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1、Could Achilles Never Catch up with A Tortoise?On LimitLife on MathematicsThe fear of Matthew Human beings have learned to challenge in the predicament of survival,which led to constant imagination and practice of the limit people canfinally soar at an altitude of 30,000 meters,and skyscrapers have b
2、een builthigher and higher.Guinness World Records manifest the continuous effortsof humans to challenge themselves.The fear of Matthew Is swimming across the English Channel a happy journey for humans torelive the ocean where life begins?Is jumping over the valley in the MaroonBellsamemoryrecall ofo
3、ur ancestrysprimitiveability?Sportscompetition allows people to pick up traces of humans glory.Does thecontinuous progress of sports competition carry peoples strong desire forlimits?Limits seem to be joking with us.It is a kind of limitation,and at the sametime it describes an approximation.It is n
4、ot far away on the one hand,andit seems impossible to reach on the other.A Story of Loan Matthew needed to borrow one million yuan.The interest for half a year:10%.Made a further concession on the interest:on a monthly basis.Interest on loanAfter half a year,Matthew should pay back the money with in
5、terest,and the total should be10%1000,000161,100,0006+=，（）The compound interest means that continuously added to the current principal.The simple interest means that theinterest received each month will notbe added to the principal of the nextmonth.According to this calculation method,the money that
6、 Matthew needed to return to his friends after six months should be610%1,000,00011,104,2606+（）The Geometric Growth!Cycle of compound interest can be calculated by week.The power changing from 6 to 26 reminded Matthew of a legend.Kings RewardDuring King Shirhams time in India,Sissa ibn Dahir,the Vizi
7、er,recommended chess,a game he invented,to the king.Since he was pleased with the game,the king wanted to reward him,so he asked Sissa to name his reward.The Vizier said,“Give me one grain of rice for the first square of thechessboard,two grains for the next square,four for the next,eight for thenex
8、t and so on for all 64 squares,with each square having double thenumber of grains as the square before.”The king discovered that,the number of grains he requested was actually an astronomical number:263122218446744073709551615+=ncyclesConspiracy or Help?10%1100(1)100(1)10nnsnn=+=+Relatively low loan
9、 interest:10%.The amount of money continues to increase infinitely?Divide half a year into n cycles,then The fear would increase with no limit?Where does the thinking of a limit take us from the macro perspectiveDoes the limit mean a practical arrival?How did ZenosParadoxes trigger a second crisis i
10、n the history ofmathematics?Where does the thinking of a limit take usfrom the macro perspective?The Second CrisisAn ancient Greek scholar,Zeno,put forward aparadox that Achilles could never catch up with atortoise.The second crisisZeno of Elea,(born c.490 B.C.E.died c.430B.C.E.),wasaGreekphilosophe
11、randmathematicianwhothoughtdeeplyabouttherelationship between motion and stillness,infinityand finiteness,continuity and discreteness.Someparadoxes he once gave suffered misunderstandingand criticism for a long time,but they had aprofound impact on the thinking and developmentof mathematics.A1 S1 A2
12、 S2 A3S3A4S4A5The Tortoise and The HareThe assumption:the tortoise would run first.The hare would never catch up with the tortoise.The distance between the tortoise and the hare21110SS=，32110SS=，43110SS=，A1 S1 A2 S2 A3S3A4S4A50n Suppose that the hare is 10 times faster than the tortoise,obviously we
13、 haveWhen the rabbit chases the turtle for the n-th time,we finally get the distance between the tortoise and the hare Question:For any n,there is111()010nnSS=21121111()()101010nnnnSSSS=The Cyclotomic MethodLets look at another problem,called thecyclotomic method,a method used by ancientChinese math
14、ematician Liu Hui to calculatethe perimeter and the area of a circle.The cyclotomic methodLiu Hui(c.225-295 AD)was a mathematician living inthe northern Wei Kingdom,and he was one of thefounders of ancient Chinese mathematical theory.He made great achievements in algebra and geometry,such as the con
15、cept of positive and negative numbers andthe solution of linear equations.Especially,he applied thethinking of a limit in the calculation of area and volume.His important books Commentary on The Nine Chapterson the Mathematical Art and Haidao suanjing(“Sea IslandMathematical Manual”)have made profou
16、nd influence onthe development of Chinese mathematics.Idea:let regular polygon approach a circle.Approximation“Cut and cut again until it is indivisible.”Area of the circle2limnnSr=0nSSlim2=nnLrA smart approach：Make a symmetrical triangle on each sideof the inscribed polygon,and the circle is wrappe
17、d by the newshape,then we haveLet the area of the circle be S0，the radius be r，the side length of the regular polygon of n sides in the circle ln，its circumference Ln，and the area Sn.Double the sides of the polygon one by one such as from n to 2n,and we get 22222222112222()()22=+=+nnnnLn ln ADnACCDn
18、lrrl rLrlnODABnSnnn=212)21(2 n 202rSr)(2202nnnnSSSSS+20rS=Reach the limit?“reaches”The Squeeze TheoremUsual logical reasoning is changing in some extent?The microscopic world of mathematicsBe as small as you want it to be.？10010The micro world and macro world.Macro：water boilsMicro:random movementsC
19、auchys mindFrench mathematician CauchyCauchys mindAugustin-Louis Cauchy(1789-1857)was aFrench mathematician and physicist whomade outstanding contributions in calculus,complex variable functions,algebra,andtheoretical physics.The calculus of complexfunctions was founded by him,and hisrigorous mathem
20、atical analysis became thefoundation of calculus theory.Cauchys mindlim:limit;:as small as it can be.Cauchy description revealed the essence of the limit.German mathematician Karl Weierstrass gave the modern definition of the limit.The definition of the limit Karl Weierstrass(1815-1897)was a Germanm
21、athematician,whomadeimportantcontributions in many fields such as calculus,real analysis,complex function,calculus ofvariations,and algebra.One of his notable studies was the creation of the-definition of a limit which allowed thenarrative of mathematical analysis to be refined.The definition of the
22、 limitWe can understand the micro concept of limit from a macro perspective.For any0,there is an integer0Nso that anynsatisfyingNn canmakeLunhold,thenLunn=lim.Back to the Tortoise and the Hare A1 A2 A3A4A5The thinking of limit:the limit can be reached.The distance between the hare and the tortoise t
23、ends to 0 but cant reach 0.Zenos description followed the macro logic.Does the limits of various human sports exist？How does mathematics describe the fundamental differencebetween limits and boundaries?Why do we say that the limitsof various sports people predict are merely the synonym forboundaries
24、?Do the limits of human sports really exist?Various LimitsThe limit is not only in the mathematical courses,but also in peoples daily life and many fields.Bungee jumpingSkydivingClimbing Mount QomolangmaDo the limits of many sports exist?Pole vault Sergey Bubka is a talented pole vaulter of the form
25、er Soviet Union.Since 1983,he has won the World Championships 6 times in a row,set 35 indoor and outdoor world records,and dominated the world pole vault field for up to 20 years.In 1985,he jumped 6 meters in Paris,which was considered an insurmountable height for pole vaulters at the time.Bubka eve
26、n turned breaking a world record into an art-improving it just a little bit at a time!Is there the height limit of humans in pole vaulting?The sport of pole vault tests the comprehensivequalities of athletes in all aspects.In February 2004,in the indoor track and field competitionheld in Ukraine,the
27、 French pole vaulter Renaud Lavilleniesuccessfully jumped over the 6-meter-16 bar,breaking therecord Bubka holding for 21 years in the world.People may have such a question Is 6 meters 14 the height limit of humans in pole vaulting?No.ItemBubbas record1100-meter run10.02 seconds2Long jump7.80 meters
28、3Run-up9.6 meters per second4Pole vault6 meters 14RacingPeople long love this sports category,especially sprints.The sprint event was held at the first ancient Olympic Games in Greece.In 1896,the 100-meter race appeared in the first modern Olympics again.It was generally believed that it would be im
29、possible for men to finish 100-meter race in 10 seconds10-second eraIn 1968,James Hines of the US set a new world recordof 9 seconds 9 in the semi-finals of the AmericanChampionships held in Sacramento,marking the entryof the human 100-meter dash into the 10-second era.Prediction methodUsing statist
30、ics to make predictions.No one would run into 9.69 seconds until 2030?The limit predictions of 9 seconds 70 was brokenOn August 16,2008,Jamaican Usain Bolt swept aloft in majesty,breaking the limit of 9 seconds 70 at the Beijing Olympics with his new record of 9 seconds 69.Biomechanical characterist
31、icsFrom the biomechanical characteristics of human running,Ariel,a doctor of biomechanics in the United States,claims that 9seconds 60 may be the speed limit of humans 100-meter dash.Because once humans exceed this speed,there may be dangerssuch as bone fracture and joint soft tissue detachment.It w
32、as Bolt again,in the 2009 World Championships inAthletes held in Berlin,who won the championship again at9 seconds 58 and broke the prediction of 9 seconds 60.Speed Limit Exists?男子百米世界纪录演进历程男子百米世界纪录演进历程国籍姓名成绩（秒）日期(按时间排序)比赛地点牙买加博尔特9.582009年8月17日德国柏林牙买加博尔特9.692008年8月16日中国北京牙买加博尔特9.722008年6月1日美国纽约牙买加鲍威
33、尔9.742007年9月9日意大利雷蒂牙买加鲍威尔9.772005年6月14日希腊雅典美国蒙哥马利9.782002年9月14日法国巴黎美国格林9.791999年6月16日希腊雅典加拿大多诺万-贝利9.841996年7月27日美国亚特兰大美国伯勒尔9.851994年6月7日瑞士洛桑美国刘易斯9.861991年8月25日日本东京美国伯勒尔9.901991年6月14日美国纽约美国刘易斯9.921988年9月24日韩国汉城美国史密斯9.931983年7月3日美国科罗拉多美国海因斯9.951968年10月14日墨西哥首都德国哈里10.01960年6月21日瑞士苏黎世美国维利-威廉姆斯10.11956年8
34、月3日德国柏林The limits of various human sports must exist.How high can a human jumpThe current record of high jump may still be broken,and there will still to rise,but it definitely has an upper bound.The height of a human jumping will not exceed 4 meters,which means 4 meters is only a boundary for human
35、jumping,it is not the limit.We humans are not the animal that can jump the highest.The mammal that jumps the highest is probably the kangaroo.It can jump to a height of 4 meters.Limit or Boundary?LHow a sequence approach its limit?Speed limit 9 seconds 60 shorter than 9.60 seconds.BoundaryTrend show
36、s the limit?男子百米世界纪录演进历程男子百米世界纪录演进历程国籍姓名成绩（秒）日期(按时间排序)比赛地点牙买加博尔特9.582009年8月17日德国柏林牙买加博尔特9.692008年8月16日中国北京牙买加博尔特9.722008年6月1日美国纽约牙买加鲍威尔9.742007年9月9日意大利雷蒂牙买加鲍威尔9.772005年6月14日希腊雅典美国蒙哥马利9.782002年9月14日法国巴黎美国格林9.791999年6月16日希腊雅典加拿大多诺万-贝利9.841996年7月27日美国亚特兰大美国伯勒尔9.851994年6月7日瑞士洛桑美国刘易斯9.861991年8月25日日本东京美国伯
37、勒尔9.901991年6月14日美国纽约美国刘易斯9.921988年9月24日韩国汉城美国史密斯9.931983年7月3日美国科罗拉多美国海因斯9.951968年10月14日墨西哥首都德国哈里10.01960年6月21日瑞士苏黎世美国维利-威廉姆斯10.11956年8月3日德国柏林Approaching trendwhen the record of 100-metersprint is of great interest,people do find anapproaching trend.If the curve goes up along this trend,but is restri
38、cted by this horizontal line m,then we will find that this curve will eventually approach its own destination,andthis destination is the limit.The limit of high jumpThe height of 4 meters in high jump is not the limit of humans,but just a boundary.However,it proves from another point of view that wh
39、ile the height record of human jump is constantly rising,the limit must exist!Hundred-meter speed limitThere will be a lower bound for the 100-meterrecord.The fastest animal should be a leopard.The fastest time of a leopard finishing 100meters is 6 seconds 13.We can use the time as alower bound for
40、humans 100-meter-runningperformance.Therefore,the human 100-meter record is decreasing(monotonousdecline of the sequence)and a lower bound on the time achievement isfound(the lower bound of the sequence).We will be sincere to say:human 100-meter speed limit must exist!Matthews troubleWith proper der
41、ivation,we can prove the following two facts.(1)The sequence 1(1)nn+is monotonically increasing.1(2)(1)4nn+Now the monotonically increasing sequence 1(1)nn+have an upper bound,which must have a limit.And the limit is 1lim(1)2.71828nnen+=This is the first time in the history of mathematics to use a limit to define a number.The End of The Story0.11lim 100(1)100110.5210+=nnenWe can easily getMatthew can completely give up his suspicion of his friend.