Quadratic Equations And Functions e Exponential And Logarithmic Functions

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 61 Quadratic Equations Student Outcomes Students solve quadratic equations with real coefficients that have complex solutions They recognize when the quadratic formula gives complex solutions and write them as a bi for real numbers a and b Summary A quadratic equation with real coefficients may have real or complex solutions Given a quadratic equation 𝑎𝑥2 𝑏𝑥 𝑐 0 the discriminant 𝑏2 4𝑎𝑐 indicates whether the equation has two distinct real solutions one real solution or two complex solutions If 𝑏2 4𝑎𝑐 0 there are two real solutions to 𝑎𝑥2 𝑏𝑥 𝑐 0 If 𝑏2 4𝑎𝑐 0 there is one real solution to 𝑎𝑥2 𝑏𝑥 𝑐 0 If 𝑏2 4𝑎𝑐 0 there are two complex solutions to 𝑎𝑥2 𝑏𝑥 𝑐 0 Exercises 1 Give an example of a quadratic equation in standard form that has a Exactly two distinct real solutions b Exactly one distinct real solution c Exactly two complex nonreal solutions 2 Suppose we have a quadratic equation 𝑎𝑥2 𝑏𝑥 𝑐 0 so that 𝑎 𝑐 0 Does the quadratic equation have one solution or two distinct solutions Are they real or complex Explain how you know 3 Solve the equation 5𝑥2 4𝑥 4 0 4 Solve the equation 2𝑥2 8𝑥 8 5 Solve the equation 9𝑥 9𝑥2 4 𝑥 𝑥2 6 Solve the equation 3𝑥2 𝑥 2 0 7 Solve the equation 6𝑥4 4𝑥2 4𝑥 2 2𝑥23𝑥2 1 8 Solve the equation 25𝑥2 100𝑥 300 0 9 Write a quadratic equation in standard form such that 6 is its only solution 10 Is it possible that the quadratic equation 𝑎𝑥2 𝑏𝑥 𝑐 0 has a positive real solution if 𝑎 𝑏 and 𝑐 are all positive real numbers 11 Is it possible that the quadratic equation 𝑎𝑥2 𝑏𝑥 𝑐 0 has a positive real solution if 𝑎 𝑏 and 𝑐 are all negative real numbers Extension 12 Show that if 𝑘 32 the solutions of 5𝑥2 7𝑥 𝑘 0 are not real numbers 13 Let 𝑘 be a real number and consider the quadratic equation 𝑘 1𝑥2 4𝑘𝑥 3 0 a Show that the discriminant of 𝑘 1𝑥2 5𝑘𝑥 2 0 defines a quadratic function of 𝑘 b Find the zeros of the function in part a and make a sketch of its graph c For what value of 𝑘 are there two distinct real solutions to the original quadratic equation d For what value of 𝑘 are there two complex solutions to the given quadratic equation e For what value of 𝑘 is there one solution to the given quadratic equation 14 We can develop two formulas that can help us find errors in calculated solutions of quadratic equations a Find a formula for the sum 𝑆 of the solutions of the quadratic equation 𝑎𝑥2 𝑏𝑥 𝑐 0 b Find a formula for the product 𝑅 of the solutions of the quadratic equation 𝑎𝑥2 𝑏𝑥 𝑐 0 c June calculated the solutions 7 and 1 to the quadratic equation 𝑥2 5𝑥 7 0 Do the formulas from parts a and b detect an error in her solutions If not determine if her solution is correct d Paul calculated the solutions 3 𝑖2 and 3 𝑖2 to the quadratic equation 𝑥2 6𝑥 7 0 Do the formulas from parts a and b detect an error in his solutions If not determine if his solutions are correct e Joy calculated the solutions 3 2 and 3 2 to the quadratic equation 𝑥2 6𝑥 7 0 Do the formulas from parts a and b detect an error in her solutions If not determine if her solutions are correct f If you find solutions to a quadratic equation that match the results from parts a and b does that mean your solutions are correct g Summarize the results of this exercise 63 Graph Quadratic Functions Student Outcomes Students graph simple quadratic equations of the form y ax h2 k completedsquare or vertex form recognizing that hk represents the vertex of the graph and use a graph to construct a quadratic equation in vertex form Students understand the relationship between the leading coefficient of a quadratic function and its concavity and slope and recognize that an infinite number of quadratic functions share the same vertex Summary When graphing a quadratic equation in vertex form 𝑦 𝑎𝑥 ℎ2 𝑘 ℎ 𝑘 are the coordinates of the vertex Exercises 1 Find the vertex of the graphs of the following quadratic equations a 𝑦 3𝑥 52 35 b 𝑦 𝑥 12 9 2 Write a quadratic equation to represent a function with the following vertex Use a leading coefficient other than 1 a 100 200 b 3 4 6 3 Use vocabulary from this lesson ie stretch shrink opens up and opens down to compare and contrast the graphs of the quadratic equations 𝑦 𝑥2 2 and 𝑦 2𝑥2 3 65 Quadratic Inequalities Exercises 1 In the following exercises solve graphically and write the solution in interval notation 2 In the following exercises solve each inequality algebraically and write any solution in interval notation 3 Explain critical points and how they are used to solve quadratic inequalities algebraically 4 Solve x22x8 both graphically and algebraically Which method do you prefer and why 5 Describe the steps needed to solve a quadratic inequality graphically 6 Describe the steps needed to solve a quadratic inequality algebraically 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 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 61 1 a x2 2x 0 0 pois b24ac 22410 2 1 b x2 2x 1 0 pois b24ac 22411 0 c x2 2x 3 0 pois b24ac 22413 2 2 a c 0 c a b24ac b24aa b24a2 0 A equação apresenta duas soluções pois Δ 0 já que b e a não elevados ao quadrado Logo resultado sempre será positivo 3 5x2 4x 4 0 Δ b2 4ac 42 454 16 80 64 2 soluções complexas 4 2x2 8x 8 2x2 8x 8 0 Δ b2 4ac 82 428 6464 0 1 solução real x bΔ 2a 80 22 2 5 9x 9x2 4 x x2 9x2 3x 9x x 4 0 10x2 8x 1 0 Δ b2 4ac 82 4104 64 160 96 2 soluções complexas 6 3x2 x 2 0 Δ b2 4ac 12 432 1 24 23 2 soluções complexas 7 6x4 4x2 4x 2 2x23x2 1 6x4 4x2 4x 2 6x4 2x2 6x4 6x4 4x2 2x2 4x 2 6x2 4x 2 0 Δ b2 4ac 42 462 16 48 32 2 soluções complexas FORONI 21 7 10 8 25x2 100x 300 0 Δ b2 4ac 1002 425300 10000 30000 20000 2 soluções complexas 9 x 6 Δ 0 a62 6b c 0 36a 6b c 0 se assumir a 1 e b 1 36 6 c 0 a 1 b 1 c 30 c 36 6 30 x2 x 30 0 10 b2 4ac 0 Somente temos 1 solução para se se a e c forem menores que b 11 12 4ac 0 b2 4ac 0 Se aplica a mesma conclusão da questão anterior 12 Δ b2 4ac 0 se c 32 5x2 4x k 0 42 4532 16 64 48 soluções complexas 13 a assumir k 1 1 1x2 51x 2 0 2x2 5x 2 0 b2 Δ b2 4ac 52 422 25 16 9 x bΔ 2a 5 3 4 x1 5 3 4 2 4 05 x2 5 3 4 8 4 2 c b2 4ac 5k2 4k 13 0 25k2 12k 1 0 25k2 12k 12 0 FORONI 13k 12 k 1213 k 092 22 62 d 25k 12k 12 0 k 092 e 25k 12k 12 0 k 092 FORONI 63 1 a 5 35 b 1 9 2 a 100 200 y a x 100² 200 b 36 y a x 3² 6 4 3 ① y x² 2 ② y 2 x² 3 O segundo grafico apresenta amplitude maior pois a 1 Também o segundo gráfico é decrescente pois a 0 65 1 a x² x 6 0 Δ b² 4ac 1² 4 1 6 1 24 25 1 25 21 1 5 2 x₁ 62 3 7 x 3 x₂ 42 2 b x² 4x 3 0 Δ b² 4ac 4² 413 16 16 0 4 0 21 4 2 2 x 2 c x² x 2 0 Δ 1² 4 1 2 1 8 9 1 9 2 1 1 3 2 x₁ 42 2 2 x 1 x₂ 22 1 d x² 2x 3 0 Δ 2² 4 1 3 4 12 16 2 16 2 1 2 4 2 1 x 3 x₁ 22 1 x₂ 62 3 FORONI 2 a x² 6x 8 0 Δ b² 4ac 6² 4 1 8 36 32 4 2 x 4 b Δ 2a 6 4 2 x₁ 82 4 x₂ 42 2 b x² x 12 0 Δ 1² 41 12 1 48 49 1 49 2 1 1 7 2 4 x 3 x₁ 62 3 x₂ 92 4 c x² 6x 4 0 Δ 6² 414 36 16 20 0775 x 1341 6 20 2 x₁ 2682 2 1341 x₂ 15322 0765 d 2x² 7x 4 0 Δ 7² 424 49 32 81 4 x 05 7 81 2 2 7 9 4 x₁ 24 05 x₂ 164 4 e x² x 6 0 Δ 1² 4 1 6 1 24 23 1 23 2 x 1 23 2 1 23 2 2 x² 2x 4 0 Δ 2² 414 4 16 12 2 12 2 x 2 12 3 Os números críticos são os valores do x onde uma desigualdade é igual a zero ou é indefinida Eles quebram os atos em FORONI intervalos 4 x² 2x 8 0 Δ 2² 418 4 32 36 4 x 4 2 2 362 2 6 2 x₁ 42 2 x₂ 82 4 71 1 a a₀ 1 a₁ 21 23 a₂ 23 23 49 l₀ 230 1 Sim l₁ 231 23 l₂ 232 49 2 a₀ 3 a₁ 23 3 9 a₂ 29 3 21 l₀ 20 13 401 3 2 4 3 3 l₁ 2113 411 3 3 Não c a₀ 1 3⁰ 1 3 a₁ 1 31 3 3 1 a₂ 1 32 9 3 3 l₁ 3⁰ 1 Não 2 a 1 500102 200 710 2 710102 700 92420 3 92410102 200 114268 4 114260102 200 136554 b V M 1 taxat V M 102m FORONI a 20 a b⁰⁰ 20 a 1 a 20 50 20 b⁴⁰ c c 140 50 20 b¹ b 52 pt 20 52t40 b 2000 a b⁰ a 2000 750 2000 b⁵ᶜ c acumini c 15 750 2000 b¹ b 7502000 75200 pt 2000 75200 t5 c 25 a b⁶ᶜ c 12 a 4287 br 9s 55 a b⁸ᶜ pt 4287 as t2 d 50 a b³⁰ c 130 a 625 br 4s 40 a b⁸ᶜ pt 625 us t3 e a p0 20 52 t40 20 40 20 52 t40 log52 2 log52 52t40 07564 t40 t3025 e p0 4287 as t2 4287 8574 4287 as t2 logas 2 logas as t2 117924 t2 t 236 28 a A2 30 p0 A A k v⁰ A k p30 A2 A k v³⁰ 12 v³⁰ v 12 130 0977 pt k 0977 t b pt 100 0977 t d 1 100 0977 t 001 0977 t log001 001 log0977 0977t t 19791 segundos g a hr demora 21 h para atingir 50 de perda enquanto que Bi leva aproximadamente 5 dias b Hg p0 100 100 k v⁰ k 100 21 h 0875 dias p0875 50 50 100 v⁰ᵈ⁷⁵ 05 v⁰ᵈ⁷⁵ v 07445 c Hg pt 100 045 t t dias Bi p10 100x 25 100 v¹⁰ 025 v¹⁰ v 087 Bi pt 100 097x g pt gt pt 5000 13t 1 05 13t 2 118 t gt 10000 11t log118 2 log118 t t 4187 p4187 5000 134187 15000 g4187 10000 114187 15000 26 3 a 20 a b⁰⁰ 20 a 1 a 20 50 20 b⁴⁰ c c 140 50 20 b¹ b 52 pt 20 52t40 b 2000 a b⁰ a 2000 750 2000 b⁵ᶜ c acumini c 15 750 2000 b¹ b 750 75 pt 2000 75200t5 c 25 a b⁶ᶜ c 12 a 4287 br 9s 55 a b⁸ᶜ pt 4287 ast2 d 50 a b³⁰ c 130 a 625 br 4s 40 a b⁸ᶜ pt 625 ust3 e a p0 20 52 t40 20 40 20 52t40 log52 2 log52 52t40 07564 t40 t 3025 e p0 4287 ast2 4287 8574 4287 ast2 logas 2 logas ast2 117924 t2 t 236 FORONI 10 Não o crescimento não segue uma linha de tendência exponencial a f0 1 f2 4 001 ki0 logo k 001 004 001 x4 logo x4 4 logo x 141 Então ft 001141t b 23 66 066 001141t logo 66001 141t log141 6666 log141 141t logo t 1222 horas 12 horas e 13 minutos c 045 001141t logo 45 141t log141 45 log141 141t logo t 1108 horas 11 horas e 5 minutos d 1 001141t logo 100 141t log141 100 log141 141t logo t 1340 horas 13 horas e 24 minutos 12 A f0 60 logo k 60 f2 195 logo 195 60x2 logo x2 325 logo x 180 A ft 60180t B f0 370 k 370 f3 1057 logo 1057 370x3 logo x3 286 logo x 142 B ft 370142t A empresa A está crescendo mais em número de clientes FORONI loga b x ax b 73 log110 log4 1 log16 x log4 x 166 log110 1 logo 166x b log14 01 b166 01 logo b 025 2 a para x0 1 y0 0 0 log1 1 logo b0 b logb 1 1 1 para xc 7 yc 1 1 log7 7 logo b1 b logb 7 logo b 7 fx log7 x b fx log7 x gx log5 x f1 0 g1 0 f2 035670 g2 043067 f3 056457 g3 068260 f10 118329 g10 143067 c x 0 y 0 f0 log7 0 infinito g0 log5 0 infinito 0 log7 x logo x 1 0 log5 x logo x 1 d ambas as funções tendem ao infinito no eixo x e tendem a 1 no eixo y 3 a 016 log7 14 logo b086 b logb 14 logo b086 025 b aproximadamente 02 b gx log12 x g14 2 f14 086 g12 1 f12 042 g1 0 f1 0 g2 1 f2 043 g4 2 f4 086 c as curvas interceptam no mesmo ponto FORONI de x 1 y 0 84 4 a f1 log2 1 0 g1 log2 13 0 f2 log2 2 1 g2 log2 23 3 f5 log2 5 232192 g5 log2 53 696 5 a gx log2 x4 logo gx log2 x4 logo fx 4 log2 x b log2 x4 4 log2 x 6 a gx log2 xx fx x log12 x logo fx x log2 x b log12 x log2 x logo log2 xx x log2 1 7 a f5 log5 5 146 logo azul g5 log5 5 1 logo verde h5 log4 5 067 logo vermelho b kx log7 x k2 log7 2 036 k3 log7 3 056 k4 log7 4 071 k5 log7 5 083 8 a f5 log5 5 146 logo verde g5 log5 5 1 logo vermelho h5 log4 5 067 logo azul b k2 log7 2 036 k3 log7 3 056 k4 log7 4 071 FORONI k5 log7 5 083 a ax x 12 x 1 x2 2x 1 x 1 x2 3x 2 b hx x22 14 x24 14 x 1 x 1 c hx log10x3 log1013 log10x log10 log x d hx 1 log x 3 e hx 3 log3 x2 3 x2 3 f hx 133 x9 g hx 3x3 x h hx senx pi2 para qualquer valor de x i hx cos x2 2cos x 2 10 a já fiz b I fgx 14 4x 1 8 x 14 324 x 314 II gfx 4x 84 1 x 32 1 x 34 III ffx 14 x 14 x x16 c I fgx 3x3 1 1 3x3 x II gfx 3x 113 1 x 1 1 x III ffx 13x 1 1 3x 1 1 3x9 2 d I fgx 1x3 1x3 II gfx 1x3 III ffx x33 x9 e I fgx x2 x2 II gfx x2 x2 III ffx x 75 1 a 25x3 6250 5x3 3125 log5 5x3 log5 3125 x 3 5 x 5 3 2 b 362x 648 62x 216 log6 62x log6 216 2x 3 x 32 c 523x5 10240 23x5 2048 log2 23x5 log2 2048 3x 5 11 3x 11 5 x 63 2 d 43x1 32 log4 43x1 log4 32 3x 1 25 x 117 e 325x 216 25x 72 log2 25x log2 72 x 123 f 5113x 120 113x 24 log11 113x log11 24 x 044 g 79x 5405 9x 77214 log9 9x log9 77214 x 302 h 333x 9 3x 312 3x 12 x 56 i log40005x log160000 2 log85x 520 105x 2 log8 85x log8 2 5x 13 x 115 2 1000 0701 1382t 1382t 142653 FORONI log1382 1382t log1382 142653 t 2245 3 50000 1032 1707t 1707t 4940271 log1707 1707t log1707 4940271 t 2017 4 1000000 0701 1382t 1382t 1426533521 log1382 1382t log1382 14265332 t 5097 5 a a função fx para 3x z não toma exponencial b fx gx fx 733x 1 73x 3 32t x 1 gx 32x 3 2x 2 2t 6 a Et St 1t Et 5000 11547t 1 0125 3148t 3148t 8 St 4000 3668t log3148 3148t log3148 8 t 0258 b Da gráficas se interceptam onde Et St onde t 0258 e Et St 93991 7 a 31000000 103m 3m 31800000 m 1572 dias b 710000000 103m 3m 710000000 m 1855 dias 8 a 103x5 7x 105 7x 103x 105 7x 7103 t 105 7x 14286 100000 log14286 14286x log14286 100000 x 232 b 34x2 2 34x 2x U 24x y 24x 313x U 46x U 1284x log1284 4 log1284 1284x x 054 c 10x2 5 100x2 x 2 10x2 5 102x2 2x 4 4x2 x2 2x 4 5 0 3x2 2x 1 FORONI a x1 36 x2 46 1 Δ b2 4ac 22 431 4 12 16 x 2 166 2 46 d 4x23x4 5x4 2x23x4 22x4 2x23x4 2x 4 2x25x8 5x4 2x2 11x 12 0 Δ 112 4212 121 96 25 x 11 25 4 11 5 4 x1 16 4 4 x2 6 4 32 9 a 2x2 8x 22x 23x 2x 3x x 3 b 32x 12 log332x log312 2x 226 x 150 10 a 10x1 10x1 1287 log1010x1 log1010x1 log101287 x 1 x 1 311 2 311 x 1 2 x 1 b 24x 4x1 342 log4 24x log4 4x1 log4 342 12xx1 42 x2 x 84 x2 x 84 0 2 Δ 12 4184 1 336 346 x 1 346 2 1 588 x1 244 x2 344 11 a μ 10x μ2 3μ 2 0 Δ 32 412 9 8 1 μ 3 1 2 3 1 μ1 42 2 μ2 22 1 1 z 10x log10 z log10 10x x 030 10 2 1 10x x 0 b μ 2x μ2 3μ 4 0 Δ 32 414 9 16 25 μ 3 25 2 3 5 μ1 82 4 μ2 22 1 1 4 2x log2 4 log2 2x x 2 2 1 2x log2 1 log2 2x x c μ ex 3μ2 8μ 3 0 Δ 82 433 64 36 100 μ 8 100 6 8 10 6 μ1 186 3 μ2 26 13 033 1 3 ex ln3 ln ex x 109861 2 033 ex ln033 ln ex x d 4x 72x 12 0 22x 72x 12 0 22x 72x 12 0 μ2 7μ 12 0 μ2 7μ 12 0 Δ 72 4112 49 48 1 μ 7 1 2 7 1 μ1 82 4 μ2 62 3 1 4 2x log2 4 log2 2x x 2 3 2x log2 3 log2 2x x e μ 10x μ2 7μ 1 0 Δ 72 411 4 4 8 μ 2 81 2 μ1 241 μ2 0415 1 μ 10x 241 10x log10 241 log10 10x x 038 12 a 2x2y 23 x 2y 3 42xy 40 2x y 0 y 2x x 22x 3 x 4x 3 3x 3 x 1 y 21 2 b 2xy1 25 2x y 6 y 6 2x 4x2y 412 x 2y 12 x 26 2x 12 x 12 4x 12 3x 125 x 416 y 6 2416 232 a 3x 32xy1 3x 6y 3 x 2y 3 32xy 33x 9 4y 3x 9 4y 32y 3 9 4y 6y 9 9 4y 6y y 0 x 20 3 3 13 a log4 4x log4 53 x 037 ln 029x 9 log029 029x log29 9 x 177 c 4x 8x1 22x 23x1 2x 3x3 3x 2x 3 x 3 d 3x 32 53 3x 52x 125 31 52x 1389 75x 1389 log75 75x log75 1389 x 061 e 075x 1330 1331 075 133x 133 056x 133 log056 056x log056 133 x 04918