e是自然对数的底数

e是自然对数的底数 ln和e是什么关系? 对数和底数是干嘛的? 三角函数的画图? ln就是loge lne=logee=1 lne=1 他俩没啥关系 一个是运算符号 一个是自然数 e的ln次方等于1 e^(ln3)=3 In=loge ln(1)=loge(1)=0 e=2.71多 e是自然对数的底数，是一个无限不循环小数。e在科学技术中用得非常多，一般不使用以10为底数的对数。学习了高等数学后就会知道，许多结果和它有紧密的联系，以e为底数，许多式子都是最简的，用它是最“自然”的，所以叫“自然对数”，因而在涉及对数运算的计算中一般使用它，是一个数学符号，没有很具体的意义。 其值是2.71828……，是这样定义的: 当n->?时，(1+1/n)^n的极限。 注:x^y表示x的y次方。 你看，随着n的增大，底数越来越接近1，而指数趋向无穷大，那结果到底是趋向于1还是无穷大呢,其实，是趋向于2.718281828……这个无限不循环小数 1，1/1!，1/2!，1 /3!，1/4!，……，1/n!，当n趋近无穷时，其极限值就为e. 对数(Logarithm 若 )。则b叫做以a为底N的对数，记作 。当a=10时称作常用对数，当a=e时，称作自然对数。 我们知道，一般对数的底可以为任意不等于1的正数。即对数的底如果为超越数e(e=2.718)我们就把这样的对数叫作自然对数，用符号“LN”表示。在这里“1”是对数“logarithm"的第一个字母，“N”是自然“nature"的第一个字母，把两个字母合在一起，就表示自然对数。 “lg”才表示以10为底的对数～～～～ ln1=0 表示e的0次方=1 ln100=4.605170…… 表示e的4.605170次方=100 personal credit business process management, and development of business processes, clear the loan object and scope, establish loan operation of assessment and accountability mechanisms. Diversity management principles the Bank's strict separation between business and consumer use of personal credit, then according to the different varieties of business to establish the appropriate access standard, post-lending management requirements. Business use personal credit should be strictly in accordance with the qilu Bank's small business credit risk management guidelines and associated working capital loans management requirements, such as assessing the management of production and operation, repayment ability, the level of risk, and so on. Income debt ratio control principle of Bank income debt ratio control mechanism, combined with the borrower's income, debt, expenditures, loans, guarantees and other factors, determines the loan amount and term, control each repayment does not exceed the amount of the borrower repayment ability. Principles of trans-regional limits Branches shall not in principle to operating income mainly comes from offsite (prefecture-level cities and administrative divisions at or above) or inward (MS) charge provided credit support for individual customers is located in different places, but except those with unique advantages such as critical. Have set up branches in different places, other bodies are not entitled to carry out the branch of credit business in the region. Branch County area has been established, other bodies are not entitled to apply for the credit within the County. Double before going through the personal credit business credit investigation, Contracting, security verification, credit check, key account manager should be double handle. Bank supports risk managers, loan review, professional 放心，绝对没有错的～以10为底的对数。 ln1=0, ln100=2 ln1=x,即 10^x=1，可得x=0 同理ln100=y 10^y=100，可得y=2 ln= log e 是以e为底的自然对数 是以e为底的对数,以10为底一般写作lg 您好～不知你是否知道对数函数，它是指数函数y=a^x(a>0且a不为1)的反函数，记作y=log a x(这里a应该写为下标，只是打不出来，请见谅～a称为底数，x称为真数，x>0)显然log a x表示的是求a的多少次幂等于x?特别地，我们把以10为底的对数称为常用对数，记作 lg x;把以e为底的对数成为自然对数。这里的e是科学界非常重要而常见的常数，e=2.718281828……。按照上述记号的定义，你应该可以知道ln e=1(因为e^1=e)。无论以什么数a(a>0且a不为1)为底，1的对数都是0(因为a^0=1)。所以ln 1=0。对于一般的正数x,求它的自然对数ln x可以查自然对数表，也可以通过科学计算器来求。解答完毕，如果还有什么不明白的地方，欢迎继续讨论，谢谢～ 这个问题属于初等函数范畴，需要具备函数极限、微积分 方面的知识基础。浏览了楼主的回答列表，我认为楼主的知识基础已经具备。 ================================ 设函数 f(x) = (1 + 1/x)^x 首先证明当 x 趋向正无穷大时，该函数有极限。其次求该极限。 取x为整数n的情况，利用二项式定理 f(n) = (1+1/n)^n =(k从0到n的求和)?n(n-1)(n-2)……(n-k+1)/(k!*n^k) =(k从0到n的求和)?(1/k!)*(1-1/n)(1-2/n)……[1-(k-1)/n] 同理写出f(n+1)的展开式，容易看出 f(n+1) > f(n) 因此 f(n)是单调递增函数 同时从f(n)的展开表达式还可以得到 f(n) ? 1 + 1 + 1/2～ + 1/3～ + …… + 1/n! 再利用 n! > 2^(n-1) ，。。。(此定理的证明从略) f(n) < 2 + 1/2 + 1/2^2 + 1/2^3 + …… + 1/2^(n-1) = 3 - 1/2^(n-1) < 3 综上所述，f(n)随n单调递增，同时有界。因此 f(n)有极限。 personal credit business process management, and development of business processes, clear the loan object and scope, establish loan operation of assessment and accountability mechanisms. Diversity management principles the Bank's strict separation between business and consumer use of personal credit, then according to the different varieties of business to establish the appropriate access standard, post-lending management requirements. Business use personal credit should be strictly in accordance with the qilu Bank's small business credit risk management guidelines and associated working capital loans management requirements, such as assessing the management of production and operation, repayment ability, the level of risk, and so on. Income debt ratio control principle of Bank income debt ratio control mechanism, combined with the borrower's income, debt, expenditures, loans, guarantees and other factors, determines the loan amount and term, control each repayment does not exceed the amount of the borrower repayment ability. Principles of trans-regional limits Branches shall not in principle to operating income mainly comes from offsite (prefecture-level cities and administrative divisions at or above) or inward (MS) charge provided credit support for individual customers is located in different places, but except those with unique advantages such as critical. Have set up branches in different places, other bodies are not entitled to carry out the branch of credit business in the region. Branch County area has been established, other bodies are not entitled to apply for the credit within the County. Double before going through the personal credit business credit investigation, Contracting, security verification, credit check, key account manager should be double handle. Bank supports risk managers, loan review, professional 之后利用初等函数中的夹挤定理，又可以进一步证明 f(x) 与f(n)类似。于是定义 x趋于正无穷大时，f(x)极限值为 e。 通过对 x取一个很大的数，可以计算出 e。x取得越大，e值越精确。 e?2.7182818284…… e 值是这样定义出的。进一步研究又表明e值有一些有趣的数学性质。 例如对于以a为底的对数函数 f(x)=loga(x)求微分， 其结果为 f'(x)= [loga(e)]/x 这个结果的简单证明过程: f'(x) = lim [f(x+Δx) - f(x)]/Δx 。其中 Δx 趋向0。 代入 f(x)及 f(x+Δx)表达式后， f'(x)= (1/x) * lim loga(1+Δx/x)^(Δx/x) f'(x) = (1/x) * lim loga(1 +1/z)^z ，其中z趋向正无穷大 所以 f'(x)=(1/x)* loga(e) 然后在利用这个结果以及反函数的微分，可以证明 指数函数的微分 为 f(x) = a^x f'(x) = loge(a) * a^x 因此定义 loge(a) = ln a 自此出现了自然对数。另外从 (a^x)' = lna * a^x 可以推出 e^x 的导数恰好是其自身 personal credit business process management, and development of business processes, clear the loan object and scope, establish loan operation of assessment and accountability mechanisms. Diversity management principles the Bank's strict separation between business and consumer use of personal credit, then according to the different varieties of business to establish the appropriate access standard, post-lending management requirements. Business use personal credit should be strictly in accordance with the qilu Bank's small business credit risk management guidelines and associated working capital loans management requirements, such as assessing the management of production and operation, repayment ability, the level of risk, and so on. Income debt ratio control principle of Bank income debt ratio control mechanism, combined with the borrower's income, debt, expenditures, loans, guarantees and other factors, determines the loan amount and term, control each repayment does not exceed the amount of the borrower repayment ability. Principles of trans-regional limits Branches shall not in principle to operating income mainly comes from offsite (prefecture-level cities and administrative divisions at or above) or inward (MS) charge provided credit support for individual customers is located in different places, but except those with unique advantages such as critical. Have set up branches in different places, other bodies are not entitled to carry out the branch of credit business in the region. Branch County area has been established, other bodies are not entitled to apply for the credit within the County. Double before going through the personal credit business credit investigation, Contracting, security verification, credit check, key account manager should be double handle. Bank supports risk managers, loan review, professional 对数的发明是16世纪末至17世纪初的事。当时在自然科学领域特别是天文学方面经常遇到十分复杂的数值计算，数学家们为了寻求化简计算的方法而发明了对数。一般认为，对数是由苏格兰数学家纳皮尔和瑞士工程师比尔吉彼此独立地发明的。但在此之前，在法国数学家许凯(15世纪)和德国数学家施蒂费尔(1487—1567)的工作中就孕育了对数的思想。他们研究等比数列与等差数列之间的关系，特别是施蒂费尔将这两种数列加以对比，指出，等比数列各项的乘、除、乘方、开方运算、相当于等差数列相应各项的加、减、乘、除运算。但是他们都没有进一步发展这种思想。 比尔吉是瑞士的一位工程师，他曾担任著名天文学家开普勒的助手，因此经常接触复杂的天文计算，于是产生了化简数值计算的强烈愿望。他受施蒂费尔工作的影响，考虑等差数列 0，10，20，…，10n和与之对应的等比数列 由此建立了一种对数体系，于1620年发表在《等差数列和等比数列表》中。不难看出，比尔吉所造的对数表，把对数的底取为 ，与现在自然对数的底e相差甚小。 比尔吉发明对数的时间大约在1610年，但他推迟了发表的时间，而纳皮尔的对数表在1614年公诸于世，早比尔吉6年。纳皮尔是苏格兰的一个贵族，他对数值计算颇有研究。他制造的“纳皮尔算筹”，化简了乘除法运算，其原理就是用加减法来代替乘除法。纳皮尔发明对数的动机是为寻求球面三角计算的简便方法，他依据一种非常独特的与质点运动有关的设想构造出所谓对数方法，其核心思想表现为算术数列与几何数列之间的联系。在他的《奇妙的对数表的描述》中阐明了对数原理(见[《奇妙的对数表的描述》])，后人称他发明的对数为纳皮尔对数，记为 ，它与自然对数的关系为 以10为底的常用对数，是由另一位英国数学家布里格斯首先采用的。在他1624年出版的《对数算术》中，载有14位的常用对数表。他还制作了正弦、正切对数表。荷兰数学家兼出版商弗拉克补充了布里格斯的对数表，他出版的几种对数表(包括三角函数对数表)很快在欧洲普及。弗拉克还最早阐明对数首数的意义。 关于以e为底的自然对数的准确涵义，是由英国一位数学教师斯佩德尔(J.Speiodell)首先指出的，他在1619年出版了关于对数的著作，包含1—1000的自然对数表。 对数传到中国的时间是17世纪中叶，中国数学家薛风祚和波兰传教士穆尼阁合作的《比例对数表》是我国最早的对数著作。 personal credit business process management, and development of business processes, clear the loan object and scope, establish loan operation of assessment and accountability mechanisms. Diversity management principles the Bank's strict separation between business and consumer use of personal credit, then according to the different varieties of business to establish the appropriate access standard, post-lending management requirements. Business use personal credit should be strictly in accordance with the qilu Bank's small business credit risk management guidelines and associated working capital loans management requirements, such as assessing the management of production and operation, repayment ability, the level of risk, and so on. Income debt ratio control principle of Bank income debt ratio control mechanism, combined with the borrower's income, debt, expenditures, loans, guarantees and other factors, determines the loan amount and term, control each repayment does not exceed the amount of the borrower repayment ability. Principles of trans-regional limits Branches shall not in principle to operating income mainly comes from offsite (prefecture-level cities and administrative divisions at or above) or inward (MS) charge provided credit support for individual customers is located in different places, but except those with unique advantages such as critical. Have set up branches in different places, other bodies are not entitled to carry out the branch of credit business in the region. Branch County area has been established, other bodies are not entitled to apply for the credit within the County. Double before going through the personal credit business credit investigation, Contracting, security verification, credit check, key account manager should be double handle. Bank supports risk managers, loan review, professional 旋涡形或螺线型是自然事物极为普遍的存在形式，比如:一缕袅袅升上蓝天的炊烟，一朵碧湖中轻轻荡开的涟漪，数只缓缓攀援在篱笆上的蜗牛和无数在恬静的夜空携拥着旋舞的繁星…… 螺线特别是对数螺线的美学意义可以用指数的形式来表达: φkρ=αe 其中，α和k为常数，φ是极角，ρ是极径，e是自然对数的底。为了讨论方便，我们把e或由e经过一定变换和复合的形式定义为“自然律”。因此，“自然律”的核心是e，其值为2.71828……，是一个无限循环数。 数，美吗, 1、数之美 人们很早就对数的美有深刻的认识。其中，公元前六世纪盛行于古希腊的毕达哥斯学派见解较为深刻。他们首先从数学和声学的观点去研究音乐节奏的和谐，发现声音的质的差别(如长短、高低、轻重等)都是由发音体数量方面的差别决定的。例如发音体(如琴弦)长，声音就长;振动速度快，声音就高;振动速度慢，声音就低。因此，音乐的基本原则在于数量关系。 毕达哥斯学派把音乐中的和谐原理推广到建筑、雕刻等其它艺术，探求什么样的比例才会产生美的效果，得出了一些经验性的。例如，在欧洲有长久影响的“黄金律”据说是他们发现的(有人说，是蔡泌于一八五四年提出了所谓的“黄金分割律”。所谓黄金分割律“就是取一根线分为两部分，使长的那部分的平方等于短的那部分乘全线段。”“如果某物的长与宽是按照这个比例所组成的，那么它就比由其它比例所组成的长方形„要美?。”)。 这派学者还把数学与和谐的原则应用于天文学的研究，因而形成所谓“诸天音乐”或“宇宙和谐”的概念，认为天上诸星体在遵照一定的轨道运动中，也产生一种和谐的音乐。他们还认为，人体的机能也是和谐的，就象一个“小宇宙”。人体之所以美，是由于它各部分——头、手、脚、五官等比例适当，动作协调;宇宙之所以美，是由于各个物质单位以及各个星体之间运行的速度、距离、周转时间等等配合协调。这些都是数的和谐。 中国古代思想家们也有类似的观点。道家的老子和周易《系辞传》，都曾尝试以数学解释宇宙生成，后来又衍为周易象数派。《周易》中贲卦的表示朴素之美，离卦的表示华丽之美，以及所谓“极其数，遂定天下之象”，都是类似数学推理的结论。儒家的荀卿也说过:“万物同宇宙而异体。无宜而有用为人，数也。”庄子把“小我”与“大我”一视同仁，“小年”与“大年”等量齐观，也略同于毕达哥拉斯学派personal credit business process management, and development of business processes, clear the loan object and scope, establish loan operation of assessment and accountability mechanisms. Diversity management principles the Bank's strict separation between business and consumer use of personal credit, then according to the different varieties of business to establish the appropriate access standard, post-lending management requirements. Business use personal credit should be strictly in accordance with the qilu Bank's small business credit risk management guidelines and associated working capital loans management requirements, such as assessing the management of production and operation, repayment ability, the level of risk, and so on. Income debt ratio control principle of Bank income debt ratio control mechanism, combined with the borrower's income, debt, expenditures, loans, guarantees and other factors, determines the loan amount and term, control each repayment does not exceed the amount of the borrower repayment ability. Principles of trans-regional limits Branches shall not in principle to operating income mainly comes from offsite (prefecture-level cities and administrative divisions at or above) or inward (MS) charge provided credit support for individual customers is located in different places, but except those with unique advantages such as critical. Have set up branches in different places, other bodies are not entitled to carry out the branch of credit business in the region. Branch County area has been established, other bodies are not entitled to apply for the credit within the County. Double before going through the personal credit business credit investigation, Contracting, security verification, credit check, key account manager should be double handle. Bank supports risk managers, loan review, professional 之把“小宇宙”和“大宇宙”互相印证。所谓“得之于手而应用于心，口不能言，有数存在焉与其间”。这种从数的和谐看出美的思想，深深地影响了后世的中国美学。 2、黄金律之美 黄金律历来被染上瑰丽诡秘的色彩，被人们称为“天然合理”的最美妙的形式比例。我们知道，黄金律不仅是构图原则，也是自然事物的最佳状态。中世纪意大利数学家费勃奈舍发现，许多植物叶片、花瓣以及松果壳瓣，从小到大的序列是以0.618:1的近似值排列的，这即是著名的“费勃奈舍数列”:1、2、3、5、8、13、21、34……动物身上的色彩图案也大体符合黄金比。舞蹈教练、体操专家选择人材制定的比列尺寸，例如肩宽和腰的比例、腰部以上与腰部以下的比列也都大体符合黄金比。 现代科学家还发现，当大脑呈现的“倍塔”脑电波的高频与低频之比是1:0.618的近似值(12.9赫兹与8赫兹之比)时，人的心身最具快感。甚至，当大自然的气温(23摄氏度)与人的体温37摄氏度之比为0.618:1时,最适宜于人的身心健康,最使人感到舒适。另外，数学家们为工农业生产的优选法，所提出的配料最佳比例、组织结构的最佳比例等等，也都大体符合黄金律。 然而，这并不意味着黄金律比“自然律”更具有美学意义。我们可以证明，当对数螺线: φkρ=αe 的等比取黄金律，即k=0.0765872，等比P1/P2=0.618时，则螺线中同一半径线上相邻极半径之比都有黄金分割关系。事实上，当函数f(X)等于e的X次方时，取X为0.4812,那么，f(X)=0.618…… 因此，黄金律被“自然律”逻辑所蕴含。换言之，“自然律”囊括了黄金律。 黄金律表现了事物的相对静止状态，而“自然律”则表现了事物运动发展的普遍状态。因此，从某种意义上说，黄金律是凝固的“自然律”，“自然律”是运动着的黄金律。 3、“自然律”之美 “自然律”是e及由e经过一定变换和复合的形式。e是“自然律”的精髓，在数学上它是函数: 1(1+——) X的X次方，当X趋近无穷时的极限。 personal credit business process management, and development of business processes, clear the loan object and scope, establish loan operation of assessment and accountability mechanisms. Diversity management principles the Bank's strict separation between business and consumer use of personal credit, then according to the different varieties of business to establish the appropriate access standard, post-lending management requirements. Business use personal credit should be strictly in accordance with the qilu Bank's small business credit risk management guidelines and associated working capital loans management requirements, such as assessing the management of production and operation, repayment ability, the level of risk, and so on. Income debt ratio control principle of Bank income debt ratio control mechanism, combined with the borrower's income, debt, expenditures, loans, guarantees and other factors, determines the loan amount and term, control each repayment does not exceed the amount of the borrower repayment ability. Principles of trans-regional limits Branches shall not in principle to operating income mainly comes from offsite (prefecture-level cities and administrative divisions at or above) or inward (MS) charge provided credit support for individual customers is located in different places, but except those with unique advantages such as critical. Have set up branches in different places, other bodies are not entitled to carry out the branch of credit business in the region. Branch County area has been established, other bodies are not entitled to apply for the credit within the County. Double before going through the personal credit business credit investigation, Contracting, security verification, credit check, key account manager should be double handle. Bank supports risk managers, loan review, professional 人们在研究一些实际问题，如物体的冷却、细胞的繁殖、放射性元素的衰变时，都要研究 1(1+——) X的X次方，当X趋近无穷时的极限。正是这种从无限变化中获得的有限，从两个相反方向发展(当X趋向正无穷大的时，上式的极限等于e=2.71828……，当X趋向负无穷大时候，上式的结果也等于e=2.71828……)得来的共同形式，充分体现了宇宙的形成、发展及衰亡的最本质的东西。 现代宇宙学表明，宇宙起源于“大爆炸”，而且目前还在膨胀，这种描述与十九世纪后半叶的两个伟大发现之一的熵定律，即热力学第二定律相吻合。熵定律指出，物质的演化总是朝着消灭信息、瓦解秩序的方向，逐渐由复杂到简单、由高级到低级不断退化的过程。退化的极限就是无序的平衡，即熵最大的状态，一种无为的死寂状态。这过程看起来像什么,只要我们看看天体照相中的旋涡星系的照片即不难理解。如果我们一定要找到亚里士多德所说的那种动力因，那么，可以把宇宙看成是由各个预先上紧的发条组织，或者干脆把整个宇宙看成是一个巨大的发条，历史不过是这种发条不断争取自由而放出能量的过程。 生命体的进化却与之有相反的特点，它与热力学第二定律描述的熵趋于极大不同，它使生命物质能避免趋向与环境衰退。任何生命都是耗散结构系统，它之所以能免于趋近最大的熵的死亡状态，就是因为生命体能通过吃、喝、呼吸等新陈代谢的过程从环境中不断吸取负熵。新陈代谢中本质的东西，乃是使有机体成功的消除了当它自身活着的时候不得不产生的全部熵。 “自然律”一方面体现了自然系统朝着一片混乱方向不断瓦解的崩溃过程(如元素的衰变)，另一方面又显示了生命系统只有通过一种有序化过程才能维持自身稳定和促进自身的发展(如细胞繁殖)的本质。正是具有这种把有序和无序、生机与死寂寓于同一形式的特点，“自然律”才在美学上有重要价值。 如果荒僻不毛、浩瀚无际的大漠是“自然律”无序死寂的熵增状态，那么广阔无垠、生机盎然的草原是“自然律”有序而欣欣向荣的动态稳定结构。因此，大漠使人感到肃穆、苍茫，令人沉思，让人回想起生命历程的种种困顿和坎坷;而草原则使人兴奋、雀跃，让人感到生命的欢乐和幸福。 e=2.71828……是“自然律”的一种量的表达。“自然律”的形象表达是螺线。螺线的数学表达式通常有下面五种:(1)对数螺线;(2)阿基米德螺线;(3)连锁螺线;(4)双曲螺线;(5)回旋螺线。对数螺线在自然界中最为普遍存在，其它螺线也与对数螺线有一定的关系，不过目前我们仍未找到螺线的通式。对数螺线是1638年经笛卡尔引进的，后来瑞士数学家雅各?伯努利曾详细研究过它，发现对数螺线的渐屈线和渐伸线仍是对数螺线，极点在对数螺线各点的切线仍是personal credit business process management, and development of business processes, clear the loan object and scope, establish loan operation of assessment and accountability mechanisms. Diversity management principles the Bank's strict separation between business and consumer use of personal credit, then according to the different varieties of business to establish the appropriate access standard, post-lending management requirements. Business use personal credit should be strictly in accordance with the qilu Bank's small business credit risk management guidelines and associated working capital loans management requirements, such as assessing the management of production and operation, repayment ability, the level of risk, and so on. Income debt ratio control principle of Bank income debt ratio control mechanism, combined with the borrower's income, debt, expenditures, loans, guarantees and other factors, determines the loan amount and term, control each repayment does not exceed the amount of the borrower repayment ability. Principles of trans-regional limits Branches shall not in principle to operating income mainly comes from offsite (prefecture-level cities and administrative divisions at or above) or inward (MS) charge provided credit support for individual customers is located in different places, but except those with unique advantages such as critical. Have set up branches in different places, other bodies are not entitled to carry out the branch of credit business in the region. Branch County area has been established, other bodies are not entitled to apply for the credit within the County. Double before going through the personal credit business credit investigation, Contracting, security verification, credit check, key account manager should be double handle. Bank supports risk managers, loan review, professional 对数螺线，等等。伯努利对这些有趣的性质惊叹不止，竟留下遗嘱要将对数螺线画在自己的墓碑上。 英国著名画家和艺术理论家荷迦兹深深感到:旋涡形或螺线形逐渐缩小到它们的中心，都是美的形状。事实上，我们也很容易在古今的艺术大师的作品中找到螺线。为什么我们的感觉、我们的“精神的”眼睛经常能够本能地和直观地从这样一种螺线的形式中得到满足呢,这难道不意味着我们的精神，我们的“内在”世界同外在世界之间有一种比历史更原始的同构对应关系吗, 我们知道，作为生命现象的基础物质蛋白质，在生命物体内参与着生命过程的整个工作，它的功能所以这样复杂高效和奥秘无穷，是同其结构紧密相关的。化学家们发现蛋白质的多钛链主要是螺旋状的，决定遗传的物质——核酸结构也是螺螺状的。 古希腊人有一种称为风鸣琴的乐器，当它的琴弦在风中振动时，能产生优美悦耳的音调。这种音调就是所谓的“涡流尾迹效应”。让人深思的是，人类经过漫长岁月进化而成的听觉器官的内耳结构也具涡旋状。这是为便于欣赏古希腊人的风鸣琴吗,还有我们的指纹、发旋等等，这种审美主体的生理结构与外在世界的同构对应，也就是“内在”与“外在”和谐的自然基础。 有人说数学美是“一”的光辉，它具有尽可能多的变换群作用下的不变性，也即是拥有自然普通规律的表现，是“多”与“一”的统一，那么“自然律”也同样闪烁着“一”的光辉。谁能说清e=2.71828……给数学家带来多少方便和成功,人们赞扬直线的刚劲、明朗和坦率，欣赏曲线的优美、变化与含蓄，殊不知任何直线和曲线都可以从螺线中取出足够的部分来组成。有人说美是主体和客体的同一，是内在精神世界同外在物质世界的统一，那么“自然律”也同样有这种统一。人类的认识是按否定之否定规律发展的，社会、自然的历史也遵循着这种辩证发展规律，是什么给予这种形式以生动形象的表达呢,螺线～ 有人说美在于事物的节奏，“自然律”也具有这种节奏;有人说美是动态的平衡、变化中的永恒，那么“自然律”也同样是动态的平衡、变化中的永恒;有人说美在于事物的力动结构，那么“自然律”也同样具有这种结构——如表的游丝、机械中的弹簧等等。 “自然律”是形式因与动力因的统一，是事物的形象显现，也是具象和抽象的共同表达。有限的生命植根于无限的自然之中，生命的脉搏无不按照宇宙的旋律自觉地调整着运动和节奏……有机的和无机的，内在的和外在的，社会的和自然的，一切都合而为一。这就是“自然律”揭示的全部美学奥秘吗,不～“自然律”永远具有不能穷尽的美学内涵，因为它象征着广袤深邃的大自然。正因为如此，它才吸引并且值的人们进行不懈的探索，从而显示人类不断进化的本质力量。(原载《科学之春》杂志1984年第4期，原题为:《自然律——美学家和艺术家的瑰宝》) personal credit business process management, and development of business processes, clear the loan object and scope, establish loan operation of assessment and accountability mechanisms. Diversity management principles the Bank's strict separation between business and consumer use of personal credit, then according to the different varieties of business to establish the appropriate access standard, post-lending management requirements. Business use personal credit should be strictly in accordance with the qilu Bank's small business credit risk management guidelines and associated working capital loans management requirements, such as assessing the management of production and operation, repayment ability, the level of risk, and so on. Income debt ratio control principle of Bank income debt ratio control mechanism, combined with the borrower's income, debt, expenditures, loans, guarantees and other factors, determines the loan amount and term, control each repayment does not exceed the amount of the borrower repayment ability. Principles of trans-regional limits Branches shall not in principle to operating income mainly comes from offsite (prefecture-level cities and administrative divisions at or above) or inward (MS) charge provided credit support for individual customers is located in different places, but except those with unique advantages such as critical. Have set up branches in different places, other bodies are not entitled to carry out the branch of credit business in the region. Branch County area has been established, other bodies are not entitled to apply for the credit within the County. Double before going through the personal credit business credit investigation, Contracting, security verification, credit check, key account manager should be double handle. Bank supports risk managers, loan review, professional 呃，，?:?b汗 除了新学的时候老师为了带领大家探索，列个表示意一下，学完后就要记得那个y=sinX的标准图像，这个以后都是随手画的 而其他各式各样的三角函数就是在这个基础上变化的，比如你说的y=2sin(x+2) 是y=sinx先横坐标不变纵坐标全部伸长到原来的2倍，然后向昨平移2个单位 呃，，这么说你也听不懂吧，可以看看下面这个课件，书上也是有的 如果说是题目要求列表的画，那就装模作样取那几个点的列个表而已，不可能是再按照原来那样真正的去找点连线的 还有最后，希望你以后上课认真听讲啊，还是那句老话上课听比你回家死裹强的多 参考资料: personal credit business process management, and development of business processes, clear the loan object and scope, establish loan operation of assessment and accountability mechanisms. Diversity management principles the Bank's strict separation between business and consumer use of personal credit, then according to the different varieties of business to establish the appropriate access standard, post-lending management requirements. Business use personal credit should be strictly in accordance with the qilu Bank's small business credit risk management guidelines and associated working capital loans management requirements, such as assessing the management of production and operation, repayment ability, the level of risk, and so on. Income debt ratio control principle of Bank income debt ratio control mechanism, combined with the borrower's income, debt, expenditures, loans, guarantees and other factors, determines the loan amount and term, control each repayment does not exceed the amount of the borrower repayment ability. Principles of trans-regional limits Branches shall not in principle to operating income mainly comes from offsite (prefecture-level cities and administrative divisions at or above) or inward (MS) charge provided credit support for individual customers is located in different places, but except those with unique advantages such as critical. Have set up branches in different places, other bodies are not entitled to carry out the branch of credit business in the region. Branch County area has been established, other bodies are not entitled to apply for the credit within the County. Double before going through the personal credit business credit investigation, Contracting, security verification, credit check, key account manager should be double handle. Bank supports risk managers, loan review, professional