2023高考數(shù)學(xué)題型技巧變式一應(yīng)俱全（上冊）

2023高考數(shù)學(xué)題型*技巧*變式一應(yīng)俱全（上冊）

專題17集合..................................................................................................6

【題型一】集合的表示........................................................................................6

【題型二】集合元素的特征-.......................................................................6

【題型三】集合的關(guān)系-......................................................................................7

【題型四】集合的運(yùn)算-...........................................................................8

【題型五】集合與排列組合概率...............................................................................8

【題型六】新定義-...........................................................................................9

【題型七】集合與圓和圓錐曲線-.............................................................................9

專題2-2中心對(duì)稱、軸對(duì)稱與周期性歸類.......................................................................12

【題型一】中心對(duì)稱性質(zhì)1：幾個(gè)復(fù)雜的奇函數(shù)...............................................................12

【題型二】中心對(duì)稱性質(zhì)2：與三角函數(shù)結(jié)合的中心對(duì)稱.......................................................13

【題型三】軸對(duì)稱...........................................................................................13

【題型四】中心對(duì)稱和軸對(duì)稱構(gòu)造出周期性...................................................................14

【題型五】畫圖：放大鏡..........................................................................15

【題型六】利用對(duì)稱解決恒成立和存在型.....................................................................16

【題型七】函數(shù)整數(shù)問題....................................................................................17

專題23零點(diǎn)................................................................................................

【題型一】水平線法：參變分離......................................................................20

【題型二】基礎(chǔ)圖像交點(diǎn)法..................................................................................20

【題型三】分段函數(shù)含參....................................................................................21

【題型四】研究直線斜率（臨界是切線）尋找交點(diǎn)關(guān)系............................................................22

【題里五】“放大鏡”函數(shù)的交點(diǎn)......................................................................22

【題型六】函數(shù)變換：..........................................................................23

【題型七】對(duì)數(shù)函數(shù)絕對(duì)值“積定法”...................................................................24

【題型八】高斯函數(shù)型......................................................................................24

【題型九】與三角函數(shù)結(jié)合..................................................................................25

【題型十】借助周期性......................................................................................26

專題2-4復(fù)合二次型和鑲嵌函數(shù)的零點(diǎn)..........................................................................30

【題型一】一元二次第合型基礎(chǔ)型：可因式分解...............................................................30

【題型二】一元二次第合型：根的分布型.....................................................................30

【題型三】一元二次復(fù)合型：參變分離與判別式、求根公式型.................................................31

【題型四】一元二次復(fù)合型（老高考）：線性規(guī)劃型.............................................................32

【題型五】一元二次復(fù)合型：函數(shù)性質(zhì)綜合型.................................................................32

【題型六】嵌套函數(shù)基礎(chǔ)型..................................................................................33

【題型七】嵌套函數(shù)常規(guī)型：無參雙坐標(biāo)系換元轉(zhuǎn)換法.........................................................34

【題型八】嵌套函數(shù)含參型：解析式含參.....................................................................34

【題型九】嵌套函數(shù)含參型：參數(shù)在方程.....................................................................35

【題型十】嵌套函數(shù)含參型：雙函數(shù)型.......................................................................36

【題型十一】嵌套函數(shù)雙復(fù)合型...............................................................................37

專題3-1導(dǎo)數(shù)求切線及公切線歸類...............................................................................41

【題型一】求切線基礎(chǔ)型：給切點(diǎn)求切線..................................................................41

【題型二】求切線基礎(chǔ)型：有切線無切點(diǎn)求切點(diǎn)...............................................................41

【題型三】求切線基礎(chǔ)：無切點(diǎn)求參.........................................................................42

【題型四】無切點(diǎn)多參......................................................................................42

【題里五】“過點(diǎn)”型切線.........................................................................43

【題型六】判斷切線條數(shù)....................................................................................44

【題型七】多函數(shù)（多曲線）的公切線...................................................................44

【題型八】切線的應(yīng)用：距離最值.....................................................................45

【題型九】切線的應(yīng)用：距離公式轉(zhuǎn)化型.....................................................................45

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【題型十】切線的應(yīng)用：恒成立求參等應(yīng)用.....................................46

【題型十一】切線的應(yīng)用：零點(diǎn)等...........................................................................47

專題3-2含參討論.............................................................................................50

【題型一】討論思維基礎(chǔ)：求導(dǎo)后一元一次型參數(shù)在常數(shù)位置.（單參）.........................................50

【題型二】討論思維基礎(chǔ)：求導(dǎo)后一元一次型參數(shù)在系數(shù)位理（單參）.........................................50

【題型三】討論思維基礎(chǔ)：求導(dǎo)后一元一被型參數(shù)在“斜率”和常數(shù)位置（雙參）..............................51

【題型四】上下平移思維基礎(chǔ)：反比例函數(shù)型.................................................................52

【題型五】上下平移：指數(shù)型................................................................................53

【題型六】上下平移：對(duì)數(shù)函數(shù)型...........................................................................54

【題型七】一元二次可因式分解型...........................................................................55

【題型八】一元二次不能因式分解：判別式+韋達(dá)定理+求根公式...............................................56

【題型九】雙線法：指數(shù)型..................................................................................57

【題型十】雙線法：對(duì)數(shù)型..................................................................................58

【題型十一】含三角函數(shù)型討論..............................................................................59

【題型十二】二階求導(dǎo)討論型................................................................................60

【題型十三】已知單調(diào)性求參.................................................................................60

【題型十四】不確定單調(diào)增或減求參..........................................................................61

【題型十五】存在單調(diào)增（減）區(qū)間...........................................................................62

【題型十六】非單調(diào)函數(shù)求參................................................................................63

專題3-3導(dǎo)數(shù)構(gòu)造函數(shù)十三種歸類...............................................................................67

【題型一】利用x"f（x）構(gòu)造型...........................................................................67

【題型二】利用f（x）/x”構(gòu)造型..........................................................................67

【題型三】利用enxf（x）構(gòu)造型............................................................................68

【題型四】用f（x）/enx構(gòu)造型.............................................................................69

【題型五】利用sinx與f（x）構(gòu)造型.......................................................................70

【題型六】利用cosx與f（x）構(gòu)造型.......................................................................71

【題型七】復(fù)雜型：en與af（x）+bg（x）等構(gòu)造型.............................................................72

【題型八】復(fù)雜型：（kx+b）與f（x）型.....................................................................72

【題型九】復(fù)雜型：與ln（kx+b）結(jié)合型....................................................................73

【題型十】復(fù)雜型：基礎(chǔ)型添加因式型.......................................................................74

【題型十一】復(fù)雜型：二次構(gòu)造..............................................................................75

【題型十二】綜合構(gòu)造......................................................................................76

【題型十三】技巧計(jì)算型構(gòu)造................................................................................77

專題3-4：超難壓軸小題?導(dǎo)數(shù)和函數(shù)歸類（1）82

【題型一】整數(shù)解..........................................................................................82

【題型二】零點(diǎn)............................................................................................82

【題型三】同構(gòu)............................................................................................83

【題型四】恒成立求參：移項(xiàng)討論型.........................................................................84

【題型五】恒成立求參：代入消參型（虛設(shè)根型）.............................................................85

【題型六】恒成立求參：構(gòu)造函數(shù)...........................................................................85

【題型七】恒成立求參：分離參數(shù)（常規(guī)）....................................................................86

【題型八】恒成立求參：分離參數(shù)（洛必達(dá)法則）.............................................................87

【題型九】恒成立求參：倍函數(shù)..............................................................................87

【題型十】恒成立求參：雙函數(shù)最值型.......................................................................88

【題里十一】數(shù)列與導(dǎo)數(shù)：..................................................................................89

專題3-5超難壓軸小題：導(dǎo)數(shù)與函數(shù)歸類（2）...................................................................92

【題型一】導(dǎo)數(shù)中的“距離”1：利用同底指數(shù)和對(duì)數(shù)關(guān)于y=x對(duì)稱關(guān)系（原函數(shù)與反函數(shù)）.....................92

【題型二】導(dǎo)數(shù)中的“距離”2：構(gòu)造型距離..................................................................93

【題型三】導(dǎo)數(shù)中的“距離”3：其他距離....................................................................93

【題型四】極值點(diǎn)偏移......................................................................................94

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【題型五】嵌套函數(shù)求參.......................................................95

【題型六】多參型1：復(fù)雜討論型............................................................................95

【題型七】多參型2：凸凹翻轉(zhuǎn)型............................................................................96

【題型八】多參型3:比值代換等代換.........................................................................96

【題型九】多參型4：韋達(dá)定理型............................................................................97

【題型十】多參型5：“二次”最值型........................................................................97

專題3-6導(dǎo)數(shù)壓軸大題歸類（1）..................................................................................................................................................................101

【題型一】求參1：端點(diǎn)值討論型............................................................................101

【題里二】求參2：“存在”型..............................................................................101

［題型三】求參3：“恒成立”型............................................................................102

【題型四】求參4：分離參數(shù)之“洛必達(dá)法則”...............................................................103

【題型五】同構(gòu)求參5：絕對(duì)值同構(gòu)求參型...................................................................103

【題型六】同構(gòu)求參6：k與刈構(gòu)造新函數(shù)型.................................................................104

【題型七】零點(diǎn)型.........................................................................................105

【題型八】不確定根型.....................................................................................106

【題型九】取整討論型.....................................................................................106

【題型十】證明不等式1：基礎(chǔ)型...........................................................................107

【題型十一】證明不等式2：數(shù)列不等式之單變量構(gòu)造型......................................................107

【題型十二】證明不等式3：數(shù)列不等式之無限求和型........................................................108

【題型十三】證明不等式4：構(gòu)造單變量函數(shù)型..............................................................109

【題型十四】證明不等式5：湊配主元型....................................................................109

專題3-7導(dǎo)致壓軸大題歸類：不等式證明歸類（2）.............................................................................................................................113

【題型一】不等式證明6:凹凸翻轉(zhuǎn)型.......................................................................113

【題型二】不等式證明7：三角函數(shù)與導(dǎo)數(shù)不等式............................................................113

【題型三】不等式證明8：極值點(diǎn)偏移之不含參型............................................................114

【題型四】不等式證明9：極值點(diǎn)偏移之含參型..............................................................115

【題型五】不等式證明10：三個(gè)“極值點(diǎn)（零點(diǎn)）”不等式....................................................115

【題型六】不等式證明I］：比值代換（整體代換等）.........................................................116

【題型七】不等式證明11：非對(duì)稱型（零點(diǎn)xl與x2系數(shù)不一致）.............................................117

【題型八】不等式證明12：韋達(dá)定理型.....................................................................117

【題型九】不等式證明13：利用第一問.....................................................................118

【題里十】不等式證明14：含ex和Inx型..................................................................118

【題型十一】不等式證明15：先放縮再證明.................................................................119

【題型十二】不等式證明16.：切線放縮證明兩根差型（剪刀模型）............................................119

【題型十三】不等式證明17：條件不等式證明...............................................................120

【題型十四】綜合證明，xl與x2型........................................................................121

專題4-1三角函數(shù)性質(zhì)、最值和W小題歸類.....................................................................125

【題型一】圖像與性質(zhì)1：“識(shí)圖”..........................................................................125

【題型二】圖像與性質(zhì)2：求周期...........................................................................126

【題型三】圖像與性質(zhì)3：正余弦函數(shù)的對(duì)稱軸..............................................................127

【題型四】圖像和性質(zhì)4：對(duì)稱中心.........................................................................128

【題型五】最值與范圍1：輔助角...........................................................................129

【題型六】最值與范圍2：一元二次正余弦有界性............................................................129

【題型七】最值與范圍3：sinx與cosx積和（差）換元型......................................................130

【題型八】址值與范圍4：分式型...........................................................................131

【題型九】最值與范圍5：絕對(duì)值型........................................................................132

【題型十】三角換元1：圓代換..............................................................................132

【題型十一】三角換元2：雙變量消元代換..................................................................133

【題型十二】三角換元3：無理根號(hào)代換....................................................................133

【題型十三】三角換元4：正切代換.........................................................................134

【題型十四】三角換元5：向量中的三角換元................................................................134

【題型十五】三角函數(shù)中w求解............................................................................135

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【題型十六】數(shù)列與三角函數(shù)..................................................136

專題4-2正余弦定理與解三角形小題1.........................................................................139

【題型一】解三角形基礎(chǔ)：角與對(duì)邊..........................................................................139

【題型二】判斷三角形形狀.................................................................................139

【題型三】最值與范圍1：先判斷角........................................................................140

【題型四】最值與范圍2：余弦定理........................................................................141

【題型五】最值與范圍3：輔助角...........................................................................141

【題型六】最值與范圍4：均值不等式......................................................................142

【題型七】最值與范圍5：周長最值..........................................................................142

【題型八】面積1:消角.....................................................................................143

【題型九】面積2：正切代換...............................................................................144

【題型十】最優(yōu)與范圍6：建系設(shè)點(diǎn)........................................................................144

【題型十一】最值與范圍7：求正切的最值范闈..............................................................145

【題型十二】圖形1：中線..................................................................................146

【題型十三】圖形2：角平分線.............................................................................146

【題型十四】圖形3：高.....................................................................................147

【題里十五】圖形4：四邊形................................................................................148

專題4-3正余弦定理與解三角形小題歸類2...................................................................151

【題型一】圖形5：“擴(kuò)展線”...............................................................................151

【題型二】向量...........................................................................................151

【題型三】四心1：外心....................................................................................152

【題型四】四心2：內(nèi)心....................................................................................153

【題型五】四心3：重心....................................................................................153

【題型六】四心4：垂心....................................................................................154

【題型七】解三角形應(yīng)用題.................................................................................155

【題里八】超難壓軸小題1.................................................................................156

【題型九】超難壓軸小題2.................................................................................157

專題4-4三角函數(shù)與解三角形大題歸類.........................................................................161

【題型一】Asin（Ox+。）圖像與性質(zhì)1:給圖求解析式和值域（最值）......................................161

【題型二】Asin（Ox+0）圖像與性質(zhì)2：二倍角降嘉公式恒等變形...........................................162

【題型三】Asin（Ox+0）圖像與性質(zhì)3：恒等變形（“打散”-重組-輔助角）.................................163

【題型四】Asin（Ox+6）圖像與性質(zhì)4：零點(diǎn)求參.........................................................164

【題型五】解三角形基礎(chǔ)：正弦定理、角與對(duì)邊..............................................................164

【題型六】解三角形基礎(chǔ)2：余弦定理變形..................................................................165

【題型七】解三角形1：山I枳最值...........................................................................166

［題型八］解三角形2：周長最值...........................................................................167

【題型九】解三角形3：邊長最值...........................................................................167

【題型十】解三角形4：不對(duì)稱型最值......................................................................168

【題型十一】解三角形5：中線.............................................................................168

【題型十二】解三角形6：角平分線........................................................................169

【題型十三】三角形存在個(gè)數(shù)...............................................................................170

【題型十四】四邊形轉(zhuǎn)化為解三角形........................................................................170

【題型十五】解三角形：四邊形求最值......................................................................172

【題型十六】三角形中證明題...............................................................................173

【題型十七】解三角形綜合.................................................................................174

【題型十八】建模應(yīng)用.....................................................................................175

專題5向量小題歸類.........................................................................................181

【題型一】向量基礎(chǔ)：“繞三角形”（基底拆分）..............................................................181

【題型二】系數(shù)未知型“繞三角形”........................................................................182

【題型三】求最值型“繞三角形”...........................................................................183

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【題型四】數(shù)星積............................................................?.............................184

【題型五】數(shù)量積最值型...................................................................................185

【題型六】向量模.........................................................................................185

【題型七】投影向量........................................................................................186

【題型八】向量技巧1：極化恒等式.........................................................................186

【題型九】向量技巧2：等和線..............................................................................187

【題型十】向量技巧3：奔馳定理與面積....................................................................188

【題型十一】解析幾何中的向量..............................................................................190

【題型十二】向量四心.....................................................................................190

［題型十三】綜合應(yīng)用.....................................................................................191

【題型十四】超難小題.......................................................................................192

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專題1?1集合

【題型一】集合的表示

【典例分析】

如圖，四個(gè)棱長為1的正方體排成一個(gè)正四棱柱，是一條側(cè)棱，P,（i=1,2,…，8）是上底面上其余的八個(gè)點(diǎn),

貝喋合懇'.產(chǎn)寶?二刃二1、2、3、八8）中的元素個(gè)數(shù)（）

A.1B.2C.4D.8

【提分秘籍】

基本規(guī)律

1.列舉法，注意元素互異性和無序性

2.描述法，注意準(zhǔn)確理解集合元素，能理解不同符號(hào)的元素

【變式演練】

1.設(shè)集合"加=空￡一=,N={x|x==+；,&=Z},則（）

A.M=NB.MtiNC.M堅(jiān)ND.MYN

2.-1?（x|2',<x<2'-\x?3^m€.v）,若川表示集合A”中元素的個(gè)數(shù)，則及I二，則

hJ+M+ki+…+hiol=.

3.已知集合M是滿足下列性質(zhì)的函數(shù)/⑷的全體：存在力零常數(shù)k,對(duì)定義域中的任意x,等式/'（心）=|

+/⑶恒成立.現(xiàn)有兩個(gè)函數(shù)：/（A）二奴+6（〃才0）,g（x）=log2x,則函數(shù)/（x）、g（A）與集合M的關(guān)系

為.

【題型二】集合元素的特征-

【典例分析】

己雕合A=《1=2|\<31共）卜，B=,1x+2..

{.x=N\-<0b,則集合懇本=xy\x=A,y=B}的元素個(gè)數(shù)為（)

無3J

A.6B.7

C.8D.9

6

【提分秘籍】

基本規(guī)律

1.研究集合問題，一定要抓住元素，看元素應(yīng)滿足的屬性。

2.研究兩（多個(gè)）集合的關(guān)系時(shí)，關(guān)鍵是將兩集合的關(guān)系轉(zhuǎn)化為元素間的關(guān)系。

【變式演練】

1.已知集合M?，1?8石，外。2。）,則下列四個(gè)元素中屬于M的元素的個(gè)數(shù)是（）

①i+&；?。?6△；③，］卜；@h-Q+4

A.4B.3C.2D.1

2.