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1、11不等关系12比较大小,一、不等关系在数学意义上，不等关系可以体现：_之间的不等关系；_之间的不等关系；_之间的不等关系；_之间的不等关系,二、比较大小1任意两个实数a，b都能比较大小：如果ab0，那么_；如果_，那么a0_；ab0_；ab0_.,2比较实数大小的方法比较两个实数a与b的大小，需归结为判断它们的差ab的_(注意：这里指差的符号，至于差的值究竟是什么，无关紧要),三、不等式的性质1若ab，则_；2若ab，c0，则_；3若ab，cb，bc，那么_.友情提示：(1)要特别注意性质2、3中_的符号，因为_的符号相异，结论恰好相反(2)所有性质中的a和b可以是_，也可以是_.,答案：常
2、量与常量变量与常量函数与函数一组变量ababbababcacbcacccc实数式子,1.不等式与等式之间主要有哪些异同？不等式与等式是生活、生产实践中最常见的关系式，其相异的性质主要在与数相乘时，不等式两边乘(除以)的数的符号不同时，结论不同；而等式则不然等式与不等式的性质对比如下表：,2.不等式的证明或比较实数大小有哪些方法及注意事项呢？证明一个不等式和比较实数的大小一样，根据题目的特点可以有不同的证明方法(1)作差法和作商法是比较实数大小和证明不等式的重要方法，但是它们又有自己的适用范围，对于不同的问题应当选择不同的方法进行解决：一般的实数大小的比较都可以采用作差法，但是我们要考虑作差后与
3、0的比较，通常要进行因式分解，配方或者其他变形操作，所以，作差后必须容易变形到能看出与0的大小关系,(2)在证明不等式时还可以利用已经证明的结论，或者利用不等式的性质对不等式进行变形，使不等式变成简单易于比较大小的形式，再比较大小得出结论，需要注意的是，有些结论的递推是双向的，而有些是单向的，例如，不等式性质中的对称性就是双向的，而传递性就是单向的，在不等式两边同乘一个数或式子的时候，必须先判断要乘的数或式子的符号，决定相乘后是否改变符号(3)有些不容易从正面证明的不等式还可以采用反证法进行证明，具体可以根据课本对性质4的推论3的证明方法和步骤，它可以把难以从正面说明的问题转化为其反面进行说明
4、.,例1对于实数a、b、c，判断下列命题的真假：(1)若ab，则acbc；(2)若ab，则ac2bc2；(3)若aabb2；(4)若ab0，则(5)若ab0，则,解析：(1)因未知c的正负或是否为零，无法确定ac与bc的大小，所以是假命题；(2)因为c20，所以只有c0时才能正确c0时，ac2bc2，所以是假命题；变式：若ac2bc2，则ab，此命题是真命题；(3)aab；ab2，命题是真命题；,变式训练1如果ab，则下列各式正确的是()Aalgxlgxb(x0)Bax2bx2Ca2b2Da2xb2x,解析：对于A：当x0时，lgxR，当lgx0时，algxblgx(x0)不成立，故应排除A；
5、对于B：xR，当x0时，ax2bx2，ax2bx2不成立，故应排除B；对于C：a2b2(ab)(ab)，又由ab可知ab0，但是ab的符号是不确定的，因此a2b2不成立，故应排除C；对于D：由指数函数的性质可知，2x0，又ab，a2xb2x成立，故选择D.答案：D,实数(或式)比较大小的依据是abab0；abab0；a0，b0时， 1ab)方法步骤是作差(商)变形判断大于或小于零(大于1或小于1)关键是变形，变形的目的在于便于判断正负常见的变形有因式分解、配方等,例2已知x1，比较x36x与x26的大小解析：(x36x)(x26)x3x26x6x2(x1)6(x1)(x1)(x26)，x1，(
6、x1)(x26)0，x36xx26.,变式训练2设mR，xR，比较x2x1与2m22mx的大小,例3比较aabb与abba(a、b为不相等的正数)的大小,变式训练3若m0，比较mm与2m的大小,例4已知a0，试比较a与 的大小,变式训练4已知a，b均为正数，nN*，比较(ab)(anbn)与2(an1bn1)的大小解析：(ab)(anbn)2(an1bn1)an1abnanbbn12an12bn1abnanban1bn1a(bnan)b(anbn)(ab)(bnan)，a、bR，nN*，且n1，当ab0时，ab0，bnan.(ab)(bnan)0.,当ba0时，aban.(ab)(bnan)0
7、时，ab0.所以(ab)(bnan)0.综上所述，(ab)(anbn)2(an1bn1)0.即(ab)(anbn)2(an1bn1),例5(一题多解)求证： ab.分析：本题可以用比较法证明；也可以用不等式性质得到证明,变式训练5已知ab，cbd.证明：证法1：由ab知ab0，由c0，(ac)(bd)(ab)(dc)0，acbd.证法2：cd.又ab，a(c)b(d)，即acbd.,例6求下面题目中 的取值范围(1)m3；(2)m2；(3)3m2.,答案：,例7设f(x)ax2bx且1f(1)2,2f(1)4，求f(2)的取值范围分析：本题是关于x的一元二次函数，可以利用换元法来求解在求解时一
8、定要注意已知条件中a、b的关系，准确把握a、b的取值范围，否则容易出错下面我们再用一种新的方法待定系数法来求解,f(2)3f(1)f(1)1f(1)2,2f(1)4，53f(1)f(1)10，故5f(2)10.,例8甲、乙两人同时从寝室到教室，甲一半路程步行，一半路程跑步；乙一半时间步行，一半时间跑步，如果两人步行速度、跑步速度均相同，则()A甲先到教室B乙先到教室C两人同时到教室 D谁先到教室不确定分析：用路程速度时间，求甲、乙两人所用的时间，再用比较法求解,答案：B,变式训练8甲、乙两水果商先后分别两次从某地购进水果，甲商每次购20000斤，乙商每次购2万元，若两次购进的价格不同，试判断甲
9、、乙两人谁的购买方式合算(即平均单价低)分析：先求出两人两次的平均单价，再用比较法解,新课标北师大版课件系列,高中英语必修2,Beijing Opera,the National Treasure,Beijing Opera dates back to the 18th century. It is a combination of acting, talking, singing, music, dancing and acrobatics.The costumes are always in bright colour.,costume,date backcombinationa nati
10、onal treasurea piercing voiceacrobaticscostume,a mixture of two or more thingsthe clothes worn in plays or filmsto have a history ofa group of valuable things,very special or important for the countrymovement of ones body using the skills of an acrobata very high and loud voice,Task 1: Listen to an
11、interview about Beijing Opera and answer the following questions.When did Beijing Opera start? What skills do Beijing Opera actors need to have?3. How many main roles are there in Beijing Opera?4. Match the main roles with their names. sheng male roles with brightly painted faces dan female roles ji
12、ng clown roles chou male roles,When did Beijing Opera start? It dates back to the late 18th century.What skills do Beijing Opera actors need to have? It is a combination of acting, talking, singing, music, dancing and acrobatics.How many main roles are there in Beijing Opera? There are four.,4. Matc
13、h the main roles with their names. sheng male roles dan female roles jing male roles with brightly painted faces chou clown roles,sheng,dan,jing,chou,Beijing Opera,history,main roles,skills needed,Task 2:Listen and use the Strategies to answer the questions.Where was Beijing Opera performed at the b
14、eginning?2. Why does it have very loud music and a piercing singing style?3. Why do performers wear brightly coloured costumes?,Where was Beijing Opera performed at the beginning? It was performed mostly on open-air stages in markets, streets, teahouses or temple courtyards.2. Why does it have very
15、loud music and a piercing singing style? In order to be heard over the crowds.3. Why do performers wear brightly coloured costumes? In order to be seen by the crowds clearly.,Beijing Opera,history,main roles,skills needed,costume,singing,Qionghua Cantonese Opera Festival,Read the passage about Canto
16、nese Opera carefully and find out information according to the diagram.2. Discuss with your group members to find out the similarities and differences between Beijing Opera and Cantonese Opera.,Cantonese Opera,history,main roles,skills needed,costume,singing,Besides Beijing Opera, there are still ot
17、her National Treasures in China. They serve as a window of China. And through this window, foreigners are able to understand Chinese culture better.,Besides Beijing Opera, there are still other National Treasures in China. They serve as a window of China. And through this window, foreigners are able to understand Chinese culture better.,Beijing Opera,history,main roles,skills needed,costume,singing,Various body movements can represent actions,riding in a carriage,riding a horse,holding a tassel in his hand,an army of thousands,four generals and four soldiers,walking with a flag on each side,
