal cost equals marginal revenue?

40 40 CHAPTER 2. MODELS IN MANAGEMENT AND ECONOMICS Background: The process of finding the maximum or minimum of a function is called optimization. A general optimization procedure consists of the following steps: a. Draw a figure if possible and label all the quantities relevant to the problem. b. Focus on the quantity to be optimized. Name it. Find a formula for the quantity to be maximized or minimized. c. Use conditions in the problem to eliminate variables in order to express the quantity to be optimized in terms of a single variable. d. Find the practical domain for the variables in step c. e. Use the method of calculus to find the maximum or the minimum. Model Problem 7. An efficiency study of the morning shift at a factory indicates that the number of units produced by an average worker t hours after 8:00 A.M. may be modeled by the formula Q(t) = t 3 + 9t t. At what time in the morning is the worker performing most efficiently?

41 2.3. MISCELLANEOUS MODELS 41 Background: To optimize a continuous function over an interval [a, b] we proceed as follows: (a) Find the critical points of f in the interval [a, b]. (b) Evaluate f at the critical points found in (a). (c) Evaluate f at the endpoints. (d) The largest value in (b)-(c) is the maximum of f and the smallest value is the minimum of f. Model Problem 8. A manufacturer can produce a pair of earrings at a cost of $3. The earrings have been selling for $ 5 per pair and at this price, consumers have been buying 4,000 pairs per month. The manufacturer is planning to raise the price of the earrings and estimates that for each $ 1 increase in the price, 400 fewer pairs of earrings will be sold each month. At what price should the manufacturer sell the earrings to maximize profit?

42 42 CHAPTER 2. MODELS IN MANAGEMENT AND ECONOMICS Background: The revenue function can be expressed in terms of p and q by the formula R(q) = pq. Now, recall from calculus that if the derivative of a function f is given then one can find f by integration. That is, f(x) = f (x)dx Model Problem 9. A manufacturer estimates that the marginal revenue of a certain commodity is R (q) = q when q units are produced. Find the demand function p as a function of q.

43 2.3. MISCELLANEOUS MODELS 43 Background: This problem requires the understanding of the content of Section 3.1. The reader is encouraged to read that section before attempting to solve the problem. Also, integration by parts is needed in the process. Model Problem 10. The Evans price-adjustment model assumes that if there is an excess demand D over supply S in any time period, the price p changes at a rate proportional to the excess,d S, that is, dp dt = k(d S). Suppose that for a certain commodity, demand is linear, and supply is cyclical D(p) = c pd S(t) = a sin (bt). Solve the differential equation to express price p(t) in terms of a, b, c, and d.

44 44 CHAPTER 2. MODELS IN MANAGEMENT AND ECONOMICS Background: The geometric series k=0 axk diverges for x 1 and converges for x < 1 with sum ax k = a 1 x. k=0 Model Problem 11. Suppose that nationwide approximately 90% of all income is spent and 10% is saved. How much total spending will be generated by a $40 billion tax rebate if savings habits do not change?

45 2.3. MISCELLANEOUS MODELS 45 Background: The graph of a function is concave up if it bends upward as we move left to right. This means that the slope of the tangent line gets either more and more positive or less and less negative. The garph is concave downward if it bends downward. This happens when the slope of the tangent line is either getting less and less positive or more and more negative. A line is neither concave up nor concave down. Model Problem 12. When a new product is advertised, more and more people try it. However, the rate at which new people try it slows as time goes on. (i) Sketch a graph of the total number of people who have tried such product against time. (ii) What do you know about the concavity of the graph?

46 46 CHAPTER 2. MODELS IN MANAGEMENT AND ECONOMICS Background: Recall that a function takes an x-value to a unique y-value. In some cases, one can consider the opposite process. That is, for each y-value there is a unique x-value. In this case we say that the given function has an inverse function. Geometrically, a function has an inverse function if and only if its graph intersects any horizontal line at most once. Algebraically, to find the formula for the inverse function, we solve for x in terms of y. Model Problem 13. The cost of producing q articles is given by the following function f(q) = q. (i) Find a formula for the inverse function f 1. (ii) Explain in practical terms what the inverse function tells you.

47 2.3. MISCELLANEOUS MODELS 47 Background: Recall the definition of the derivative: Thus, for h small, we can write or f f(a + h) f(a) (a) = lim. h 0 h f (a) f(a + h) f(a) h f(a + h) hf (a) + f(a). Model Problem 14. The Rule of 70 is a rule of thumb to estimate how long it takes money in a bank to double. Suppose the money is in an account earning i% annual interest rate, compounded yearly. The Rule of 70 says that the time it takes the amount of money to double is approximately 70 i years, assuming i is small. Find a local linearization of ln (1 + x), and use it to explain why this rule works.

48 48 CHAPTER 2. MODELS IN MANAGEMENT AND ECONOMICS Background: Optimization problem. Model Problem 15. A company manufactures only one product. The quantity, q, of this product produced per month depends on the amount of capital, K, invested and the amount of labor, L, available each month. Assume that q can be expressed as a Cobb-Douglas production function: q = ck α L β where c, α, β, are positive constants, with 0 < α < 1 and 0 < β < 1. In this problem we will see how the Russian government could use a Cobb-Douglas function to estimate how many people a newly privatized industry might employ. A company in such an industry has only a small amount of capital available to it and needs to use all of it, so K is fixed. Suppose L is measured in man-hours per month, and that each man-hour costs the company w rubles (Russian currency). Suppose the company has no other costs besides labor, and that each unit of the good can be sold for a fixed price of p rubles. How many man-hours of labor per month should the company use in order to maximize its profit?

49 2.3. MISCELLANEOUS MODELS 49 Background: Geometric series. See Problem 11. Model Problem 16. This problem illustrates how banks create credit and can thereby lend out more money than has been deposited. Suppose that initially $100 is deposited in a bank. Experience has shown bankers that on the average only 8% of the money deposited is withdrawn by the owner at any time. Consequently, bankers feel free to lend out 92% of their deposits. Thus, $ 92 of the original $ 100 is loaned out to other customers( to start a business, for example). This $ 92 will become someone s else income and, sooner or later, will be redeposited in the bank. Then 92% of $ 92, is loaned out again and eventually redeposited. Of the $92(0.92)=$ 84.64, the bank again loans out 92%, and so on. (i) Find the total amount of money deposited in the bank as a result of these transactions. (ii) The total amount of money deposited divided by the original deposit is called the credit multiplier. Calculate the credit multiplier for this example and explain what this number tells us.

50 50 CHAPTER 2. MODELS IN MANAGEMENT AND ECONOMICS Background: Every solution to the equation can be written in the form dy dt = ky y = y(0)e kt where k > 0 represents growth, whereas k < 0 represents decay. Model Problem 17. A bank account earns interest continuously at a rate of 5% of the current balance per year. Assume that the initial deposit is $1,000, and no other deposits or withdrawals are made. (i) Write the differential equation satisfied by the balance in the account. (ii) Solve the differential equation and graph the solution.

51 Chapter 3 Models in Physics and Engineering Sciences In many models, we will have information relating a rate of change of a dependent variable with respect to one or more independent variables and are interested in discovering the function relating the variables. 3.1 Models Based on 1st Order Differential Equations Many problems in applied mathematics and engineering are described by ordinary differential equations: (i) Decay of a radioactive material: dx = ky. (ii) Newton s Second Law: m d2 x dt = F (t). 2 dy By a first order differential equation we mean an equation of the form dy = F (x, y) (3.1) dx Suppose first that F (x, y) = f(x) g(y). In this case, g(y)dy = f(x)dx. Integrating to obtain g(y)dy = f(x)dx + C where C is a constant of integration. method of seperation of variables. This method of solution is called the Class Activity 3.1 Find the general solution to the equation dy dx = ky. 51

52 52 CHAPTER 3. MODELS IN PHYSICS AND ENGINEERING SCIENCES

53 3.1. MODELS BASED ON 1ST ORDER DIFFERENTIAL EQUATIONS 53 Next, suppose that (3.1) is rewritten in the form dy + p(x)y = q(x) (3.2) dx We call (3.2) a first order linear differential equation. Let s try and solve the above equation. Let r(x) be a function such that that is, d(r(x)y) dx Comparing this with (3.2) to obtain = r(x)q(x) r(x)y + r (x)y = r(x)q(x). r (x) = p(x)r(x) or Integrating to obtain dr r = p(x)dx. ln r = p(x)dx + C Class Activity 3.2 (i) Show that r(x) = Ke p(x)dx. (ii) Show that K can be chosen to be 1 and hence r(x) = e p(x)dx. We call r(x) an integrating factor. (iii) Show that y = 1 r(x) q(x)r(x)dx + C r(x). (iv) Find the general solution to the first order ODE y y = x

54 54 CHAPTER 3. MODELS IN PHYSICS AND ENGINEERING SCIENCES Class Activity 3.3(Population Growth) The first population growth was formulated by Thomas Malthus in In this model, the births and deaths are assumed to be proportinal to both the total size of population N(t) and to time over small time intervals δt. Show that N(t) satisfies a first order linear differential equation. Solve the equation.

55 3.1. MODELS BASED ON 1ST ORDER DIFFERENTIAL EQUATIONS 55 Class Activity 3.4 Infusion of glucose into the bloodstream is modeled by the differential equation dg dt = c ag. Here G is the amount of glucose in the bloodstream at time t, c the constant rate of infusion, and a is a positive constant governing the removal rate from the bloodstream. Determine the predicted glucose level in the bloodstream, and show that G c a as t.

56 56 CHAPTER 3. MODELS IN PHYSICS AND ENGINEERING SCIENCES Project IV Consider a population growth problem of the form where k satisfies the equation dp dt = kp k = r(m P ), r > 0 is a constant, and M is the maximum population. (a) Find an explicit formula for P. (b) What is the value of P (t) in the long run? (c) Graph the function P (t).

57 3.2. MODELS BASED ON 2ND ORDER LINEAR DIFFERENTIAL EQUATIONS Models Based on 2nd Order Linear Differential Equations By a homogeneous linear second order differential equation with constant coefficients we mean an equation of the form ay + by + cy = 0 (3.3) One usually assumes a solution of the form u(x) = e λx (for constant λ.) Substitution of this expression into (3.3) gives aλ 2 e λx + bλe λx + ce λx = 0 Dividing by e λx > 0 gives the characteristic equation aλ 2 + bλ + c = 0 (3.4) To solve this equation for λ we need to consider the following three cases: If b 2 4ac > 0 then (3.4) has two distinct solutions λ 1 = b b 2 4ac 2a and λ 2 = b+ b 2 4ac 2a. Thus, C 1 e λ1x and C 2 e λ2x, where C 1 and C 2 are arbitrary constants, are two solutions of (3.3). The general solution to (3.3) is the function y = C 1 e λ1x + C 2 e λ 2x If b 2 4ac = 0 then (3.4) has only one solution, λ = b 2a. By substitution, we can check that both y = C 1 e λx and y = C 2 xe λx are solutions to (3.3) so that the general solution is given by y = (C 1 + C 2 x)e λx If b 2 4ac < 0 then (3.4) has complex conjugate roots λ 1 = b i 4ac b 2 2a λ 2 = b+i 4ac b 2 2a and y = C 2 e b is given by y = C 1 e b 2a x cos ( and. By substitution, we can check that both y = C 1 e b 2a x cos ( 4ac b 2 2a x sin ( 4ac b 2 )x are solutions to (3.4) and the general solution 2a 4ac b 2 2a 2a )x )x+c 2 e b 4ac b 2a x 2 sin ( )x = Ce b 4ac b 2a x 2 sin ( x + ω) 2a 2a

58 58 CHAPTER 3. MODELS IN PHYSICS AND ENGINEERING SCIENCES Class Activity 3.5 (i) Find the solution of y 3y + 2y = 0 satisfying y(0) = 1 and y (0) = 0. (ii) Find the solution of y + y = 0 satisfying y(0) = 1 and y (0) = 0. (iii) Find the solution of y 2y + y = 0 satisfying y(0) = 2 and y(1) = 0.

59 3.2. MODELS BASED ON 2ND ORDER LINEAR DIFFERENTIAL EQUATIONS59 Class Activity 3.6(Oscilation of a Mass on a Spring) Consider a mass m attached to the end of a spring hanging from the ceiling. We assume that the mass of the spring is negligible in comparison with the mass m. When the system is left undisturbed, no net force acts on the mass. The force of gravity is balanced by the force the spring exerts on the mass, and the spring is in the equilibrium position. Pulling down on the mass stretches the spring, increasing the tension, so the combination of gravity and spring force is upward so that we feel a force pulling upward. Pushing up the spring, decreases the tension, so the combination of gravity and spring force is downward. What happens if we push the mass upward and then release it? That is describe the motion of the mass with respect to time. (Hint: Hooke s Law states that the net force F exerted on the mass m satisfies the equation F = ks where k > 0 is a constant that depends on the physical proporties of the particular spring; s is the displacement from the equilibrium position. The negative sign in the formula means that the net force is in the opposite direction to the displacement.)

60 60 CHAPTER 3. MODELS IN PHYSICS AND ENGINEERING SCIENCES Project V (Electrical Circuits) The charge Q (in Amperes) on a capacitor in a circuit with inductance L (in henries), capacitance C (in farads), and resistance R (in ohms) satisfies the differential equation L d2 Q dt 2 (i) Find an expression for Q(t). (ii) What happens to Q(t) as t? Capacitor + R dq dt + 1 C Q = 0. L Q C +Q R Resistor Inductor

61 3.3. MODELS BASED ON 1ST ORDER LINEAR DIFFERENCE EQUATIONS Models Based on 1st Order Linear Difference Equations Consider a sequence of numbers {x 0, x 1, x 2, x 3, } for which an explicit form of the nth term is not given. An equation which expresses a value of a sequence as a function of the other terms in the sequence is called a difference equation. In particular, an equation which expresses x n in terms of x n 1 is called a first-order difference equation. Class Activity 3.7 Radium is a radioactive element which decays at a rate of 1% every 25 years. This means that the amount left at the beginning of any given 25 year period is equal to the amount at the beginning of the previous 25 year period minus 1% of that amount. Let x 0 be the initial amount of radium and x n be the amount of radium still remaining after 25n years (n = 0, 1, 2 ). (i) Find the relationship between x n and x n 1. (ii) Find an explicit formula for x n in terms of n and x 0. (iii) The half-life of a radioactive element is the number of years required for one-half of an initial amount to decay. Find the half-life of Radium.