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MTH 209 Week 5 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. . Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2 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. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 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). Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 Section 12.1 Composite and Inverse Functions Copyright © 2013, 2009, and 2005 Pearson Education, Inc. Objectives • Composition of Functions • One-to-One Functions • Inverse Functions • Tables and Graphs of Inverse Functions The compositionsg f andf g represent evaluating functions f and g in two different g f, ways. When evaluating function f is f g g, performed first followed by function whereas for functions g is performed first followed by function f. Example Evaluate g f (3). a. f ( x) x2 ; g ( x) 2 x 3 Solution a. g f (3) g ( f (3)) b. g Try Q’s pg 806 2 f ( x ) 2 x ; g ( x ) x 2x 1 b. g (9 ) f (3) 32 9 15 g (9) 2(9) 3 f (3) g ( f (3)) g ( 6) 25 13,15,19 f (3) 2(3) 6 g (6) 62 2(6) 1 Example Try Q’s pg 806 23,25 Use Table 12.1 and 12.2 to evaluate the expression. f g (3) Table 9.2 Table 9.1 x 0 1 2 3 x 0 1 2 3 f(x) 1 2 5 7 g(x ) 1 0 1 2 f g (3) f ( g (3)) f ( 2) 5 Example Try Q’s pg 806 29 Use the graph below to evaluate ( g f )(2). Example Try Q’s pg 806 37,39 Determine whether each graph represents a one-toone function. a. b. The function is not one-to-one. The function is one-to-one. Example Try Q’s pg 807 43,49 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. f ( x) 4 x x g ( x) 4 Example Try Q’s pg 807 63,71 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. x 4 y 3 x 3 4y x3 y 4 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. x 1 2 3 4 5 f(x ) 4 8 12 16 20 The table below shows a table of values for the inverse of f. x 4 8 12 16 20 f1(x) 1 2 3 4 5 Graphs of Inverse Functions f ( x) x 2 f 1 ( x) x 2 Section 12.3 Logarithmic Functions Copyright © 2013, 2009, and 2005 Pearson Education, Inc. Objectives • The Common Logarithmic Function • The Inverse of the Common Logarithmic Function • Logarithms with Other Bases Common Logarithmic Functions A common logarithm is an exponent having base 10. Denoted log or log10. Example Try Q’s pg 835 27,31 Evaluate each expression, if possible. 2 log( 1000) log100 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 10k for some real number k. 10,000 = 104 If x = 10k, then log x = k; log x = log 10,000 = log 104 = 4 Example Try Q’s pg 835 17,23, 29,43 Simplify each common logarithm. 1 log1000 log a. b. c. log 55 100 Solution a. log1000 c. log 55 1000 103 log1000 103 3 b. log 1 100 1 10 2 100 1 log log102 2 100 The power of 10 is not obvious, use a calculator. 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 . Graphs The graph of y = log x shown is a one-to-one function because it passes the horizontal line test. Example Try Q’s pg 835 25,35,37,41 Use inverse properties to simplify each expression. x 4 log10 a. b.10log8 2 Solution a. log10x 4 2 x2 4 b.10log8 Because 10logx = x for any positive real number x, 10log8 8 Example Graph each function f and compare its graph to y = log x. a. log( x 3) b. log( x) 2 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. Example Try Q’s pg 835 47,49 Graph each function f and compare its graph to y = log x. a. log( x 3) b. log( x) 2 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. Example Try Q’s pg 836 105 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/cm2. What decibel level is this? Solution To find the decibel level, evaluate f(10-10). f ( x) 160 10log x f (1010 ) 160 10 log(1010 ) 160 10(10) 60 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. Example Try Q’s pg 835 81,83 Simplify each logarithm. 1 a. log 2 16 b. log 2 32 Solution a. 16 2 4 log2 16 log2 24 4 b. 1 25 32 1 log2 log2 25 5 32 Natural Logarithms The base-e logarithm is referred to as a natural logarithm and denoted either logex or ln x. A natural logarithm is an exponent having base e. To evaluate natural logarithms we usually use a calculator. Example Try Q’s pg 836 61,63 Approximate to the nearest hundredth. 1 ln a. ln 20 b. 4 Solution a.ln 20 b. ln 1 4 Example Try Q’s pg 835-6 55,75,85,87 Simplify each logarithm. a. log6 36 b. log 3 9 2 Solution a. log 6 36 36 62 log 6 36 log 6 62 2 b. log 3 92 92 (32 )2 34 log3 34 4 Example Try Q’s pg 835-6 39,57,59,89 Simplify each expression. a. eln 6 b. 10log(4 x16) Solution a. e ln 6 6 because elnk = k for all positive k. b. 10log(4 x16) 4x 16 for x > 4 because 10logk = k for all positive k. Section 14.1 Sequences Copyright © 2013, 2009, and 2005 Pearson Education, Inc. Objectives • Basic Concepts • Representations of Sequences • Models and Applications f x a , a 0 and a 1, SEQUENCES x 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. The nth term, or general term, of a sequence is an = f(n). 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 a1 = f(1) = 5(1) + 3 = 8 a2 = f(2) = 5(2) + 3 = 13 a3 = f(3) = 5(3) + 3 = 18 a4 = f(4) = 5(4) + 3 = 23 The first four terms are 8, 13, 18, and 23. Example (cont) Try Q’s pg 915 9,11,13 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 a1 = f(1) = (4)1-1 + 2 = 3 a2 = f(2) = (4)2-1 + 2 = 6 a3 = f(3) = (4)3-1 + 2 = 18 a4 = f(4) = (4)4-1 + 2 = 66 The first four terms are 3, 6, 18, and 66. 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. 10 8 6 4 2 X 0 The terms of the sequence are 2, 4, −6, 8, and −10. Y -2 -4 -6 -8 -10 1 2 3 4 5 6 7 8 9 10 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 an = 20n for n = 1, 2, 3, …, 6 Numerical Representation n an 1 20 2 40 3 60 4 80 5 6 100 120 Example (cont) Graphical Representation Plot the points (1, 20), (2, 40), (3, 60), (4, 80), (5, 100), (6, 120). 150 Y 135 120 Hours 105 90 75 60 45 30 15 0 X 1 2 3 4 5 6 7 Months Overtime 8 9 10 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 an = 700(1.09)n-1. (See Example 4.) Solution Numerical Representation n an 1 2 700 763 3 831.6 7 4 906.5 2 5 6 988.1 1077.04 1 Example (cont) Try Q’s pg 916 29, 39,45 1200 Insect Population (per acre) Graphical Representation Plot the points (1, 700), (2, 763), (3, 831.67), (4, 906.52), (5, 988.11), and (6, 1077.04). Y 1080 960 840 720 600 480 360 240 120 0 X 1 2 3 4 5 6 7 8 9 10 Year These results indicate that the insect population gradually increases. Because the growth factor is 1.09, the population is increasing by 9% each year. Section 14.2 Arithmetic and Geometric Sequences Copyright © 2013, 2009, and 2005 Pearson Education, Inc. Objectives • Representations of Arithmetic Sequences • Representations of Geometric Sequences • Applications and Models f x aSEQUENCE , a 0 and a 1, ARITHMETIC x An arithmetic sequence is a linear function given by an = dn + c whose domain is the set of natural numbers. The value of d is called the common difference. 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. Example (cont) Try Q’s pg 923-24 11,17,25 Determine whether f is an arithmetic sequence. If it is, identify the common difference d. b. n f(n) c. 20 Y 18 1 2 3 4 5 −8 −5 −2 1 4 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. 16 14 12 10 8 6 4 2 0 X 1 2 3 4 5 6 7 8 The sequence shown in the graph is not an arithmetic sequence because the points are not collinear. That is, there is no common difference. Example Try Q’s pg 924 29,31 Find the general term an for each arithmetic sequence. a. a1 = 4 and d = −3 b. a1 = 5 and a8 = 33 Solution a. Let an = dn + c for d = −3, we write an = −3n + c, and to find c we use a1 = 4. a1 = −3(1) + c = 4 or c = 7 33 5 Thus, an = −3n + 7. d 4. 8 1 b. The common difference is Therefore, an = 4n + c. To find c we use a1 = 5. a1 = 4(1) + c = 5 or c = 1. Thus an = 4n + 1. GENERAL TERM OF AN ARITHMETIC f x a , a 0 and a 1, SEQUENCE x The nth term an of an arithmetic sequence is given by an = a1 + (n – 1)d, where a1 is the first term and d is the common difference. Example Try Q’s pg 924 35 If a1 = 2 and d = 5, find a17 Solution To find a17 apply the formula an = a1 + (n – 1)d. a17 = 2 + (17 – 1)5 = 82 f x SEQUENCE a , a 0 and a 1, GEOMETRIC x A geometric sequence is given by an = a1(r)n-1, where n is a natural number and r ≠ 0 or 1. The value of r is called the common ratio, and a1 is the first term of the sequence. 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. n f(n) 1 36 2 3 4 5 12 4 4/3 4/9 10 Y 9 8 7 6 5 4 3 2 1 0 X 1 2 3 4 5 6 7 8 Example Try Q’s pg 924 41,45,53 Determine whether f is a geometric sequence. If it is, identify the common ratio. a. f(n) = 4(1.7)n-1 This sequence is geometric because f(x) = 4(1.7)n -1 defines a exponential function. The common ratio is 1.7. b. n f(n) 1 2 3 36 12 4 4 5 4/3 4/9 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. Example Find a general term an for each geometric sequence. a. a1 = 4 and r = 5 b. a1 = 3, a3 = 12, and r < 0. Solution a. Let an = a1(r)n-1. Thus, an = 4(5)n-1 b. an = a1(r)n-1 a3 = a1(r)3-1 12 = 3r2 4 = r2 r = ±2 It is specified that r < 0, so r = −2 and an = 3(−2)n-1. Example Try Q’s pg 925 61 If a1 = 2 and r = 4, find a9 Solution To find a9 apply the formula an = a1(r)n-1 with a1 = 2, r = 4, and n = 9. a9 = 2(4)9-1 a9 = 2(4)8 a9 = 131,072 Section 14.3 Series Copyright © 2013, 2009, and 2005 Pearson Education, Inc. Objectives • Basic Concepts • Arithmetic Series • Geometric Series • Summation Notation 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. and e.accurate Series are instrumental in calculating approximations of numbers like Slide 65 FINITEf SERIES x a , a 0 and a 1, x A finite series is an expression of the form a1 + a2 + a3+ ∙∙∙ + an. Example The table represents the number of Lyme disease cases reported in Connecticut from 1999 – 2005, where n = 1 corresponds to 1999. n an 1 2 3 4 5 6 7 3215 3773 3597 4631 1403 1348 1810 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 a1 + a2 + a3+ ∙∙∙ + a9. Example (cont) n an Try Q’s pg 933 41 1 2 3 4 5 6 7 3215 3773 3597 4631 1403 1348 1810 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 a1 + a2 + a3+ ∙∙∙ + a9. This represents the total number of Lyme Disease cases reported over 9 years from 1999 through 2007. SUM OF THE FIRST n TERMS OF f x a , a 0 and a 1, AN ARITHMETIC SEQUENCE x The sum of the first n terms of an arithmetic sequence denoted Sn, is found by averaging the first and nth terms and then multiplying by n. That is, a1 an Sn = a1 + a2 + a3 + ∙∙∙ + an n= 2 . 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 an = 45,000 + 2500(n – 1). The first and fifth years’ salaries are a1 = 45,000 + 2500(1 – 1) = 45,000 a5 = 45,000 + 2500(5 – 1) = 55,000 Example (cont) Try Q’s pg 934 49 Thus the total amount earned during this 5-year period is 45, 000 55, 000 S5 5 2 $250, 000. The sum can also be found using n S n 2a1 n 1 d 2 5 S5 2 45, 000 5 1 2500 $250, 000. 2 Example Try Q’s pg 933 13 Find the sum of the series 4 7 10 58. Solution The series has n = 19 terms with a1 = 4 and a19 = 58. We can then use the formula to find the sum. a1 an Sn n 2 4 58 S19 19 2 589 SUM OF THE FIRST x n TERMS OF A GEOMETRIC f x a , a 0 and a 1, SEQUENCE If its first term is a1 and its common ration is r, then the sum of the first n terms of a geometric sequence is given by 1 rn Sn a1 , 1 r provided r ≠ 1. Example Try Q’s pg 933 17,19 Find the sum of the series 5 + 15 + 45 + 135 + 405. Solution The series is geometric with n = 5, a1 = 5, and r = 3, so 1 35 S5 5 605. 1 3 Example Try Q’s pg 934 23 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 1 I n 1 Let a1 = 4000, I = 0.08, and n = 35. Sn a1 I The future value of the annuity is 1 0.0835 1 4000 0.08 $689, 267. SUMMATION NOTATION n a k 1 k a1 a2 a3 an Example Find3 each sum. a. 4k k 1 6 3 c. 3k 6 b. 4 k 1 k 1 Solution 3 a. 4k 4(1) 4(2) 4(3) k 1 = 4 8 12 24 3 b. 4 4 4 4 12 k 1 Example (cont) Try Q’s pg 934 27,29,31 Find3 each sum. a. 4k k 1 3 b. 4 k 1 6 c. 3k 6 k 1 Solution 6 c. 3k 6 3 1 6 3 2 6 3 3 6 3 4 6 3 5 6 3 6 6 k 1 9 12 15 18 21 24 99 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.