1. MTH 209 Week 5 Third

2. Final Exam logistics  Here is what I've found out about the final exam in MyMathLab (running from a week ago to 11:59pm five days after class tonight.. Slide 2 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

3. Final Exam logistics  There will be 50 questions.  You have only one attempt to complete the exam.  Once you start the exam, it must be completed in that sitting. (Don't start until you have time to complete it that day or evening.)  You may skip and get back to a question BUT return to it before you hit submit. You must be in the same session to return to a question.  There is no time limit to the exam (except for 11:59pm five nights after the last class).  You will not have the following help that exists in homework:  Online sections of the textbook  Animated help  Step-by-step instructions  Video explanations  Links to similar exercises  You will be logged out of the exam automatically after 3 hours of inactivity. Your session will end.  IMPORTANT! You will also be logged out of the exam if you use your back button on your browser. You session will end. Slide 3 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

4. Due for this week…  Homework 3 (on MyMathLab – via the Materials Link)  The fifth night after class at 11:59pm.  Do the MyMathLab Self-Check for week 5.  Learning team hardest problem assignment.  Complete the Week 5 study plan after submitting week 5 homework.  Participate in the Chat Discussions in the OLS (yes, one more time). Slide 4 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

5. Section 12.1 Composite and Inverse Functions Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

6. Objectives Composition of Functions One-to-One Functions Inverse Functions Tables and Graphs of Inverse Functions

7. The compositions and represent evaluating functions f and g in two different ways. When evaluating function f is performed first followed by function g, whereas for functions g is performed first followed by function f.

8. Example Evaluate a.b. Solution a. b. Try Q’s pg 806 13,15,19

9. Example Use Table 12.1 and 12.2 to evaluate the expression. Table 9.1 x0123 f(x)f(x)1257 x0123 g(x)g(x) 11 012 Table 9.2 Try Q’s pg 806 23,25

10. Example Use the graph below to evaluate Try Q’s pg 806 29

12. Example Determine whether each graph represents a one-to- one function. a.b. The function is not one-to-one. The function is one-to-one. Try Q’s pg 806 37,39

14. Example State the inverse operations for the statement. Then write a function f for the given statement and a function g for its inverse operations. Multiply x by 4. Solution The inverse of multiplying by 4 is to divide by 4. Try Q’s pg 807 43,49

16. Example Find the inverse of the one-to-one function. f(x) = 4x – 3 Solution Step 1: Let y = 4x – 3 Step 2: Write the formula as x = 4y – 3 Step 3: Solve for y. Try Q’s pg 807 63,71

17. Tables and Graphs of Inverse Functions Inverse functions can be represented with tables and graphs. The table below shows a table of values for a function f. x12345 f(x)f(x) 48121620 x48121620 f - 1 (x) 12345 The table below shows a table of values for the inverse of f.

18. Graphs of Inverse Functions

19. Section 12.3 Logarithmic Functions Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

20. Objectives The Common Logarithmic Function The Inverse of the Common Logarithmic Function Logarithms with Other Bases

21. Common Logarithmic Functions A common logarithm is an exponent having base 10. Denoted log or log 10.

22. Example Evaluate each expression, if possible. a. b. Solution a. Is x positive? No, x = –1000 is negative, log x is undefined. b. Is x positive? Yes, x = 10,000 Write x as 10 k for some real number k. 10,000 = 10 4 If x = 10 k, then log x = k; log x = log 10,000 = log 10 4 = 4 Try Q’s pg 835 27,31

23. Example Simplify each common logarithm. a.b.c. log 55 Solution a. b. c. log 55 The power of 10 is not obvious, use a calculator. Try Q’s pg 835 17,23, 29,43

24. Graphs The graph of a common logarithm increases very slowly for large values of x. For example, x must be 100 for log x to reach 2 and must be 1000 for log x to reach 3. The graph passes through the point (1, 0). Thus log 1 = 0. The graph does not exist for negative values of x. The domain of log x includes only positive numbers. The range of log x includes all real numbers. When 0 < x < 1, log x outputs negative values. The y-axis is a vertical asymptote, so as x approaches 0, log x approaches .

25. Graphs The graph of y = log x shown is a one-to-one function because it passes the horizontal line test.

26. Example Use inverse properties to simplify each expression. a. b. Solution a.b. Because 10 logx = x for any positive real number x, Try Q’s pg 835 25,35,37,41

27. Example Graph each function f and compare its graph to y = log x. a.b. Solution a. Use the knowledge of translations to sketch the graph. The graph of log(x – 3) is similar to the graph of log x, except it is translated 3 units to the right.

28. Example Graph each function f and compare its graph to y = log x. a.b. Solution b. Use the knowledge of translations to sketch the graph. The graph of log(x) + 2 is similar to the graph of log x, except it is translated 2 units upward. Try Q’s pg 835 47,49

29. Example Sound levels in decibels (dB) can be computed by f(x) = 160 + 10 log x, where x is the intensity of the sound in watts per square centimeter. Ordinary conversation has an intensity of 10 -10 w/cm 2. What decibel level is this? Solution To find the decibel level, evaluate f(10 -10 ). Try Q’s pg 836 105

30. Logarithms with Other Bases Common logarithms are base-10 logarithms, but we can define logarithms having other bases. For example base-2 logarithms are frequently used in computer science. A base-2 logarithm is an exponent having base 2.

31. Example Simplify each logarithm. a.b. Solution Try Q’s pg 835 81,83

32. Natural Logarithms The base-e logarithm is referred to as a natural logarithm and denoted either log e x or ln x. A natural logarithm is an exponent having base e. To evaluate natural logarithms we usually use a calculator.

33. Example Approximate to the nearest hundredth. a.b. Solution Try Q’s pg 836 61,63

35. Example Simplify each logarithm. a.b. Solution Try Q’s pg 835-6 55,75,85,87

37. Example Simplify each expression. a.b. Solution because e lnk = k for all positive k. for x > 4 because 10 logk = k for all positive k. Try Q’s pg 835-6 39,57,59,89

38. Section 14.1 Sequences Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

39. Objectives Basic Concepts Representations of Sequences Models and Applications

40. A finite sequence is a function whose domain is D = {1, 2, 3, …, n} for some fixed natural number n. An infinite sequence is a function whose domain is the set of natural numbers. SEQUENCES The nth term, or general term, of a sequence is a n = f(n).

41. Example Write the first four terms of each sequence for n = 1, 2, 3, and 4. a. f(n) = 5n + 3 b. f(n) = (4) n-1 + 2 Solution a. f(n) = 5n + 3 a 1 = f(1) = 5(1) + 3 = 8 a 2 = f(2) = 5(2) + 3 = 13 a 3 = f(3) = 5(3) + 3 = 18 a 4 = f(4) = 5(4) + 3 = 23 The first four terms are 8, 13, 18, and 23.

42. Example (cont) Write the first four terms of each sequence for n = 1, 2, 3, and 4. a. f(n) = 5n + 3 b. f(n) = (4) n-1 + 2 Solution b. f(n) = (4) n-1 + 2 a 1 = f(1) = (4) 1-1 + 2 = 3 a 2 = f(2) = (4) 2-1 + 2 = 6 a 3 = f(3) = (4) 3-1 + 2 = 18 a 4 = f(4) = (4) 4-1 + 2 = 66 The first four terms are 3, 6, 18, and 66. Try Q’s pg 915 9,11,13

43. Example Use the graph to write the terms of the sequence. Solution The points (1, 2), (2, 4), (3, −6), (4, 8), and (5, −10) are shown in the graph. The terms of the sequence are 2, 4, −6, 8, and −10.

44. Example An employee at a parcel delivery company has 20 hours of overtime each month. Give symbolic, numerical, and graphical representations for a sequence that models the total amount of overtime in a 6 month period. Solution Symbolic Representation Let a n = 20n for n = 1, 2, 3, …, 6 Numerical Representation n123456 anan 20406080100120

45. Example (cont) Graphical Representation Plot the points (1, 20), (2, 40), (3, 60), (4, 80), (5, 100), (6, 120). Months Hours Overtime

46. Example Suppose that the initial population of adult female insects is 700 per acre and that r = 1.09. Then the average number of female insects per acre at the beginning of the year n is described by a n = 700(1.09) n-1. (See Example 4.) Solution Numerical Representation n123456 anan 700763831.6 7 906.5 2 988.1 1 1077.04

47. Example (cont) Graphical Representation Plot the points (1, 700), (2, 763), (3, 831.67), (4, 906.52), (5, 988.11), and (6, 1077.04). These results indicate that the insect population gradually increases. Because the growth factor is 1.09, the population is increasing by 9% each year. Year Insect Population (per acre) Try Q’s pg 916 29, 39,45

48. Section 14.2 Arithmetic and Geometric Sequences Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

49. Objectives Representations of Arithmetic Sequences Representations of Geometric Sequences Applications and Models

50. An arithmetic sequence is a linear function given by a n = dn + c whose domain is the set of natural numbers. The value of d is called the common difference. ARITHMETIC SEQUENCE

51. Example Determine whether f is an arithmetic sequence. If it is, identify the common difference d. a. f(n) = 7n + 4 Solution a. This sequence is arithmetic because f(x) = 7n + 4 defines a linear function. The common difference is d = 7.

52. Example (cont) Determine whether f is an arithmetic sequence. If it is, identify the common difference d. b. c. nf(n)f(n) 1−8 2−5 3−2 41 54 The table reveals that each term is found by adding +3 to the previous term. This represents an arithmetic sequence with the common difference of 3. The sequence shown in the graph is not an arithmetic sequence because the points are not collinear. That is, there is no common difference. Try Q’s pg 923-24 11,17,25

53. Example Find the general term a n for each arithmetic sequence. a. a 1 = 4 and d = −3b. a 1 = 5 and a 8 = 33 Solution a. Let a n = dn + c for d = −3, we write a n = −3n + c, and to find c we use a 1 = 4. a 1 = −3(1) + c = 4 or c = 7 Thus, a n = −3n + 7. b. The common difference is Therefore, a n = 4n + c. To find c we use a 1 = 5. a 1 = 4(1) + c = 5 or c = 1. Thus a n = 4n + 1. Try Q’s pg 924 29,31

54. The nth term a n of an arithmetic sequence is given by a n = a 1 + (n – 1)d, where a 1 is the first term and d is the common difference. GENERAL TERM OF AN ARITHMETIC SEQUENCE

55. Example If a 1 = 2 and d = 5, find a 17 Solution To find a 17 apply the formula a n = a 1 + (n – 1)d. a 17 = 2 + (17 – 1)5 = 82 Try Q’s pg 924 35

56. A geometric sequence is given by a n = a 1 (r) n-1, where n is a natural number and r ≠ 0 or 1. The value of r is called the common ratio, and a 1 is the first term of the sequence. GEOMETRIC SEQUENCE

57. Example Determine whether f is a geometric sequence. If it is, identify the common ratio. a. f(n) = 4(1.7) n-1 b.c. nf(n)f(n) 136 212 34 44/3 54/9

58. Example Determine whether f is a geometric sequence. If it is, identify the common ratio. a. f(n) = 4(1.7) n-1 b. nf(n)f(n) 136 212 34 44/3 54/9 This sequence is geometric because f(x) = 4(1.7) n -1 defines a exponential function. The common ratio is 1.7. The table reveals that each successive term is one-third the previous. This sequence represents a geometric sequence with a common ration of r = 1/3. Try Q’s pg 924 41,45,53

59. Example Find a general term a n for each geometric sequence. a. a 1 = 4 and r = 5 b. a 1 = 3, a 3 = 12, and r < 0. Solution a. Let a n = a 1 (r) n-1. Thus, a n = 4(5) n-1 b. a n = a 1 (r) n-1 a 3 = a 1 (r) 3-1 12 = 3r 2 4 = r 2 r = ±2 It is specified that r < 0, so r = −2 and a n = 3(−2) n-1.

60. Example If a 1 = 2 and r = 4, find a 9 Solution To find a 9 apply the formula a n = a 1 (r) n-1 with a 1 = 2, r = 4, and n = 9. a 9 = 2(4) 9-1 a 9 = 2(4) 8 a 9 = 131,072 Try Q’s pg 925 61

61. Example A swimming pool on a warm sunny day begins with a high chlorine content of 5 parts per million. (Assume that each day 40% of the chlorine dissipates.) a. Write a sequence that models the amount of chlorine in the pool at the beginning of the first 3 days, assuming that no additional chlorine is added and that the other days are just as warm and sunny. Solution a.Because 40% dissipates, 60% remains in the water at the beginning of the next day. If the concentration at the beginning of the first day is 5 parts per million, then at the beginning of the second day it is 5 ∙ 0.6 = 3 parts per million and at the start of the third day it is 3 ∙ 0.6 = 1.8 parts per million. The first three terms are 5, 3, 1.8. Thus this is a geometric sequence, with common ratio 0.6.

62. Example A swimming pool on a warm sunny day begins with a high chlorine content of 5 parts per million. (Assume that each day 40% of the chlorine dissipates.) b. Write the general term for this sequence. Solution b.The initial amount is a 1 = 5 and the common ratio is r = 0.6, so the sequence can be represented by a n = 5(0.6) n – 1.

63. Section 14.3 Series Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

64. Objectives Basic Concepts Arithmetic Series Geometric Series Summation Notation

65. Introduction A series is the summation of the terms in a sequence. Series are used to approximate functions that are too complicated to have a simple formula. Series are instrumental in calculating accurate approximations of numbers like Slide 65

66. A finite series is an expression of the form a 1 + a 2 + a 3 + ∙∙∙ + a n. FINITE SERIES

67. Example The table represents the number of Lyme disease cases reported in Connecticut from 1999 – 2005, where n = 1 corresponds to 1999. a.Write a series whose sum represents the total number of Lyme Disease cases reported from 1999 to 2005. Find its sum. b. Interpret the series a 1 + a 2 + a 3 + ∙∙∙ + a 9. n1234567 anan 3215377335974631140313481810

68. Example (cont) a.Write a series whose sum represents the total number of Lyme Disease cases reported from 1999 to 2005. Find its sum. The required series and sum are given by: 3215 + 3773 + 3597 + 4631 + 1403 + 1348 + 1810 = 19,777. b. Interpret the series a 1 + a 2 + a 3 + ∙∙∙ + a 9. This represents the total number of Lyme Disease cases reported over 9 years from 1999 through 2007. n1234567 anan 3215377335974631140313481810 Try Q’s pg 933 41

69. The sum of the first n terms of an arithmetic sequence denoted S n, is found by averaging the first and nth terms and then multiplying by n. That is, S n = a 1 + a 2 + a 3 + ∙∙∙ + a n = SUM OF THE FIRST n TERMS OF AN ARITHMETIC SEQUENCE

70. Example A worker has a starting annual salary of $45,000 and receives a $2500 raise each year. Calculate the total amount earned over 5 years. Solution The arithmetic sequence describing the salary during year n is computed by a n = 45,000 + 2500(n – 1). The first and fifth years’ salaries are a 1 = 45,000 + 2500(1 – 1) = 45,000 a 5 = 45,000 + 2500(5 – 1) = 55,000

71. Example (cont) Thus the total amount earned during this 5-year period is The sum can also be found using Try Q’s pg 934 49

72. Example Find the sum of the series Solution The series has n = 19 terms with a 1 = 4 and a 19 = 58. We can then use the formula to find the sum. Try Q’s pg 933 13

73. If its first term is a 1 and its common ration is r, then the sum of the first n terms of a geometric sequence is given by provided r ≠ 1. SUM OF THE FIRST n TERMS OF A GEOMETRIC SEQUENCE

74. Example Find the sum of the series 5 + 15 + 45 + 135 + 405. Solution The series is geometric with n = 5, a 1 = 5, and r = 3, so Try Q’s pg 933 17,19

75. Example A 30-year-old employee deposits $4000 into an account at the end of each year until age 65. If the interest rate is 8%, find the future value of the annuity. Solution Let a 1 = 4000, I = 0.08, and n = 35. The future value of the annuity is Try Q’s pg 934 23

76. SUMMATION NOTATION

77. Example Find each sum. a.b.c. Solution a. b.

78. Example (cont) Find each sum. a.b.c. Solution c. Try Q’s pg 934 27,29,31

79. End of week 5  You again have the answers to those problems not assigned  Practice is SOOO important in this course.  Work as much as you can with MyMathLab, the materials in the text, and on my Webpage.  Do everything you can scrape time up for, first the hardest topics then the easiest.  You are building a skill like typing, skiing, playing a game, solving puzzles.