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本科生毕业论文题目:例谈变形技巧在数学解题中的应用所在院系:专业:姓名:学号:指导教师:数学与信息技术学院数学与应用数学1120510125论文完成日期:2013年5月2日—2— 目录摘要……………………………………………………………………1一、变形的相关理论…………………………………………………2二、变形技巧在一元二次方程中的应用……………………………3三、变形技巧在因式分解中的应用…………………………………5四、变形技巧在不等式中的应用……………………………………7五、变形技巧在三角函数中的应用…………………………………9参考文献………………………………………………………………11—2— 摘要:变形是数学解题的一种基本方法,变形能力的强弱制约着解题能力的高低.本文主要探讨变形技巧在一元二次方程、因式分解、不等式和三角函数解题中的应用.掌握并灵活运用好变形技巧,可以将复杂问题简单化,减少麻木性,提高解题效率.关键词:数学解题；变形技巧；一元二次方程；因式分解；基本不等式；三角函数中图分类号:O119文献标识码:A-16- 例谈变形技巧在数学解题中的应用陈海霞（1120510125）数学是个有机的整体,各部分之间相互联系,相互渗透,从而构成相互交错的立体空间,对各部分知识间的灵活掌握,更需要融会贯通.[1]近些年,数学题目越来越新颖,技巧性强,对有些题目进行适当变形,把复杂的数学问题简单化,从而顺利求得问题的答案.掌握并灵活运用好各类问题的变形技巧,有助于培养学生的逻辑推理能力,运算能力和空间想象能力,同时,用变形的方法,有助于把握数学问题的本质,它既是教师常用的一种重要数学方法,也是学生解题时一种非常有效的思想方法.此外,数学的学习内容是有意义的,富有挑战性的,要重视学生的学习能力和学习方法,充分利用数学变形技巧进行解题,不断提升学生的数学素质.[2]一、变形的相关理论变形是数学解题的一种常用方法,变形能力的强弱制约着解题能力的高低.[1]变形是为了达到某种目的或需要而采取的一种手段,是化归、转化和联想的准备阶段,它属于技能性的知识,既灵活又多变,一个公式,一个法则,它的表达形式多种多样,也存在技巧与方法,在实践中反复操作才能把握,-16- 能够让学生更好的理解变形技巧,乃至灵活运用.变形的一般形式主要有以下三种:1.等价变形等价变形就是利用等价关系进行的变形,在等价关系的条件下,通过等价变换的方式使数学问题得到解决,等价变形的本质就是在保持原来各种量之间的关系不变的情况下,只是改变它们的表达形式.常见的等价变形依据有:根据特定概念的定义,对数式,指数式的相互转化,如对数函数,可以等价变形为;根据等式与不等式的基本性质,比如移项,系数化为;根据计算的结果,将具体方程或不等式的形式转化为其具体的解或解集等.2.恒等变形恒等变形是在等价变形的思想指导下进行的,它的变形形式有代数式恒等变形、多项式恒等变形、分式恒等变形、三角函数恒等变形、对数式恒等变形等.若将两个代数式子中的字母换成任意相同的数值,这两个代数式的值都相等,我们就称这两个代数式恒等,表示两个代数式恒等的式子叫做恒等式.如是一个恒等式,把式子变为的这步变形,使变形的式子恒等,我们把这样的变形叫做恒等变形.3.同解变形同解变形是在等价转化思想的指导下,通过等价的变换,使得原来的等式与变形的等式有相同的解.-16- 方程的同解变形的一般形式有:交换其中任意两个方程的位置,其余不变;将任一个方程乘以一个非零的常数,其余不变;将任一个方程的常数倍加到另一个方程上,其余不变.需要注意的是:①方程两边同时加上或减去同一个分式不是同解变形,如方程的两边都加上,得,原方程的解为,而变形后的方程无解.②方程两边同时乘以不是同解变形,如方程的两边都乘以,得,即,此方程的解为任何实数,而原方程的解为.③方程两边同时乘以或除以同一个整式不是同解变形,如方程的两边都乘以,得,原方程的解为,而变形后的方程解为,.④方程两边平方不是同解变形,如方程,两边平方,得,原方程的解为,而变形后的方程解为,.二、变形技巧在一元二次方程中的应用学生在平时学习中不善于积累变形经验,在稍复杂的问题面前常因变形方向不清,导致问题难以解决,有些含有或可转化为一元二次方程的代数问题,能对方程进行适当变形并施以代换,常常可使问题化繁为简.[3]下面列举说明.例1已知,是方程的两根,求的值.分析:作为方程两根,,地位是平等的,而所求式子中,的次数相差悬殊,应设法将的次数降下来,由-16- ,得,从左向右次数降低了,对可进行连续降次,最终降为一次,即,于是,所以只要求出即可.解:因为是方程的根,所以,即,则,所以,又因为,是方程的两根,由韦达定理得,于是.本题若按步就搬地求出,的值,则计算较复杂,而且容易出错,而通过变形的技巧先从结论出发,转换思维,则可以提高解题的效率,节省时间,把握好问题之间的潜在问题.例2已知,是一元二次方程的两个根,求的值.[3]分析:观察所要求的表达式,表达式较复杂,即使求出,的值代入,计算也较难进行,所以应考虑将表达式变形成与有关的式子,巧妙运用韦达定理,不必分别求出和的值.解:由,是一元二次方程的两个根,可得,,及,,则-16- .在解决一元二次方程的代数问题时,要认真观察已知条件和所要求的式子,考察它们之间有什么关联,再充分利用已知条件来解决所要求的问题.同时是要灵活应用韦达定理:即如果,为方程的两个根,则,.在解这类问题时,可以从已知条件出发,也可以从结论入手,关键是要善于发现所要求式子的特点.三、变形技巧在因式分解中的应用多项式的因式分解,方法多样,技巧性强,有些多项式乔装打扮,貌似不能因式分解,但经过适当变形,创造条件,便可以进行因式分解.[4]因式分解的主要方法有符号变形、加减变形、换元变形、拆项变形、化简变形等,利用这些常见的变形方法解决一些具体的因式分解的问题.掌握了这些变形方法后,这类因式分解问题就可以迎刃而解了.1.换元变形例3分解因式.分析:直接展开项数较多,也不利于进一步因式分解,-16- 可以将考虑将四个因子两两结合,并且使得两两结合之后的表达式尽可能接近,比如将与结合,与结合,得到与,显然它们有相同的项,还可以考虑将作为相同的项,两种情形都应将相同的项作为一个整体,为计算方便,可作适当的换元.解:,若令,则上式子变形为,最后再将代入可得.若将看成一整体,并令其为,则上式变形为,原式解因式为.换元变形常用于较复杂的多项式,并且其中有相同的部分,将相同的项看成整体进行换元,掌握换元法,进行适当变形,能灵活应用于其他复杂的多项式因式分解中.2.拆项变形例4分解因式.分析:拆项变形是一种常见的分解因式的方法,-16- 拆项变形之后通常分组分解,观察表达式,容易想到把前两项组合并提取,得,但这个表达式不能继续分解下去了,需要调整,假如小括号中不是减,而是减就简单了,则可以考虑将与一次项结合,将一次项拆开,拆成;或者考虑将与常数项结合,将常数项拆开,拆成.这样拆项,使复杂问题简单化,更容易使问题得到解决.解法一:拆一次项=====.解法二:拆常数项=====.本题若先提取前两项的公因子,导致无法继续分解下去,善于观察所求分解的表达式的特点,找出此题的关键是拆项,拆项后与结合进行分解.寻求多种方法进行解题,体会解决问题策略的多样性,增强应用数学的意识,提高解决问题的能力.-16- 四、变形技巧在不等式中的应用不等式就是用不等号将两个解析式连结起来所成的式子,也就是在一个式子中,数的关系不全是等号，含不等符号的式子，那它就是一个不等式.在利用不等式求解函数最值问题时,有些问题可以直接利用公式求解,有些题目必须进行必要的变形才能利用均值不等式求解;在解不等式问题时,还可以通过画图进行分析求解.下面简单介绍不等式常用的变形技巧.例5已知,求函数的最大值.[5]分析:本题无法直接运用均值不等式求解,考虑用两种方法,求出此题的最大值,由,得,再巧乘常数,或由,得,再巧提常数,最后利用均值不等式得最值.解法一:因为,得,所以.当且仅当时,即时取等号.解法二:因为,得,所以,当且仅当时,即时取等号,所以当时,取得最大值为.形如或等的表达式主要有两种变形方法,巧乘常数或巧提常数,可以使问题更易解决,掌握所用变形方法,在实践中灵活使用.-16- 例6解不等式.分析:可以采用两种方法来解决这道题,代数法和图像法.利用代数法求解时,分和两种情形讨论;利用图像法求解时,设为反比例函数,其图像为双曲线,是一次函数,其图像为直线,求的解集,即求双曲线在直线上方时的范围.解法一:用代数法,分下列两种情形:①当时,不等式两边都乘以,得,即,解得,又,所以不等式的解集为.②当时,不等式两边都乘以,得,即,解得或,又,所以不等式的解集.综合①②得,不等式的解集为或.解法二:用图像法.设,,画出这两个函数在同一平面直角坐标系内的图像,联立方程,,求出其交点,分别为和,观察图像可得:当或时,双曲线在直线上方,即,于是不等式的解集为或.代数法主要适用于计算题,能够充分的体现数学思维的严谨性；图像法则更适用于选择题、填空题等类型,-16- 能很直观的让人理解和接受.无论是用代数法解题，还是用图像法解题,都要对问题进行分析，找出恰当的方法，适当地对题目进行变形，使问题更有效率的得到解决.例7若,且满足,求的最小值.[6]分析:本题要求的值,自然想到将已知条件转化为,但转化后也不能求得的最小值,通过“1”的代换,,得=,再利用均值不等式得到最小值.解:由,且,得==,当且仅当,即且时,等号成立,所以的最小值为.以“”进行变形,是比较常见并且灵活的方法,在数学问题的求解过程中,我们要善于捕捉“”,适时将“”进行变形,获得理想的解题方法,减少了使用基本不等式的次数,有效地避免了等号不能同时取到的麻烦.五、变形技巧在三角函数中的应用三角函数变形主要为三角恒等变换,三角恒等变换在数学中涉及广泛,三角公式众多,方法灵活多变,若能熟练掌握三角恒等变换的技巧,不但能加深对三角公式的记忆与内在联系的理解,而且还能发展数学逻辑思维能力,提高数学知识的综合运用能力.[7]下面通过例题体会三角函数的变形技巧.-16- 1.变换角的形式例8求的值.分析:本题涉及到的角有三个,注意到这三个角的关系是两两相差一个特殊角,选择一个适当的角为基本量,将其余的角与这个基本量组成和差关系,改变原来的角的形式,再运用两角和正弦、余弦公式,进行变形.解:令,则原式.对含有不同角的三角函数式,通常利用各种角之间的数值关系,将它们互相表示，熟记掌握三角函数的一些转化公式,这样使解题变得更容易,要加强对运算能力的培养,学会主动寻求合理、简捷的运算途径,加强解题训练,提高运算的准确性和实效性.[7]2.代数方法变形例9锐角,满足条件,则下列结论中正确的是().-16- A.B.C.D.分析:本题通过换元转化法,将三角问题转化为代数问题来解决,用代数方法对三角函数式因式分解,进行等量代换变形,令=,=,则有,化简整理得,即,所以,由,均为锐角,可得,于是,因此选D.换元法是数学中一个非常重要而且应用十分广泛的解题方法.此题通过对复杂的三角函数式子做等量代换，将三角问题变形成代数问题，使问题变得更加简捷.该方法虽然十分简单而且方便，但要真正地掌握和灵活使用，还需要在以后的学习和实践中不断归纳和总结.由于中学数学的改革及社会发展的需求,提高我们的应试能力和解决问题的能力,数学变形技巧作为一种解题的手段越来越被人们广泛使用.但是它并没有一定的规则,所以需要我们在平时的学习中加以运用和积累.本文通过对变形技巧在一元二次方程、因式分解、不等式、三角函数解题中应用,利用具体的例子来阐述说明变形技巧的作用.熟练掌握基本的变形技巧,能够提高解决问题的能力,增强对数学学习的兴趣和学好数学的信心.-16- 参考文献:[1]陈东磊.浅谈数学中的变形技巧科教文汇[J],2012(05):108-109[2]中华人民共和国教育部.义务教育数学课程标准2011版[M],北京师范大学出版社,2012[3]徐德义.一元二次方程变形的应用[J].初中数学教与学,2002(10):14-15[4]朱德祥.方法、能力、技巧[M].昆明:云南教育出版社,1989[5]董开福.《中学数学教材分析（第一版）》[M].昆明.云南出版社,1999[6]袁良佐.加“0”与乘“1”.中学生数学[J].2002,6:15-23[7]李根水.中学数学解题方法与技巧——变形与变换[M]北京师范大学出版社,1989内部资料请勿外传9JWKffwvG#tYM*Jg&6a*CZ7H$dq8KqqfHVZFedswSyXTy#&QA9wkxFyeQ^!djs#XuyUP2kNXpRWXmA&UE9aQ@Gn8xp$R#͑Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmUE9aQ@Gn8xp$R#͑Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z8vG#tYM*Jg&6a*CZ7H$dq8KqqfHVZFedswSyXTy#&QA9wkxFyeQ^!djs#XuyUP2kNXpRWXmA&UE9aQ@Gn8xp$R#͑Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^G89AmUE9aQ@Gn8xp$R#͑Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z8vG#tYM*Jg&6a*CZ7H$dq8KqqfHVZFedswSyXTy#&QA9wkxFyeQ^!djs#XuyUP2kNXpRWXmA&UE9aQ@Gn8xp$R#͑Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmUE9aQ@Gn8xp$R#͑Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNuGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$U*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89Amv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$U*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz84!z89Amv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$U*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$U*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNuGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tn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