Expotential And Logarithmic Sequences And Series Probability

75 Exponential and Logarithmic Equations Student Outcomes Students apply properties of logarithms to solve exponential equations Students relate solutions to fx gx to the intersection points on the graphs of y fx and y gx in the case where f and g are constant or exponential functions Exercises 1 Solve the following equations a 2 5𝑥3 6250 b 3 62𝑥 648 c 5 23𝑥5 10240 d 43𝑥1 32 e 3 25𝑥 216 f 5 113𝑥 120 g 7 9𝑥 5405 h 3 33𝑥 9 i log400 85𝑥 log160000 2 Mary came up with the model 𝑓𝑡 07011382𝑡 for the first bean activity When does her model predict that she would have 1000 beans 3 Jack came up with the model 𝑔𝑡 10331707𝑡 for the first bean activity When does his model predict that he would have 50000 beans 4 If instead of beans in the first bean activity you were using fair pennies when would you expect to have 1000000 5 Let 𝑓𝑥 2 3𝑥 and 𝑔𝑥 3 2𝑥 a Which function is growing faster as 𝑥 increases Why b When will 𝑓𝑥 𝑔𝑥 6 The growth of a population of E coli bacteria can be modeled by the function 𝐸𝑡 50011547𝑡 and the growth of a population of Salmonella bacteria can be modeled by the function 𝑆𝑡 40003668𝑡 where 𝑡 measures time in hours a Graph these two functions on the same set of axes At which value of 𝑡 does it appear that the graphs intersect b Use properties of logarithms to find the time 𝑡 when these two populations are the same size Give your answer to two decimal places 7 Chain emails contain a message suggesting you will have bad luck if you do not forward the email to others Suppose a student started a chain email by sending the message to 10 friends and asking those friends to each send the same email to 3 more friends exactly one day after receiving the message Assuming that everyone that gets the email participates in the chain we can model the number of people who receive the email on the 𝑛th day by the formula 𝐸𝑛 103𝑛 where 𝑛 0 indicates the day the original email was sent a If we assume the population of the United States is 318 million people and everyone who receives the email sends it to 3 people who have not received it previously how many days until there are as many emails being sent out as there are people in the United States b The population of Earth is approximately 71 billion people On what day will 71 billion emails be sent out 8 Solve the following exponential equations a 103𝑥5 7𝑥 b 3 𝑥 5 24𝑥2 c 10𝑥25 1002𝑥2𝑥2 d 4𝑥23𝑥4 25𝑥4 9 Solve the following exponential equations a 2𝑥𝑥 8𝑥 b 3𝑥𝑥 12 10 Solve the following exponential equations a 10𝑥1 10𝑥1 1287 b 24𝑥 4𝑥1 342 11 Solve the following exponential equations a 10𝑥2 310𝑥 2 0 Hint Let 𝑢 10𝑥 and solve for 𝑢 before solving for 𝑥 b 2𝑥2 32𝑥 4 0 c 3𝑒𝑥2 8𝑒𝑥 3 0 d 4𝑥 72𝑥 12 0 e 10𝑥2 210𝑥 1 0 12 Solve the following systems of equations a 2𝑥2𝑦 8 42𝑥𝑦 1 b 22𝑥𝑦1 32 4𝑥2𝑦 2 c 23𝑥 82𝑦1 92𝑦 33𝑥9 13 Because 𝑓𝑥 log𝑏𝑥 is an increasing function we know that if 𝑝 𝑞 then log𝑏𝑝 log𝑏𝑞 Thus if we take logarithms of both sides of an inequality then the inequality is preserved Use this property to solve the following inequalities a 4𝑥 5 3 b 2 7 𝑥 9 c 4𝑥 8𝑥1 d 3𝑥2 532𝑥 e 3 4 𝑥 4 3 𝑥1 73 Logarithmic Functions Student Outcomes Students graph the functions fx logx gx log2x and hx lnx by hand and identify key features of the graphs of logarithmic functions Summary The function 𝑓𝑥 log𝑏𝑥 is defined for irrational and rational numbers Its domain is all positive real numbers Its range is all real numbers The function 𝑓𝑥 log𝑏𝑥 goes to negative infinity as 𝑥 goes to zero It goes to positive infinity as 𝑥 goes to positive infinity The larger the base 𝑏 the more slowly the function 𝑓𝑥 log𝑏𝑥 increases By the change of base formula log1 𝑏𝑥 log𝑏𝑥 Exercises 1 The function 𝑄𝑥 log𝑏𝑥 has function values in the table at right a Use the values in the table to sketch the graph of 𝑦 𝑄𝑥 b What is the value of 𝑏 in 𝑄𝑥 log𝑏𝑥 Explain how you know c Identify the key features in the graph of 𝑦 𝑄𝑥 𝒙 𝑸𝒙 01 166 03 087 05 050 100 000 200 050 400 100 600 129 1000 166 1200 179 Consider the logarithmic functions 𝑓𝑥 log𝑏𝑥 𝑔𝑥 log5𝑥 where 𝑏 is a positive real number and 𝑏 1 The graph of 𝑓 is given at right a Is 𝑏 5 or is 𝑏 5 Explain how you know b Compare the domain and range of functions 𝑓 and 𝑔 c Compare the 𝑥intercepts and 𝑦 intercepts of 𝑓 and 𝑔 d Compare the end behavior of 𝑓 and 𝑔 Consider the logarithmic functions 𝑓𝑥 log𝑏𝑥 𝑔𝑥 log1 2𝑥 where 𝑏 is a positive real number and 𝑏 1 A table of approximate values of 𝑓 is given below a Is 𝑏 1 2 or is 𝑏 1 2 Explain how you know b Compare the domain and range of functions 𝑓 and 𝑔 c Compare the 𝑥intercepts and 𝑦intercepts of 𝑓 and 𝑔 d Compare the end behavior of 𝑓 and 𝑔 On the same set of axes sketch the functions 𝑓𝑥 log2𝑥 and 𝑔𝑥 log2𝑥3 a Describe a transformation that takes the graph of 𝑓 to the graph of 𝑔 b Use properties of logarithms to justify your observations in part a On the same set of axes sketch the functions 𝑓𝑥 log2𝑥 and 𝑔𝑥 log2 𝑥 4 a Describe a transformation that takes the graph of 𝑓 to the graph of 𝑔 b Use properties of logarithms to justify your observations in part a 𝒙 𝒇𝒙 1 4 086 1 2 043 1 0 2 043 4 086 On the same set of axes sketch the functions 𝑓𝑥 log1 2𝑥 and 𝑔𝑥 log2 1 𝑥 a Describe a transformation that takes the graph of 𝑓 to the graph of 𝑔 b Use properties of logarithms to justify your observations in part a The figure below shows graphs of the functions 𝑓𝑥 log3𝑥 𝑔𝑥 log5𝑥 and ℎ𝑥 log11𝑥 a Identify which graph corresponds to which function Explain how you know b Sketch the graph of 𝑘𝑥 log7𝑥 on the same axes The figure below shows graphs of the functions 𝑓𝑥 log1 3𝑥 𝑔𝑥 log1 5 𝑥 and ℎ𝑥 log 1 11𝑥 a Identify which graph corresponds to which function Explain how you know b Sketch the graph of 𝑘𝑥 log1 7𝑥 on the same axes For each function 𝑓 find a formula for the function ℎ in terms of 𝑥 Part a has been done for you a If 𝑓𝑥 𝑥2 𝑥 find ℎ𝑥 𝑓𝑥 1 b If 𝑓𝑥 𝑥2 1 4 find ℎ𝑥 𝑓 1 2 𝑥 c If 𝑓𝑥 log 𝑥 find ℎ𝑥 𝑓10𝑥 3 when 𝑥 0 d If 𝑓𝑥 3𝑥 find ℎ𝑥 𝑓log3𝑥2 3 e If 𝑓𝑥 𝑥3 find ℎ𝑥 𝑓 1 𝑥3 when 𝑥 0 f If 𝑓𝑥 𝑥3 find ℎ𝑥 𝑓𝑥 3 g If 𝑓𝑥 sin𝑥 find ℎ𝑥 𝑓 𝑥 𝜋 2 h If 𝑓𝑥 𝑥2 2𝑥 2 find ℎ𝑥 𝑓cos𝑥 For each of the functions 𝑓 and 𝑔 below write an expression for i 𝑓𝑔𝑥 ii 𝑔𝑓𝑥 and iii 𝑓𝑓𝑥 in terms of 𝑥 Part a has been done for you a 𝑓𝑥 𝑥2 𝑔𝑥 𝑥 1 i 𝑓𝑔𝑥 𝑓𝑥 1 𝑥 12 ii 𝑔𝑓𝑥 𝑔𝑥2 𝑥2 1 iii 𝑓𝑓𝑥 𝑓𝑥2 𝑥22 𝑥4 b 𝑓𝑥 1 4 𝑥 8 𝑔𝑥 4𝑥 1 c 𝑓𝑥 𝑥 1 3 𝑔𝑥 𝑥3 1 d 𝑓𝑥 𝑥3 𝑔𝑥 1 𝑥 e 𝑓𝑥 𝑥 𝑔𝑥 𝑥2 Extension Consider the functions 𝑓𝑥 log2𝑥 and 𝑥 𝑥 1 a Use a calculator or other graphing utility to produce graphs of 𝑓𝑥 log2𝑥 and 𝑔𝑥 𝑥 1 for 𝑥 17 b Compare the graph of the function 𝑓𝑥 log2𝑥 with the graph of the function 𝑔𝑥 𝑥 1 Describe the similarities and differences between the graphs c Is it always the case that log2𝑥 𝑥 1 for 𝑥 2 Consider the functions 𝑓𝑥 log2𝑥 and 𝑥 𝑥 1 3 a Use a calculator or other graphing utility to produce graphs of 𝑓𝑥 log2𝑥 and ℎ𝑥 𝑥 1 3 for 𝑥 28 b Compare the graph of the function 𝑓𝑥 log2𝑥 with the graph of the function ℎ𝑥 𝑥 1 3 Describe the similarities and differences between the graphs c Is it always the case that log2𝑥 𝑥 1 3 for 𝑥 2 1 1 Venn diagrams 2 Venn diagrams 3 Venn diagrams 4 Venn diagrams 2 1 F C 8 24 140 228 1 140 2 24 140 166 2 just divide the numbers above by 400 002 006 035 057 3 1 A 017 049 016 018 C 1 017 049 016 018 2 NA NNA T C 016 018 034 NC 049 017 066 T 065 035 1 4 1 012 4 or 2 and 5 028 3 081 6 The probability that the car is not a sedan 1 The sequence changes only when it reaches an odd week so a2m1 10 3m m 0 a2n a2n1 60 10 3m m 503 17 considering the nearest integer it takes 17 weeks 2 a Its a geometric sequence since the ratio of two consecutive terms is always the same namely 08 b After the mth bounce hm 1508m h5 15085 49 ft 3 a The difference of differences is constant and equals 6 Thus its a quadratic sequence b The general formula is q am2 bm c The second difference is 6 so 2a 6 a 3 Taking the difference between the first and second terms 9 312 03 b1 0 b 6 From the m0 term c8 q 3m2 3m 8 4 The second difference is 2 so its a quadratic sequence with 2a 2 a 1 The first difference is 3 112 02 b1 0 b 2 m 0 q 2 c 2 qm m2 2m 2 qm 100 gives 100 m2 2m 2 m2 2m 98 0 m 2 sqrt4 392 2 1 sqrt99 1 3sqrt11 The negative root has no meaning so it will take m 1 311 9 months to reach the goal 1a 5x3 3125 log55x3 log53125 x3 5 x 2 b 62x 276 log662x log6276 2x 3 x 32 c 23x5 2048 log223x5 log22048 3x5 11 x 2 d log243x1 log232 23x1 5 6x2 5 6x 7 x 76 e 25x 72 5x log272 x log2725 f 113x 24 3x log11 log24 x log243 log11 g 9x 54057 x log9 log5405 log7 x log5405 log7log9 a 33x 93 32 312 332 3x 32 x 12 i 85x log160000 log400 log4002 log400 2 235x 2 215x 2 x 115 2 fx 1000 1000 07011382x 1000 1382x x log1000 log0701 log1382 3 log0701 log1382 x 2245 3 50000 10331707x 50000 1033 1707x x log50000 1033 log1707 x 2017 4 1000000 corresponds to 100000000 pennies By Marys model 108 0701 1382x 108 1382x 0701 log108 0701 x log 1382 x 8 log0701log1382 5803 By Jacks model the calculation is the same so 108 1033 1707x x 8 log1033 log1707 3439 5 a f grows faster because it has the larger base 3 b 23x 3 2x log2 23x log2 32x 1 x log2 3 log2 3 x 1 log2 3 x 1 log2 3 x 1 6 a From the plots they intersect around 18 hours b 500 11547x 4000 3668x 11547x 8 3668x x log 11547 log 8 x log 3668 x log 11547 log 3668 log 8 x log 8 log 11547 log 3668 181 7 a 318105 103m log318 5 m log 3 m log318 5 7572 days log 3 b 77103 103m log 77103 m log 3 m 1855 days 8 a 3x 5 x log 7 x3 log 7 5 x 5 3 log 7 b x 15 log2 3 4x 2 x 4 log2 315 2 x 2 4 log2 315 c 10x2 5 1002x2 x 2 10 4x2 2x 4 3x2 2x 1 0 x 2 4 12 6 x 1 x 13 d 4x2 3x 4 22x2 6x 8 4x2 3x 4 25x 4 22x2 6x 8 25x 4 2x2 7x 12 0 x 77 727 96 x 1 x 83 6 x 9 a 2xx 8x 2x 23x x2 3x 0 x x 3 0 x 0 x 3 b 3xx 12 3x2 12 x2 log12 log3 x log 12 log 3 10 a 10x1 10x1 10x 10 101 1287 x log 1287 log 10 101 b 4x 2 4 342 4x 57 x log 57 log 4 11 a u 10x u2 3u 2 0 u 3 9 8 2 u 2 u 1 10x 2 x log 2 10x 1 x 0 b u 2x u2 3u 4 0 u 3 9 16 u 4 u 1 2x 4 x 2 2x 1 no solution c ex u 3u2 8u 3 0 u 8 64 36 u 3 u 13 ex 3 x ln 3 ex 13 no solution d Exponentials are never negative so theres no solution e u 10x u2 2u 7 0 u 2 4 8 2 u 1 2 10x 1 2 x log 1 2 1 2 0 no solution 12 a 2x 2y 23 x 2y 3 1 4x 2y 20 4x 2y 0 2 From 2 y 2x x 22x 3 3x 3 x 1 y 2 b 22x y 1 25 2x y 1 5 1 22x 4y 21 2x 4y 1 2 From 2 2x 1 4y 1 4y y 1 5 y 1 x 52 c 23x 26y 3 3x 6y 3 1 34y 33x 9 4y 3x 9 2 1 into 2 gives 4y 6y 3 9 2y 6 y 3 x 7 13 a x log 4 log 53 x log 53 log 4 b x log 27 log 9 x log 9 log 27 c 22x 23x7 2x 3x 3 x 3 d x 2 log 3 3 2x log 5 x log 3 2 log 5 3 log 5 2 log 3 x log 1259 log 75 e 34x 43x7 43x 43x7 x x 7 2x 1 x 12 Consider the first spinner goes from 1 to 3 2 Let the first number that on the first spinner and the second number the one appearing and spinner 2 The sample space is 11 12 13 14 15 16 21 22 23 24 25 26 31 32 33 34 35 36 a total of 18 outcomes 3 Event Outcomes Probability 11 13 15 21 23 25 918 12 31 33 35 12 14 16 32 34 36 618 13 16 25 34 318 16 12 22 32 44 24 34 918 12 16 26 36 11 118 4 1 2 3 4 5 5 Trial Spinner 1 Spinner 2 Points 1 2 1 1 1 3 1 3 4 2 3 1 3 2 6 4 5 7 7 3 3 3 1 4 4 6 3 4 1 2 2 2 2 2 4 3 4 6 5 1 1 1 1 1 2 2 2 As três tabelas são iguais então fiz todas juntos A primeira coluna de cada coluna é a primeiro tabela i por ai vai 6 Tate spinner 1 Directions Throw the coin twice and spin the spinner once Two heads or two tails are worth 2 points and the spinny gives the number of point it shows One head and one tail do not score 1 Outcome is maximum score of 5 2 Outcome is minimum score of 1 3 Outcome is one head one tail and an odd number 4 Outcome is two heads and 2 5 Outcome is the score is an even number 7 The sample space is hh1 hh2 hh3 xx1 xx2 xx3 hx1 hx2 hx3 xh1 xh2 xh3 Event Outcomes Probability 1 hh3 xx3 212 16 2 hx1 xh1 212 16 3 hx1 xh1 xh3 hx3 412 13 4 hh2 112 5 hx2 xh2 hh2 xx2 412 13 8 Twins Points 1 hx 2 2 2 hx 1 1 3 xx 3 5 4 xh 3 3 5 hh 2 4 71 Exponential Functions Student Outcomes Students create exponential functions to model realworld situations Students use logarithms to solve equations of the form 𝑓𝑡 𝑎 𝑏𝑐𝑡 for 𝑡 Students decide which type of model is appropriate by analyzing numerical or graphical data verbal descriptions and by comparing different data representations Summary To model exponential data as a function of time Examine the data to see if there appears to be a constant growth or decay factor Determine a growth factor and a point in time to correspond to 𝑡 0 Create a function 𝑓𝑡 𝑎 𝑏𝑐𝑡 to model the situation where 𝑏 is the growth factor every 1 𝑐 years and 𝑎 is the value of 𝑓 when 𝑡 0 Logarithms can be used to solve for 𝑡 when you know the value of 𝑓𝑡 in an exponential function Exercises 1 Does each pair of formulas described below represent the same sequence Justify your reasoning a 𝑎𝑛1 2 3 𝑎𝑛 𝑎0 1 and 𝑏𝑛 2 3 𝑛 for 𝑛 0 b 𝑎𝑛 2𝑎𝑛1 3 𝑎0 3 and 𝑏𝑛 2𝑛 13 4𝑛 1 3 for 𝑛 1 c 𝑎𝑛 1 3 3𝑛 for 𝑛 0 and 𝑏𝑛 3𝑛2 for 𝑛 0 2 Alex is saving her babysitting money She has 500 in the bank and each month she deposits another 200 Her account earns 2 interest compounded monthly a Complete the table showing how much money she has in the bank for the first four months Month Amount in dollars 1 2 3 4 b Write a recursive sequence for the amount of money she has in her account after 𝑛 months 3 Assume each table represents values of an exponential function of the form 𝑓𝑡 𝑎𝑏𝑐𝑡 where 𝑏 is a positive real number and 𝑎 and 𝑐 are real numbers Use the information in each table to write a formula for 𝑓 in terms of 𝑡 for parts ad a 𝒕 𝒇𝒕 b 𝒕 𝒇𝒕 0 20 0 2000 4 50 5 750 c 𝒕 𝒇𝒕 d 𝒕 𝒇𝒕 6 25 3 50 8 55 6 40 5 Rewrite the expressions for each function in parts ad to determine the annual growth or decay rate e For parts a and c determine when the value of the function is double its initial amount f For parts b and d determine when the value of the function is half of its initial amount 4 When examining the data in Example 1 Juan noticed the population doubled every five years and wrote the formula 𝑃𝑡 1002 𝑡 5 Use the properties of exponents to show that both functions grow at the same rate per year 5 The growth of a tree seedling over a short period of time can be modeled by an exponential function Suppose the tree starts out 3 feet tall and its height increases by 15 per year When will the tree be 25 feet tall 6 Loggerhead turtles reproduce every 24 years laying approximately 120 eggs in a clutch Studying the local population a biologist records the following data in the second and fourth years of her study Year Populatio n 2 50 4 1250 a Find an exponential model that describes the loggerhead turtle population in year 𝑡 b According to your model when will the population of loggerhead turtles be over 5000 Give your answer in years and months 7 The radioactive isotope seaborgium266 has a halflife of 30 seconds which means that if you have a sample of 𝐴 grams of seaborgium266 then after 30 seconds half of the sample has decayed meaning it has turned into another element and only 𝐴 2 grams of seaborgium266 remain This decay happens continuously a Define a sequence 𝑎0 𝑎1 𝑎2 so that 𝑎𝑛 represents the amount of a 100gram sample that remains after 𝑛 minutes b Define a function 𝑎𝑡 that describes the amount of a 100gram sample of seaborgium266 that remains after 𝑡 minutes c Do your sequence from part a and your function from part b model the same thing Explain how you know d How many minutes does it take for less than 1 g of seaborgium266 to remain from the original 100 g sample Give your answer to the nearest minute 8 Strontium90 magnesium28 and bismuth all decay radioactively at different rates Use data provided in the graphs and tables below to answer the questions that follow Strontium90 grams vs time hours Radioactive Decay of Magnesium28 𝑹 grams 𝒕 hours 1 0 05 21 025 42 0125 63 00625 84 a Which element decays most rapidly How do you know b Write an exponential function for each element that shows how much of a 100 g sample will remain after 𝑡 days Rewrite each expression to show precisely how their exponential decay rates compare to confirm your answer to part a 100 50 25 125 625 3125 0 20 40 60 80 100 120 0 10 20 30 Bismuth grams Time days 9 The growth of two different species of fish in a lake can be modeled by the functions shown below where 𝑡 is time in months since January 2000 Assume these models will be valid for at least 5 years Fish A 𝑓𝑡 500013𝑡 Fish B 𝑔𝑡 1000011𝑡 According to these models explain why the fish population modeled by function 𝑓 will eventually catch up to the fish population modeled by function 𝑔 Determine precisely when this will occur 10 When looking at US minimum wage data you can consider the nominal minimum wage which is the amount paid in dollars for an hour of work in the given year You can also consider the minimum wage adjusted for inflation Below is a table showing the nominal minimum wage and a graph of the data when the minimum wage is adjusted for inflation Do you think an exponential function would be an appropriate model for either situation Explain your reasoning Year Nominal Minimum Wage 1940 030 1945 040 1950 075 1955 075 1960 100 1965 125 1970 160 1975 210 1980 310 1985 335 1990 380 1995 425 2000 515 2005 515 2010 725 000 200 400 600 800 1000 1935 1945 1955 1965 1975 1985 1995 2005 2015 Minimum Wage in 2012 Dollars Year US Minimum Wage Adjusted for Inflation 11 A dangerous bacterial compound forms in a closed environment but is immediately detected An initial detection reading suggests the concentration of bacteria in the closed environment is one percent of the fatal exposure level Two hours later the concentration has increased to four percent of the fatal exposure level a Develop an exponential model that gives the percentage of fatal exposure level in terms of the number of hours passed b Doctors and toxicology professionals estimate that exposure to two thirds of the bacterias fatal concentration level will begin to cause sickness Offer a time limit to the nearest minute for the inhabitants of the infected environment to evacuate in order to avoid sickness c A prudent and more conservative approach is to evacuate the infected environment before bacteria concentration levels reach 45 of the fatal level Offer a time limit to the nearest minute for evacuation in this circumstance d To the nearest minute when will the infected environment reach 100 of the fatal level of bacteria concentration 12 Data for the number of users at two different social media companies is given below Assuming an exponential growth rate which company is adding users at a faster annual rate Explain how you know Social Media Company A Social Media Company B Year Number of Users Millions Year Number of Users Millions 2010 60 2009 370 2012 195 2012 1057 103 Probability events Student Outcomes Students represent events by shading appropriate regions in a Venn diagram Given a chance experiment with equally likely outcomes students calculate counts and probabilities by addingsubtracting given counts or probabilities Students interpret probabilities in context Sumary In a probability experiment the events can be represented by circles in a Venn diagram Combinations of events using and or and not can be shown by shading the appropriate regions of the Venn diagram The number of possible outcomes can be shown in each region of the Venn diagram alternatively probabilities may be shown The number of outcomes in a given region or the probability associated with it can be calculated by adding or subtracting the known numbers of possible outcomes or probabilities Exercises 1 On a flight some of the passengers have frequentflier status and some do not Also some of the passengers have checked baggage and some do not Let the set of passengers who have frequentflier status be 𝐹 and the set of passengers who have checked baggage be 𝐶 On the Venn diagrams provided shade the regions representing the following instances 1 Passengers who have frequentflier status and have checked baggage 2 Passengers who have frequentflier status or have checked baggage 3 Passengers who do not have both frequentflier status and checked baggage 4 Passengers who have frequentflier status or do not have checked baggage 2 For the scenario introduced in Problem 1 suppose that of the 400 people on the flight 368 have checked baggage 228 have checked baggage but do not have frequentflier status and 8 have neither frequentflier status nor checked baggage 1 Using a Venn diagram calculate the following 1 The number of people on the flight who have frequentflier status and have checked baggage 2 The number of people on the flight who have frequentflier status 2 In the Venn diagram provided below write the probabilities of the events associated with the regions marked with a star 3 When an animal is selected at random from those at a zoo the probability that it is North American meaning that its natural habitat is in the North American continent is 065 the probability that it is both North American and a carnivore is 016 and the probability that it is neither American nor a carnivore is 017 1 Using a Venn diagram calculate the probability that a randomly selected animal is a carnivore 2 Complete the table below showing the probabilities of the events corresponding to the cells of the table North American Not North American Total Carnivore Not Carnivore Total 4 This question introduces the mathematical symbols for and or and not Considering all the people in the world let 𝐴 be the set of Americans citizens of the United States and let 𝐵 be the set of people who have brothers The set of people who are Americans and have brothers is represented by the shaded region in the Venn diagram below This set is written 𝐴 𝐵 read 𝐴 intersect 𝐵 and the probability that a randomly selected person is American and has a brother is written 𝑃𝐴 𝐵 The set of people who are Americans or have brothers is represented by the shaded region in the Venn diagram below This set is written 𝐴 𝐵 read 𝐴 union 𝐵 and the probability that a randomly selected person is American or has a brother is written 𝑃𝐴 𝐵 The set of people who are not Americans is represented by the shaded region in the Venn diagram below This set is written 𝐴𝐶 read 𝐴 complement and the probability that a randomly selected person is not American is written 𝑃𝐴𝐶 Now think about the cars available at a dealership Suppose a car is selected at random from the cars at this dealership Let the event that the car has manual transmission be denoted by 𝑀 and let the event that the car is a sedan be denoted by 𝑆 The Venn diagram below shows the probabilities associated with four of the regions of the diagram 1 What is the value of 𝑃𝑀 𝑆 2 Complete this sentence using and or or 𝑃𝑀 𝑆 is the probability that a randomly selected car has a manual transmission is a sedan 3 What is the value of 𝑃𝑀 𝑆 4 Complete this sentence using and or or 𝑃𝑀 𝑆 is the probability that a randomly selected car has a manual transmission is a sedan 5 What is the value of 𝑃𝑆𝐶 6 Explain the meaning of 𝑃𝑆𝐶 93 Binomial Theorem Student Outcomes Students observe patterns in the coefficients of the terms in binomial expansions They formalize their observations and explore the mathematical basis for them Students use the binomial theorem to solve problems in a geometric context Exercises 1 Consider the binomial 2𝑢 3𝑣6 a Find the term that contains 𝑣4 b Find the term that contains 𝑢3 c Find the third term 2 Consider the binomial 𝑢2 𝑣36 a Find the term that contains 𝑣6 b Find the term that contains 𝑢6 c Find the fifth term 3 Find the sum of all coefficients in the following binomial expansion a 2𝑢 𝑣10 b 2𝑢 𝑣10 c 2𝑢 3𝑣11 d 𝑢 3𝑣11 e 1 𝑖10 f 1 𝑖10 g 1 𝑖200 h 1 𝑖201 4 Expand the binomial 1 2𝑖 6 5 Show that 2 2𝑖 20 2 2𝑖 20 is an integer 6 We know 𝑢 𝑣2 𝑢2 2𝑢𝑣 𝑣2 𝑢2 𝑣2 2𝑢𝑣 Use this pattern to predict what the expanded form of each expression would be Then expand the expression and compare your results a 𝑢 𝑣 𝑤2 b 𝑎 𝑏 𝑐 𝑑2 7 Look at the powers of 101 up to the fourth power on a calculator Explain what you see Predict the value of 1015 and then find the answer on a calculator Are they the same 8 Can Pascals triangle be applied to 1 𝑢 1 𝑣 𝑛 given 𝑢 𝑣 0 9 The volume and surface area of a sphere are given by 𝑉 4 3 𝜋𝑟3 and 𝑆 4𝜋𝑟2 Suppose we increase the radius of a sphere by 0001 units from 𝑟 to 𝑟 0001 a Use the binomial theorem to write an expression for the increase in volume 𝑉𝑟 0001 𝑉𝑟 as the sum of three terms b Write an expression for the average rate of change of the volume as the radius increases from 𝑟 to 𝑟 0001 c Simplify the expression in part b to compute the average rate of change of the volume of a sphere as the radius increases from 𝑟 to 𝑟 0001 d What does the expression from part c resemble e Why does it make sense that the average rate of change should approximate the surface area Think about the geometric figure formed by 𝑉𝑟 0001 𝑉𝑟 What does this represent f How could we approximate the volume of the shell using surface area And the average rate of change for the volume g Find the difference between the average rate of change of the volume and 𝑆𝑟 when 𝑟 1 10 The area and circumference of a circle of radius 𝑟 are given by 𝐴𝑟 𝜋𝑟2 and 𝐶𝑟 2𝜋𝑟 Suppose we increase the radius of a sphere by 0001 units from 𝑟 to 𝑟 0001 a Use the binomial theorem to write an expression for the increase in area volume 𝐴𝑟 0001 𝐴𝑟 as a sum of three terms b Write an expression for the average rate of change of the area as the radius increases from 𝑟 to 𝑟 0001 c Simplify the expression in part b to compute the average rate of change of the area of a circle as the radius increases from 𝑟 to 𝑟 0001 d What does the expression from part c resemble e Why does it make sense that the average rate of change should approximate the area of a circle Think about the geometric figure formed by 𝐴𝑟 0001 𝐴𝑟 What does this represent f How could we approximate the area of the shell using circumference And the average rate of change for the area g Find the difference between the average rate of change of the area and 𝐶𝑟 when 𝑟 1 101 Terminology Student Outcomes Students determine the sample space for a chance experiment Given a description of a chance experiment and an event students identify the subset of outcomes from the sample space corresponding to the complement of an event Given a description of a chance experiment and two events students identify the subset of outcomes from the sample space corresponding to the union or intersection of two events Students calculate the probability of events defined in terms of unions intersections and complements for a simple chance experiment with equally likely outcomes Sumary Sample Space The sample space of a chance experiment is the collection of all possible outcomes for the experiment Event An event is a collection of outcomes of a chance experiment For a chance experiment in which outcomes of the sample space are equally likely the probability of an event is the number of outcomes in the event divided by the number of outcomes in the sample space Some events are described in terms of or and or not Exercises Consider a second scenario card that Alan created for his game Scenario Card 2 Tools Spinner 1 Spinner 2 a spinner with six equal sectors Place the number 1 in a sector the number 2 in a second sector the number 3 in a third sector the number 4 in a fourth sector the number 5 in a fifth sector and the number 6 in the last sector Directions chance experiment Spin Spinner 1 and spin Spinner 2 Record the number from Spinner 1 and record the number from Spinner 2 Five Events of Interest Outcome is an odd number on Spinner 2 Outcome is an odd number on Spinner 1 and an even number on Spinner 2 Outcome is the sum of 7 from the numbers received from Spinner 1 and Spinner 2 Outcome is an even number on Spinner 2 Outcome is the sum of 2 from the numbers received from Spinner 1 and Spinner 2 Player Scoring Card for Scenario 2 Turn Outcome from Spinner 1 Outcome from Spinner 2 Points 1 2 3 4 5 1 Prepare Spinner 1 and Spinner 2 for the chance experiment described on this second scenario card Recall that Spinner 2 has six equal sectors 2 What is the sample space for the chance experiment described on this scenario card 3 Based on the sample space determine the outcomes and the probabilities for each of the events on this scenario card Complete the table below Event Outcomes Probability Outcome is an odd number on Spinner 2 Outcome is an odd number on Spinner 1 and an even number on Spinner 2 Outcome is the sum of 7 from the numbers received from Spinner 1 and Spinner 2 Outcome is an even number on Spinner 2 Outcome is the sum of 2 from the numbers received from Spinner 1 and Spinner 2 4 Assign the numbers 15 to the events described on the scenario card Five Events of Interest Scenario 2 Outcome is an odd number on Spinner 2 Outcome is an odd number on Spinner 1 and an even number on Spinner 2 Outcome is the sum of 7 from the numbers received from Spinner 1 and Spinner 2 Outcome is an even number on Spinner 2 Outcome is the sum of 2 from the numbers received from Spinner 1 and Spinner 2 5 Determine at least three final scores based on the numbers you assigned to the events Player Scott Trial Outcome from Spinner 1 Outcome from Spinner 2 Points see Problem 4 1 2 3 4 5 Player Scott Trial Outcome from Spinner 1 Outcome from Spinner 2 Points see Problem 4 1 2 3 4 5 Player Scott Trial Outcome from Spinner 1 Outcome from Spinner 2 Points see Problem 4 1 2 3 4 5 6 Alan also included a fair coin as one of the scenario tools Develop a scenario card Scenario Card 3 that uses the coin and one of the spinners Include a description of the chance experiment and descriptions of five events relevant to the chance experiment Scenario Card 3 Tools Fair coin head or tail Spinner 1 Directions chance experiment Five Events of Interest 7 Determine the sample space for your chance experiment Then complete the table below for the five events on your scenario card Assign the numbers 15 to the descriptions you created Event Outcomes Probability 8 Determine a final score for your game based on five turns Turn Points 1 2 3 4 5 91 Sequences Student Outcomes Students recognize when a table of values represents an arithmetic or geometric sequence Patterns are present in tables of values They choose and define the parameter values for a function that represents a sequence Summary A sequence is a list of numbers or objects in a special order An arithmetic sequence goes from one term to the next by adding or subtracting the same value A geometric sequence goes from one term to the next by multiplying or dividing by the same value Looking at the difference of differences can be a quick way to determine if a sequence can be represented as a quadratic expression Exercises Solve the following problems by finding the functionformula that represents the 𝑛th term of the sequence 1 After a knee injury a jogger is told he can jog 10 minutes every day and that he can increase his jogging time by 3 minutes every two weeks How long will it take for him to be able to jog one hour a day 2 A ball is dropped from a height of 15 feet The ball then bounces to 80 of its previous height with each subsequent bounce a Explain how this situation can be modeled with a sequence Week Daily Jog Time 1 10 2 10 3 13 4 13 5 16 6 16 b How high to the nearest tenth of a foot does the ball bounce on the fifth bounce 3 Consider the following sequence 8 17 32 53 80 113 a What pattern do you see and what does that pattern mean for the analytical representation of the function b What is the symbolic representation of the sequence 4 Arnold wants to be able to complete 100 militarystyle pull ups His trainer puts him on a workout regimen designed to improve his pullup strength The following chart shows how many pullups Arnold can complete after each month of training How many months will it take Arnold to achieve his goal if this pattern continues Month PullUp Count 1 2 2 5 3 10 4 17 5 26 6 37 1a b From Q4 1 and log12x logax 1 log124 loga4 1 loga4 a 4 Qx log14x c Since it goes to infinity when approaching zero b 1 It cross the xaxis at x 1 as should be and equals 1 at the point corresponding to the reciprocal of the base 2a b 5 because y1 after x5 b The domains are equal namely 0 as are the ranges which are c Logarithmic functions are not defined for x0 so theres yintercept Also for all b logb1 0 the xintercept d Both go to infinity for very large x 3a b 12 because otherwise it would be greater than 1 at x 14 b The domains and range are equal 0 and respectively c Yintercepts never exist for a logarithmic function Xintercepts are x 1 d Both have b 1 so they go to a stretch fx by a factor of 3 b gx 3fx 3 log2x log2x3 a Horizontally stretch fx by a factor of 4 b gx fx4 log2x4 a Its an identity both are the same function b gx log21x log2x log21x 7 a Blue log3x Green log5x Red log11x The smaller the base the faster the function grows so log3x log5x log11x for x 1 b Não adianta eu fayr aqui já que precisa ser no mesmo eixo dos outros dus gráficos devem sempre estar entre o verde e o vermelho cujo o eixo em x 1 e atinge y 1 em x 7 8 a Blue log111x Red log15x Green log12x Same reasoning as before but now decaying b Agora devet estão entre o azul e o vermelho cruzar em x 1 e atingir y 1 em x 7 9 a h x 12 x 1 x2 2x 1 x 1 x2 3x 2 b h sqrtx22 x22 sqrtx24 x24 12 sqrt2x2 1 c hx log10x13 13 log10x 13 1 log x d h 3 log3x2 3 x2 3 e h 1x33 1x2 f h 3x3 x g h sinx π2 cosx h h cos2x 2 cosx 2 10 b 1 fgx 124x 1 8 x 12 8 x 3 12 11 gfx 414 x 2 7 x 3 7 111 14 12 x 8 8 x16 10 c 1 fgx 3x3 1 x 11 gfx 3x 13 1 x 111 ffx 33x 1 1 d 1 fgx 1x3 1x3 11 gfx 1x3 111 ffx x33 x9 1 11 fgx x2 x2 11 gfx x2 x2 111 ffx x x 11 b The domain of g is 1 while fs is 0 f can become negative and g is nonnegative From 1 to 17 the shapes are fairly similar c For large x the square root surpass the logarithm 12b k is defined for all real numbers whole k only for positive numbers From 0 to y k goes from to q while k curves the yaxis and then goes to As with the square root the graphs are similar for x 1 Once again the logarithm will be separated for very large x 1a Yes a0 1 a1 23 231 a2 232 am 23m bm b No we have a1 23 3 9 and b1 3 c No am 13 3m 3m1 bm 2a 1 500 2 71009 500 1 0021212 200 3 92442 71009 1 0021212 200 4 1 14308 114308 1 0021212 200 b Mm1 200 Mm 1 0021212 with M1 500 and m 1 3a From 20 a b0 a 20 50 20 b 4c 52 b4c 5214 bc b 52 c 14 fx 20 5274 b f0 2000 a 2000 f5 750 gives 750 2000 b5c 38 b5c b 38 c 15 fx 2000 3873 c 25 a b6c 55 a b8c Taking the ratio 115 b2c b 115 c 12 Plugging into the first equation 25 a 1153 a 1251331 25 31251331 fx 31251331 11572 d 50 a b3c 40 a b6c Taking the ratio 45 b3c b 45 c 13 From the first equation 50 a 4532 a 2504 1252 fx 1252 4533 a For a let fx 40 40 20 5274 2 5274 Using log2 on both sides 1 74 log2 52 1 74 log2 5 1 4log25 1 For c let fx 62501331 2 115x2 Applying log2 1 x2 log2115 x 2log2115 176 For b let fx 1000 1000 2000 3875 12 3875 log2 12 75 log2 38 log22 75 log2 83 5 7 log2 8 log2 3 7 5 353 For d we have 1254 1252 4533 12 4533 1 73 log2 4 log2 5 7 3log25 2 932 5 With the information given we have a3 b115 and c1 Rx 3 115 25 37 15 253 115 log 253 x log 115 x log 253 log 115 1517 years 6 a 50 a b 2c 25 b2c 1250 a b 4c b 5 c 1 50 a 52 a 5025 2 The number of turtles is mx 25 b Letting mx5000 5000 25 2500 5 log 5 2500 x 486 7 a0 100 a125 a2 254 am am1 4 b ax 100 14 x c Yes because when x is an integer m its value is am d ax 1 1100 14 x log 1100 x log 14 x log 100 log 4 2 log 4 3 A bit after 3 minutes theres less than 1g left 8 Note o primeiro gráfico não é o que a questão fala então respondi usando só a tabela eo seguinte gráfico a Magnesium28 since its halflife is 21 hours opposed to Bismuth which taves 5 days b It taxes 2124 days for Magnesium to decay so its function is Mx 100 12 87 x Bismuths rate is Bx 100 12 x5 9 For large enough x the function with the highest growth rate will surpass the other one indeprndly of the initial number Equating f and g 5000 13 x 10000 11 x 2 1311 x 1711 x log 2 x log 1311 x log 2 log 1311 415 months 10 No neither case varies following a constant ratio even approximately 11 a In this case a1 b4 and c12 fx 4 x2 b True thirds is 23 100 2003 2003 4 x2 log 2003 x2 log 4 x 2 log 2003 log 4 606 hours 006 hours 00660 6 hours and 4 minutes 4 min c 45 4 x2 x 2 log 45 549 h 5 h and 29 min log 4 log 1 d 100 4 x2 x 2 log 100 664 h 6 h and 38 min log 4 12 Company A has more than tripled its users in only two years while B got proportionally less users in more time Company A is faster 1a The general term is 6 m2u6m3uvm so we look for the term with m4 64 2 2u2 3uv4 154u281v44860u2v4 b For m3 633 2u3 3uv3 208u3 27uv3 4320u3 v3 c For m2 624 2u4 3uv2 1516u49u22760u4v2 2a Now we have 6 mu26muv3m Let m2 so 624 u24 uv32 15u8 v6 b Let m3 633 u23 uv33 20 u6 v9 c Let m4 642 u22 uv34 15u4 v12 3a 2uv10sumk010 10 k2u10kvk sumk010 10 k210k u10kvk no the sum of coefficients in the term not containing u or v so if we make uv1 on the equation we get 310 sumk010 10 k 210k as desired The sum is 310 59049 b The sum is 2110 1 c 2311 111 1 d 1311 211 2048 e 1i10 1i25 2i5 32i The sum is 32 f 1i10 1i25 2i5 32i the sum is 32 g 1i200 1i450 450 450 h 1i200 450 1i the sum is 2450 41sqrt2i6summ06 6 m sqrt2im 1 6 sqrt2 i 6 sqrt22 6 sqrt2 i3 6 sqrt2 i4 6 sqrt2 i5 sqrt2 i6 5 4 3 2 7 1 6 sqrt2 i 30 40 sqrt2 i 60 24 sqrt2 i 8 23 10 sqrt2 i 52 sqrt2 i20 2 sqrt2 i20 summ020 20 m 220m sqrt2 im summ020 20 m 220m sqrt2 im summ020 20 m 220m sqrt2 im sqrt2 im Put i2 1 and 121 and repeat for i3i i3 i higher powers i41 i41 so only the even terms are nonzero due to the square brackets But this also means that there will be no square roots on any term making each integer so that the sum is an integer 6a uv pm w2 uv2 w2 2uvw u2 v2 w2 2uv 2uw 2vw Calculating by the definition uvwuvw u2 uv uw vu v2 vw wu wv w2 u2 v2 w2 2uv 2uw 2vw The expressions are equal b ab cd2 ab2 cd2 2abcd a2 b2 c2 d2 2ab 2cd 2ac 2ad 2bc 2bd By definition abcdabcd a2 ab ac ad ba b2 bc bd ca cb c2 cd da db dc d2 a2 b2 c2 d2 2ab 2ac 2ad 2bc 2bd 2cd Once again they are equal 7 101 101 102 10201 103 1030307 104 104060401 The nth power of 101 is the numbers on the n1th row of Pascals triangle separated by zeros Then 1015 should be 1051010501 as confirmed on the calculator 8 Yes It does not matter if the number is integer real complex etc the coefficients are always the same and can be calculated using Pascals triangle 9a Vr 0001 43 π r 00013 43 π r3 0003 r2 0000001 r 0000000001 Vr 0001 Vr 43 π 0003 r2 106 r 109 b The average is Vr 0001 Vr r 0001 r ΔV ΔV 43 π 0003 r2 106 r 109 0001 c ΔV 43 π 3 r2 103 r 106 4 π r2 43 π 103 r 106 d Its the surface area plus correction terms e The volume of a spherical shell with radius r and thickness 0001 so its volume should be close to the surface area f It would be the surface area multiplied by the thickness The average rate of change of the volume can be approximated by the surface area g ΔV 1 S1 43 π 103 106 10a Ar 0001 Ar π r 00012 π r2 π r2 0002 r 106 r2 π 0002 r 106 ΔA π 0002 r 106 0001 c ΔA 2 π r 0001 π d Its the perimeter plus another term e Ar 0001 Ar is the area of a thin circle having finite thickness f It would be the perimeter of the circumference times the width of the shell The average rate of change can be approximated the perimeter g ΔA1 C1 2π 0001 2π 0001