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            【福建成人高考】高起點數學文科難點二

            福建成人高考信息網www.laiwobuy.com 發布時間: 2018年11月10日

            難點2:充要條件的判定


            充分條件、必要條件和充要條件是重要的數學概念,主要用來區分命題的條件p和結論q之間的關系.本節主要是通過不同的知識點來剖析充分必要條件的意義,讓考生能準確判定給定的兩個命題的充要關系.


            ●難點磁場


            (★★★★★)已知關于x的實系數二次方程x2+ax+b=0有兩個實數根α、β,證明:|α|<2且|β|<2是2|a|<4+b且|b|<4的充要條件.


            ●案例探究


            [例1]已知p:|1-|≤2,q:x2-2x+1-m2≤0(m>0),若?p是?q的必要而不充分條件,求實數m的取值范圍.


            命題意圖:本題以含絕對值的不等式及一元二次不等式的解法為考查對象,同時考查了充分必要條件及四種命題中等價命題的應用,強調了知識點的靈活性.


            知識依托:本題解題的閃光點是利用等價命題對題目的文字表述方式進行轉化,使考生對充要條件的難理解變得簡單明了.


            錯解分析:對四種命題以及充要條件的定義實質理解不清晰是解此題的難點,對否命題,學生本身存在著語言理解上的困難.


            技巧與方法:利用等價命題先進行命題的等價轉化,搞清晰命題中條件與結論的關系,再去解不等式,找解集間的包含關系,進而使問題解決.


            解:由題意知:


            命題:若?p是?q的必要而不充分條件的等價命題即逆否命題為:p是q的充分不必要條件.


            p:|1-|≤2-2≤-1≤2-1≤≤3-2≤x≤10


            q:x2-2x+1-m2≤0[x-(1-m)][x-(1+m)]≤0*


            ∵p是q的充分不必要條件,


            ∴不等式|1-|≤2的解集是x2-2x+1-m2≤0(m>0)解集的子集.


            又∵m>0


            ∴不等式*的解集為1-m≤x≤1+m


            ∴,∴m≥9,


            ∴實數m的取值范圍是[9,+∞.


            [例2]已知數列{an}的前n項Sn=pn+q(p≠0,p≠1),求數列{an}是等比數列的充要條件.


            命題意圖:本題重點考查充要條件的概念及考生解答充要條件命題時的思維的嚴謹性.


            知識依托:以等比數列的判定為主線,使本題的閃光點在于抓住數列前n項和與通項之間的遞推關系,嚴格利用定義去判定.


            錯解分析:因為題目是求的充要條件,即有充分性和必要性兩層含義,考生很容易忽視充分性的證明.


            技巧與方法:由an=關系式去尋找an與an+1的比值,但同時要注意充分性的證明.


            解:a1=S1=p+q.


            當n≥2時,an=Sn-Sn-1=pn-1(p-1)


            ∵p≠0,p≠1,∴=p


            若{an}為等比數列,則=p


            ∴=p,


            ∵p≠0,∴p-1=p+q,∴q=-1


            這是{an}為等比數列的必要條件.


            下面證明q=-1是{an}為等比數列的充分條件.


            當q=-1時,∴Sn=pn-1(p≠0,p≠1),a1=S1=p-1


            當n≥2時,an=Sn-Sn-1=pn-pn-1=pn-1(p-1)


            ∴an=(p-1)pn-1(p≠0,p≠1)


            =p為常數


            ∴q=-1時,數列{an}為等比數列.即數列{an}是等比數列的充要條件為q=-1.


            ●錦囊妙計


            本難點所涉及的問題及解決方法主要有:


            (1)要理解“充分條件”“必要條件”的概念:當“若p則q”形式的命題為真時,就記作pq,稱p是q的充分條件,同時稱q是p的必要條件,因此判斷充分條件或必要條件就歸結為判斷命題的真假.


            (2)要理解“充要條件”的概念,對于符號“”要熟悉它的各種同義詞語:“等價于”,“當且僅當”,“必須并且只需”,“……,反之也真”等.


            (3)數學概念的定義具有相稱性,即數學概念的定義都可以看成是充要條件,既是概念的判斷依據,又是概念所具有的性質.


            (4)從集合觀點看,若AB,則A是B的充分條件,B是A的必要條件;若A=B,則A、B互為充要條件.


            (5)證明命題條件的充要性時,既要證明原命題成立(即條件的充分性),又要證明它的逆命題成立(即條件的必要性).


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