1. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 1 of 73 Chapter 11 Difference Equations and Mathematical Models

2. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 2 of 73 Outline 11.1 Introduction to Difference Equations I 11.2 Introduction to Difference Equations II 11.3 Graphing Difference Equations 11.4 Mathematics of Personal Finance 11.5 Modeling with Difference Equations

3. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 3 of 73 11.1 Introduction to Difference Equations I 1.Difference Equation 2.Three-Step Procedure 3.Using Technology

4. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 4 of 73 Difference Equation An equation of the form y n = ay n - 1 + b where a, b and y 0 are specified real numbers is called a difference equation. The starting value, y 0, is called the initial value.

5. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 5 of 73 Example Difference Equation Suppose a savings account contains $40 and earns 6% interest, compounded annually. At the end of each year a $3 withdrawal is made. Determine the difference equation that describes how to compute each year's balance from the previous year's balance.

6. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 6 of 73 Example Difference Equation (2) The balance for the next year equals the previous year's balance plus the interest earned on the previous year's balance minus the withdrawal. y n = y n - 1 +.06y n - 1 - 3 = 1.06y n - 1 - 3. Balance+ Interest for year- Withdrawal y 0 = $40(.06)40$3 y 1 = 39.40(.06)39.40 3 y 2 = 38.76(.06)38.76 3

7. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 7 of 73 Three-Step Procedure For the difference equation y n = ay n - 1 + b: 1. Generate the first few terms. 2. Graph the terms. Plot the points (n, y n ) for n = 0, 1, 2, … 3. Solve the difference equation. The solution is

8. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 8 of 73 Example Three-Step Procedure Study the difference equation y n =.2y n - 1 + 4.8 with y 0 = 1. 1. Generate the first few terms. y 0 = 1 y 1 =.2(1) + 4.8 = 5 y 2 =.2(5) + 4.8 = 5.8 y 3 =.2(5.8) + 4.8 = 5.96 y 4 =.2(5.96) + 4.8 = 5.992.

9. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 9 of 73 Example Three-Step Procedure (2) 2. Graph these terms (0,1), (1, 5), (2, 5.8), (3, 5.96) and (4,5.992).

10. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 10 of 73 Example Three-Step Procedure (3) 3. Solve the equation. Here a =.2 and b = 4.8. Therefore,

11. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 11 of 73 Using Technology An Excel spreadsheet can be used to evaluate and graph the first few terms of a difference equation. 1. Enter 0 into cell A2, enter 1 in cell A3, select the two cells, and drag the fill handle down to A8. 2. Enter the value for y 0 into cell B2. 3. Enter the formula for y n into cell B3 using B2 in place of y n - 1, select cell B2 and B3 and drag the fill handle down to B8. 4. Highlight cells A2:B8 and use the Chart Wizard to make an XY(Scatter) type chart.

12. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 12 of 73 Example Using Technology Use an Excel spreadsheet to compute the first few terms and the graph of y n =.2y n - 1 + 4.8 with y 0 = 1.

13. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 13 of 73  A difference equation is an equation of the form y n = ay n - 1 + b, where a, b, and y 0 are specified, and determines a sequence of numbers in which each of the numbers (that is, y 0, y 1,…) is obtained from the preceding number by multiplying the preceding number by a and adding b. The first number in the sequence, y 0, is called the initial value. Summary Section 11.1 - Part 1

14. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 14 of 73  The graph of a difference equation is obtained by graphing the points (0, y 0 ), (1, y 1 ), (2, y 2 ), …  The value of the n th term of a difference equation y n = ay n - 1 + b (y 0 given and a 1) can be obtained directly (that is, without generating the preceding terms) with the formula Summary Section 11.1 - Part 2

15. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 15 of 73 11.2 Introduction to Difference Equations II 1.Difference Equation II 2.Interest 3.Amount with Interest 4.Consumer Loan

16. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 16 of 73 Difference Equation II The difference equation y n = ay n - 1 + b (y 0 given) has solution

17. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 17 of 73 Example Difference Equation II a) Solve the difference equation y n = y n - 1 + 2, y 0 = 3. b) Find y 100. a) Here a = 1, b = 2 and y 0 = 3. Therefore, y n = 3 + 2n. b) y 100 = 3 + 2(100) = 203.

18. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 18 of 73 Interest Let an amount of money be deposited in a savings account. If interest is paid only on the initial deposit (and not on accumulated interest), then the interest is called simple. If interest is paid on the current amount in the account (initial deposit and accumulated interest), then the interest is called compound.

19. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 19 of 73 Example Simple Interest Find the amount y n at the end of n years if $40 is deposited into a savings account earning 6% simple interest. [new balance] = [previous balance] + [interest]. Simple interest is computed on the original balance. So y n = y n - 1 +.06y 0 = y n - 1 +.06(40) = y n - 1 + 2.40. Since a = 1, y n = 40 + 2.4n.

20. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 20 of 73 Example Compound Interest Find the amount y n at the end of n years if $40 is deposited into a savings account earning 6% interest compounded annually. [new balance] = [previous balance] + [interest]. Compound interest is computed on the previous balance, y n - 1. So y n = y n - 1 +.06y n - 1 = 1.06y n - 1. Since a 1, y n = 40(1.06) n.

21. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 21 of 73 Amount with Interest If y 0 dollars is deposited at interest rate i per period, then the amount after n periods is Simple interest:y n = y 0 + (iy 0 )n Compound interest: y n = y 0 (1 + i) n. The initial value y 0 is called the principal. If the annual interest rate is r compounded k times per year, then i = r/k.

22. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 22 of 73 Consumer Loan A consumer loan is one in which an item is bought and paid for with a series of equal payments until the original cost plus interest is paid off. Each time period, part of the payment goes toward paying off the interest and part goes toward reducing the balance of the loan. A consumer loan used to purchase a house is called a mortgage.

23. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 23 of 73 Example Consumer Loan A consumer loan of $2400 carries an interest rate of 12% compounded annually and a yearly payment of $1000. a) Find the difference equation for the balance, y n, owed after n years. b) Compute the balances after 1, 2 and 3 years.

24. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 24 of 73 Example Consumer Loan (a) a) At the end of each year [new balance] = [previous balance] + [interest] - [payment] Since interest is compound, it is computed on the previous balance, y n - 1. y n = y n - 1 +.12 y n - 1 - 1000 = 1.12 y n - 1 - 1000

25. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 25 of 73 Example Consumer Loan (b) b) y n = 1.12 y n - 1 - 1000 y 1 = 1.12(2400) - 1000 = $1688 y 2 = 1.12(1688) - 1000 = $890.56 y 3 = 1.12(890.56) - 1000 $0

26. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 26 of 73  The value of the n th term of a difference equation y n = y n - 1 + b (y 0 given) can be obtained directly (that is, without generating the preceding terms) with the formula y n = y 0 + bn.  Money deposited in a savings account is called the principal. Summary Section 11.2 - Part 1

27. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 27 of 73  Suppose y 0 dollars is deposited at a yearly simple interest rate r. The balance after n years, y n, satisfies the difference equation y n = y n - 1 + ry 0.  Suppose y 0 dollars is deposited at a yearly interest rate r compounded k times a year. The period interest rate is i = r/k. The balance after n interest periods, y n, satisfies the difference equation y n = (1 + i)y n - 1. Summary Section 11.2 - Part 2

28. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 28 of 73 11.3 Graphing Difference Equations 1.Vertical Direction 1.Monotonic Graphs 2.Oscillating Graphs 3.Test 1 4.Constant Graphs 2.Long-run Behavior 1.Asymptotic 2.Unbounded 3.Test 2 3.5 Step Procedure

29. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 29 of 73 Vertical Direction The vertical direction of a graph refers to the up-and-down motion of successive terms. A graph is increasing if it rises when read from left to right - that is, if the terms get successively larger. A graph is decreasing if it falls when read from left to right - that is, if the terms get successively smaller.

30. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 30 of 73 Monotonic Graphs A graph that is either increasing or decreasing is called monotonic. The following are examples of monotonic graphs. Increasing graphs Decreasing graphs

31. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 31 of 73 Oscillating Graphs A graph that changes its direction with every term is called oscillating. The following are examples of oscillating graphs.

32. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 32 of 73 Test 1 If a > 0, then the graph of y n = ay n - 1 + b is monotonic. If a < 0, then the graph is oscillating. Furthermore, when a < 0, the graph oscillates about the line y = b/(1 - a).

33. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 33 of 73 Example Test 1 Discuss the vertical direction of the graph of y n = -.8y n - 1 + 9, y 0 = 50. Since a < 0, the graph is oscillating. The formula for y n yields y n = 5 + 45(-.8) n. The graph oscillates about the line y = 5.

34. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 34 of 73 Constant Graphs A graph that always remains at the same height is called constant. The following is an example of a constant graph.

35. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 35 of 73 Constant Graphs (2) The graph of y n = ay n - 1 + b (a 1) is constant if y 0 = b/(1 - a). In this case, y n = y 0 = b/(1 - a).

36. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 36 of 73 Example Constant Graph Sketch the graph of the difference equation y n = 2y n - 1 - 1, y 0 = 1. Therefore, the graph is constant with y n = 1.

37. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 37 of 73 Long-run Behavior Long-run behavior refers to the eventual behavior of the graph. If the graph approaches a horizontal line, then it is called asymptotic. If the graph goes indefinitely high or indefinitely low, it is called unbounded.

38. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 38 of 73 Asymptotic Graph The following are examples of asymptotic graphs. increasing asymptotic decreasing asymptotic oscillating asymptotic

39. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 39 of 73 Unbounded Graph The following are examples of unbounded graphs. increasing unbounded decreasing unbounded oscillating unbounded

40. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 40 of 73 Test 2 If |a| < 1, then the graph of y n = ay n - 1 + b is asymptotic to the line y = b/(1 - a). If |a| > 1, then the graph is unbounded and moves away from the line y = b/(1 - a). When |a| 1, the graph is said to be repelled by the line y = b/(1 - a).

41. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 41 of 73 Example End Behavior Discuss the graphs of y n = 1.4y n - 1 - 8 with y 0 > 20 and with y 0 < 20. Since a = 1.4 > 0, Test 1 predicts that the graphs are monotonic. Since |a| = |1.4| = 1.4 > 1, Test 2 predicts that the graphs are unbounded. Here b/(1 - a) = -8/(1 - 1.4) = 20, so the graph is repelled by the line y = 20.

42. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 42 of 73 Example End Behavior (2) The graphs are shown below. y 0 > 20 y 0 < 20

43. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 43 of 73 Summary Test 1 and Test 2

44. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 44 of 73 5 Step Procedure To sketch the graph of y n = ay n - 1 + b, a 0, + 1: 1. Draw the line y = b/(1 - a) as a dashed line. 2. Plot y 0. If y 0 is on the line y = b/(1 - a), the graph is constant and this procedure terminates. 3. If a is positive, write MONO, since the graph is then monotonic. If a is negative, write OSC, since the graph is then oscillating.

45. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 45 of 73 5 Step Procedure (2) 4. If |a| 1, write REPEL, since the graph is repelled from the line y = b/(1 - a). 5. Use all the information to sketch the graph.

46. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 46 of 73 Example Sketch the Graph Sketch the graph of y n = -1.5y n - 1 + 5, y 0 = 2.6. 1. b/(1 - a) = 5/(1 - (-1.5)) = 2. So draw the line y = 2.

47. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 47 of 73 Example Sketch the Graph (2 and 3) 2. y 0 = 2.6, which is not on the line y = 2. So the graph is not a constant. 3. a = -1.5 < 0. Write OSC above the dashed line.

48. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 48 of 73 Example Sketch the Graph (4 and 5) 4. |a| = |-1.5| = 1.5 > 1. Write REPEL below the dashed line. 5. The graph begins at 2.6, oscillates about, and is repelled by the line y = 2.

49. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 49 of 73 Summary Section 11.3 - Part 1 All the following refer to the graph of y n = ay n - 1 + b.  Increasing graph - rises when read from left to right.  Decreasing graph - falls when read from left to right.  Monotonic graph - either increasing or decreasing.

50. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 50 of 73 Summary Section 11.3 - Part 2  Oscillating graph - changes its direction from rising to falling with every term.  Attracted - moves closer and closer to the line y = b/(1 - a).  Repelled - moves farther and farther away from the line y = b/(1 - a) without bound.  If a > 0, the graph is monotonic.  If a < 0, the graph is oscillating.

51. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 51 of 73 Summary Section 11.3 - Part 3  If |a| < 1, the graph is attracted to the line y = b/(1 - a).  If |a| > 1, the graph is repelled from the line y = b/(1 - a).

52. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 52 of 73 11.4 Mathematics of Personal Finance 1.Mortgage 2.Difference Equation for Mortgage 3.Annuities 4.Difference Equation for Annuities

53. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 53 of 73 Mortgage A loan to pay for a new house is called a mortgage. It is repaid with interest in monthly installments over a number of years, usually 25 or 30. The monthly payments are computed so that after exactly the correct length of time the unpaid balance is 0.

54. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 54 of 73 Difference Equation for Mortgage Let y n be the unpaid balance on the mortgage after n months. In particular, y 0 is the initial amount borrowed. Let i denote the monthly interest rate and R the monthly mortgage payment.

55. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 55 of 73 Example Mortgage Find the monthly payment for a 30-year mortgage for $140,000 at 7.5% interest compounded monthly. n = 30(12) = 360. Since the loan is paid off in 30 years, y 360 = 0. i =.075/12 =.00625. The difference equation is y n = 1.00625y n - 1 - R, y 0 = 140000.

56. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 56 of 73 Example Mortgage (2) Now, a = 1.00625 and b = -R. So b/(1 - a) = -R/(1 - 1.00625) = 160R. The solution to the difference equation is y n = 160R + (140000 - 160R)(1.00625) n. y 360 = 0 = 160R + (140000 - 160R)(1.00625) 360 0 = 160R + (140000 - 160R)(9.421533905) R = $978.90

57. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 57 of 73 Annuity A bank account into which equal sums are deposited at regular intervals, either weekly, monthly, quarterly, or annually, is an example of an annuity. The money earns interest and accumulates for a certain number of years, after which it becomes available to the investor.

58. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 58 of 73 Difference Equation for Annuity Let y n be the amount of money in the annuity after n time periods. In particular, y 0 = 0 is the initial amount in the account. Let i denote the period interest rate and D the deposit each time period.

59. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 59 of 73 Example Annuity How much money must be deposited at the end of each quarter into an annuity at 8% interest compounded quarterly in order to have $10,000 after 15 years. n = 15(4) = 60. Since $10,000 is desired in 15 years, y 60 = 10000. i =.08/4 =.02. The difference equation is y n = 1.02y n - 1 + D, y 0 = 0.

60. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 60 of 73 Example Annuity (2) Now, a = 1.02 and b = D. So b/(1 - a) = D/(1 - 1.2) = -50D. The solution to the difference equation is y n = -50D + (0 - (-50D))(1.02) n. y 60 = 10000 = -50D + 50D(1.02) 60 10000 = -50D + 50D(3.281030788) D = $87.68

61. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 61 of 73 Summary Section 11.4 - Part 1  A mortgage is a loan that is paid off in equal payments. The sequence of successive balances (that is, amounts still owed) satisfies a difference equation of the form y n = (1 + i)y n - 1 - R, where i is the interest rate per period and R is the payment per period.

62. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 62 of 73 Summary Section 11.4 - Part 2  An annuity is a bank account into which a sequence of equal deposits is made that satisfies a difference equation of the form y n = (1 + i)y n - 1 + D, where i is the interest rate per period and D is the deposit per period.

63. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 63 of 73 11.5 Modeling with Difference Equations 1.Proportionality 2.Radioactive Decay 3.Spread of Information 4.Supply and Demand

64. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 64 of 73 Proportionality To say that quantities are proportional is the same as saying that one quantity is equal to a constant times the other quantity. The constant is called the constant of proportionality.

65. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 65 of 73 Example Radioactive Decay Experiment shows that cobalt 60, a radioactive form of cobalt used in cancer therapy, decays yearly at the rate of.12 times the amount present at the beginning of the year. a) Write the difference equation for y n, the amount present after n years. b) Sketch the graph of the difference equation.

66. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 66 of 73 Example Radioactive Decay (a)

67. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 67 of 73 Example Radioactive Decay (b) Since a =.88 is positive and less than 1, the graph is monotonic and attracted to the line y = b/(1 - a) = 0/(1 -.88) = 0.

68. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 68 of 73 Example Spread of Information Suppose the local radio and TV stations in a town of 50,000 people start broadcasting a certain piece of news. The number of people learning the news each hour is proportional to the number who had not yet heard it by the end of the preceding hour. Assume the constant of proportionality is.3. a) Write a difference equation describing the spread of the news through the population of the town n hours after the broadcast. b) Sketch the graph.

69. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 69 of 73 Example Spread of Information (a) Let y n be the number of people (in thousands) who have heard the news after n hours. The number of people who have not heard the news after n - 1 hours is 50 - y n - 1.

70. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 70 of 73 Example Spread of Information (b) Initially no one heard the news so y 0 = 0. Since a =.7 is positive and less than 1, the graph is monotonic and attracted to the line y = b/(1 - a) = 15/(1-.7) = 50.

71. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 71 of 73 Example Supply and Demand This year's level of production and price for most agricultural products greatly affect the level of production and price next year. Let the current crop of soybeans in a certain country be 80 million bushels. Let q n be the quantity of soybeans grown n years from now, and let p n be the market price in n years. Experience has shown that q n and p n are related by p n = 20 -.1q n and q n = 5p n - 1 - 10. Sketch the quantity produced n years from now.

72. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 72 of 73 Example Supply and Demand (2) First we need the difference equation for q n. Use the first equation to find p n - 1 and substitute it into the second equation. p n = 20 -.1q n means that p n - 1 = 20 -.1q n - 1. q n = 5p n - 1 - 10 = 5(20 -.1q n - 1 ) - 10 = -5q n - 1 + 90. a = -.5, b = 90 and b/(1 - a) = 90/(1 - (-.5)) = 60. Since a is negative and |a| < 1, the graph is oscillating and is attracted to the line y = 60.

73. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 73 of 73 Example Supply and Demand (3)