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.

Transcription

1 MTH 209 Week 5

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.. Copyright 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2

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. Copyright 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3

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). Copyright 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4

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

8 Example Try Q s pg ,15,19 g (3). Evaluate 2 a. f ( x) x ; g( x) 2x 3 b. Solution a. g f (3) g( f( 3) ) f g(9) 15 f x x g x x x 2 f (3) 3 9 g(9) 2(9) 3 2 ( ) 2 ; ( ) 2 1 b. g f (3) g( f( 3) ) g(6) 25 f (3) 2(3) 6 2 g(6) 6 2(6) 1

9 Example Try Q s pg ,25 Use Table 12.1 and 12.2 to evaluate the expression. f g (3) Table 9.1 x f(x) f g (3) f( g( 3) ) f 5 (2) Table 9.2 x g(x )

10 Example Try Q s pg Use the graph below to evaluate ( g f)(2).

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12 Example Try Q s pg ,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.

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14 Example Try Q s pg ,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) 4x gx ( ) x 4

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16 Example Try Q s pg ,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 4y 3 x 3 4y x 3 4 y

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. x f(x ) The table below shows a table of values for the inverse of f. x f - 1 (x)

18 Graphs of Inverse Functions f ( x) x 2 f 1 ( x) x 2

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 Try Q s pg ,31 Evaluate each expression, if possible. a. log( 1000) 2 b. log100 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

23 Example Try Q s pg ,23, 29,43 Simplify each common logarithm. a. log b. c. log 55 log 100 Solution a. log1000 b. 1 log log log log c. log 55 The power of 10 is not obvious, use a calculator.

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 Try Q s pg ,35,37,41 Use inverse properties to simplify each expression. a. 2 4 log10 x log8 b. 10 Solution a. b. 2 4 log10 x x log8 Because 10 logx = x for any positive real number x, log8 10 8

27 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.

28 Example Try Q s pg ,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.

29 Example Try Q s pg Sound levels in decibels (db) can be computed by f(x) = log x, where x is the intensity of the sound in watts per square centimeter. Ordinary conversation has an intensity of w/cm 2. What decibel level is this? Solution To find the decibel level, evaluate f(10-10 ). f f ( x) log x (10 ) log(1 0 ) ( 10) 60

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 Try Q s pg ,83 Simplify each logarithm. a. b. 1 log216 log2 32 Solution 4 a log2 16 log b log2 log

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 Try Q s pg ,63 Approximate to the nearest hundredth. 1 a. ln 20 b. ln 4 Solution a.ln 20 1 b. ln 4

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35 Example Try Q s pg ,75,85,87 Simplify each logarithm. a. b. 2 log 36 log Solution a. log log636 log6 6 2 b. log (3 ) log3 3 4

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37 Example Try Q s pg ,57,59,89 Simplify each expression. ln 6 a. e b. 10 log(4 x 16) Solution ln6 a. e 6 because e lnk = k for all positive k. log(4 x 16) b. 10 4x 16 for x > 4 because 10 logk = k for all positive k.

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

39 Objectives Basic Concepts Representations of Sequences Models and Applications

40 SEQUENCES x, 0 and 1, f x a a a 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 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 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) 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 Solution b. f(n) = (4) n a 1 = f(1) = (4) = 3 a 2 = f(2) = (4) = 6 a 3 = f(3) = (4) = 18 a 4 = f(4) = (4) = 66 The first four terms are 3, 6, 18, and 66.

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 Y X -8-10

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 n a n

45 Hours Example (cont) Graphical Representation Plot the points (1, 20), (2, 40), (3, 60), (4, 80), (5, 100), Y 150 (6, 120) Months Overtime X

46 Example Suppose that the initial population of adult female insects is 700 per acre and that r = 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 n a n

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

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 x, 0 and 1, ARITHMETIC SEQUENCE f x a a a 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.

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. Y 20 b. c. n f(n) 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. Try Q s pg ,17, The sequence shown in the graph is not an arithmetic sequence because the points are not collinear. That is, there is no common difference. X

53 Example Try Q s pg ,31 Find the general term a n for each arithmetic sequence. a. a 1 = 4 and d = 3 b. 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 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. d 4.

54 GENERAL TERM OF AN ARITHMETIC SEQUENCE x, 0 and 1, f x a a a 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.

55 Example Try Q s pg 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

56 x, 0 and 1, GEOMETRIC SEQUENCE f x a a a 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.

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. Y 10 n f(n) /3 5 4/9 X

58 Example Try Q s pg ,45,53 Determine whether f is a geometric sequence. If it is, identify the common ratio. a. f(n) = 4(1.7) n-1 b. n f(n) /3 5 4/9 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.

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) = 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 Try Q s pg 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

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

62 Objectives Basic Concepts Arithmetic Series Geometric Series Summation Notation

63 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. Series are instrumental in calculating accurate approximations of numbers like Slide 65

64 FINITE SERIES x, 0 and 1, f x a a a A finite series is an expression of the form a 1 + a 2 + a a n.

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

66 Example (cont) Try Q s pg n a n a. Write a series whose sum represents the total number of Lyme Disease cases reported from 1999 to Find its sum. The required series and sum are given by: = 19,777. b. Interpret the series a 1 + a 2 + a a 9. This represents the total number of Lyme Disease cases reported over 9 years from 1999 through 2007.

67 SUM OF THE FIRST n TERMS OF x, 0 and 1, f x a a a AN ARITHMETIC SEQUENCE 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, n S n = a 1 + a 2 + a a n = a a n 2 1.

68 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, (n 1). The first and fifth years salaries are a 1 = 45, (1 1) = 45,000 a 5 = 45, (5 1) = 55,000

69 Example (cont) Try Q s pg Thus the total amount earned during this 5-year period is 45, , 000 S5 5 $250, The sum can also be found using n S 2 n a1 n 1 d 2 5 S , $250,000. 2

70 Example Try Q s pg 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. a1 an Sn n 2 S

71 SUM OF THE FIRST f x a x n TERMS OF A GEOMETRIC SEQUENCE, a 0 and a 1, 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 S provided r 1. n 1 r 1 r, n a1

72 Example Try Q s pg ,19 Find the sum of the series Solution The series is geometric with n = 5, a 1 = 5, and r = 3, so S

73 Example Try Q s pg 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 n Let a 1 = 4000, I = 0.08, and n = 35. I n 1 The future value of the annuity is I 1 1 S a $689,267.

74 SUMMATION NOTATION n k 1 a a a a a k n

75 Example Find each sum. 3 a. 4k b c. 3k 6 k 1 Solution k 1 k 1 3 a. b. k 1 4k 4(1) 4(2) 4(3) = k

76 Example (cont) Find each sum. 3 a. 4k b c. 3k 6 Solution c. k 1 6 k 1 k k Try Q s pg ,29,31 k

77 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.