高中數(shù)學(xué)講義

2018.09杭州

第一講集合......................................................................6

一、集合的含義與表示.........................................................6

二、集合間基本關(guān)系...........................................................6

三、集合基本運(yùn)算.............................................................7

※四、集合知識(shí)進(jìn)階...........................................................8

第二講函數(shù)......................................................................10

一、函數(shù)及其表示............................................................11

1、函數(shù)和映射的概念.....................................................11

2、函數(shù)的基本性質(zhì).......................................................12

派3、函數(shù)知識(shí)進(jìn)階.......................................................13

二、累、指數(shù)、對(duì)數(shù)及其函數(shù)..................................................16

1、一些基本基本概念.....................................................16

2、哥函數(shù)及其基本性質(zhì)...................................................17

3、指數(shù)函數(shù)及其性質(zhì).....................................................18

4、對(duì)數(shù)與對(duì)數(shù)函數(shù).......................................................19

第三講三角函數(shù)................................................................23

一、任意角的概念與弧度制....................................................23

二、任意角的二角函數(shù)........................................................23

三、單位圓與三角函數(shù)線......................................................24

四、基本關(guān)系與誘導(dǎo)公式......................................................25

1、同角三角函數(shù)的基本關(guān)系..............................................25

2、誘導(dǎo)公式第一組一?基本變換公式.......................................26

3、誘導(dǎo)公式第二組一兩角和(差)的三角函數(shù).............................27

4、誘導(dǎo)公式第三組：倍角、半角和萬(wàn)能公式...............................30

5、誘導(dǎo)公式第四組:和差化積及積化和差.................................32

五、三角函數(shù)的圖像研究及變換...............................................33

1、正弦函數(shù).............................................................33

2、余弦函數(shù).............................................................34

3、正弦型函數(shù)y=Asin(w+0),(A>0,3>0)的圖像和性質(zhì)........34

六、正弦定理和余弦定理......................................................37

1、正弦定理.............................................................37

2、余弦定理.............................................................38

3、正、余弦定理的應(yīng)用...................................................40

第四講向量基本概念及運(yùn)算......................................................41

一、平面向量的概念..........................................................41

二、向量的線性運(yùn)算..........................................................42

1、向量的加法...........................................................42

2、向量的減法...........................................................43

3、數(shù)乘向量.............................................................43

4、向量共線的條件.......................................................43

5、平面向量基本定理.....................................................44

三、向量知識(shí)進(jìn)階............................................................44

1、數(shù)量積...............................................................44

派2、向量積.............................................................45

3、向量在三角形中的應(yīng)用................................................45

第五講數(shù)列.....................................................................47

一、數(shù)列的基本概念..........................................................47

二、等差數(shù)列................................................................48

三、等比數(shù)列................................................................49

※四、數(shù)列知識(shí)進(jìn)階..........................................................50

第六講邏輯推理和算法..........................................................56

一、邏輯原理................................................................56

1、什么叫邏輯...........................................................56

2、邏輯的基本規(guī)律.......................................................56

3、命題.................................................................60

4、形式化...............................................................61

5、命題條件.............................................................61

6、對(duì)命題的布爾操作.....................................................62

二、推理方法................................................................62

1、演繹推理.............................................................62

2、歸納推理.............................................................G4

三、數(shù)學(xué)歸納法..............................................................66

※四、哥德?tīng)柌煌陚湫?.......................................................67

1、哥德?tīng)柌煌陚湫远ɡ淼膬?nèi)容............................................67

2、哥德?tīng)柌煌陚湫远ɡ硖岢龅谋尘?.......................................67

3、哥德?tīng)柌煌陚涠ɡ淼囊饬x..............................................68

五、算法初步................................................................69

1、算法的基本概念.......................................................69

2、算法的基本邏輯結(jié)構(gòu)...................................................70

3、基本算法語(yǔ)句.........................................................70

第七講不等式證明..............................................................72

一、不等式的性質(zhì)............................................................72

二、證明不等式的常見(jiàn)方法....................................................72

三、利用一元二次方程解不等式...............................................74

四、初級(jí)均值不等式..........................................................74

第八講※重要不等式※..........................................................76

一、均值不等式..............................................................76

1、AG不等式的柯西推理模式..............................................76

2、利用琴生不等式來(lái)證明AG不等式.......................................78

3、利用柯西不等式來(lái)證明........................................78

4、利用AG不等式來(lái)證明HG不等式........................................79

5、利用柯西不等式來(lái)證明4“《0.....................................79

二、柯西不等式..............................................................80

三、排序不等式..............................................................81

1、調(diào)整法...............................................................82

2、阿貝爾變換法.........................................................82

四、琴生不等式..............................................................83

第九講直線和圓................................................................85

一、什么是解析幾何？........................................................85

二、關(guān)于直線的研究..........................................................86

1、直線的傾斜角和斜率...................................................86

2、兩點(diǎn)間的距離公式.....................................................86

3、直線的方程...........................................................87

4、兩條直線之間的關(guān)系...................................................87

三、關(guān)于圓的研究............................................................89

1、圓的標(biāo)準(zhǔn)方程和一般方程..............................................89

2、點(diǎn)與圓的位置關(guān)系.....................................................89

3、直線（圓）與圓的位置關(guān)系............................................90

4、※圓的面積L算問(wèn)題...................................................90

四、關(guān)于橢圓的研究..........................................................92

1、橢圓的定義及其標(biāo)準(zhǔn)方程..............................................92

2、橢圓的幾何性質(zhì).......................................................94

五、雙曲線和拋物線..........................................................94

1、雙曲線的定義及其標(biāo)準(zhǔn)方程............................................94

2、雙曲線的幾何性質(zhì).....................................................95

3、拋物線的定義、標(biāo)準(zhǔn)方程及其性質(zhì)......................................96

六、軌跡知識(shí)進(jìn)階※..........................................................98

1、極坐標(biāo)系.............................................................98

2、一些軌跡的極坐標(biāo)方程................................................98

第十講導(dǎo)數(shù)的概念與運(yùn)算.......................................................100

一、極限的相關(guān)知識(shí).........................................................100

1、極限的定義..........................................................100

2、函數(shù)的左右極限......................................................101

3、判定極限存在的兩個(gè)準(zhǔn)則.............................................101

4、兩個(gè)重要極限........................................................102

二、導(dǎo)數(shù)的相關(guān)知識(shí).........................................................102

1、導(dǎo)數(shù)的定義..........................................................102

2、導(dǎo)函數(shù)..............................................................103

3、導(dǎo)數(shù)的基本性質(zhì)......................................................103

4、基本函數(shù)的導(dǎo)數(shù)計(jì)算..................................................103

三、導(dǎo)數(shù)在研究函數(shù)中的簡(jiǎn)單應(yīng)用............................................105

1、利用導(dǎo)數(shù)判斷函數(shù)的單調(diào)性...........................................106

2、利用導(dǎo)數(shù)研究函數(shù)的極值.............................................106

第十一講空間關(guān)系.............................................................107

一、空間中的平行關(guān)系.......................................................107

1、關(guān)于平面的三個(gè)公理..................................................107

2、平面基本性質(zhì)的推論..................................................107

3、線線關(guān)系與線面平行..................................................107

4、面面平行的判定與性質(zhì)...............................................108

二、空間中的垂直關(guān)系.......................................................109

1、線和面的垂直關(guān)系....................................................109

2、面和面的垂直關(guān)系....................................................110

三、空間向量...............................................................110

1、關(guān)于空間向量的基本知識(shí).............................................110

2、空間向量基本定理....................................................111

3、空間向量的夾角......................................................111

4、兩個(gè)向量的數(shù)量積....................................................111

5、空間向量的坐標(biāo)計(jì)算..................................................112

6、空間向量平行和垂直..................................................112

7、空間向量的模和夾角..................................................112

四、空間幾何體.............................................................113

1、祖胞原理............................................................113

2、圓錐的體積L算......................................................113

3、球的體積計(jì)算........................................................115

4、球的表面積..........................................................116

第十二講計(jì)數(shù)與概率統(tǒng)計(jì)原理...................................................117

一、基本計(jì)數(shù)原理...........................................................117

二、排列....................................................................118

三、組合....................................................................118

四、概率統(tǒng)計(jì)的基本概念.....................................................120

1、隨機(jī)抽樣............................................................120

2、統(tǒng)計(jì)數(shù)據(jù)的數(shù)字特征..................................................120

3、隨機(jī)事件概率........................................................121

五、離散型隨機(jī)變量及其概率................................................122

第十三講復(fù)數(shù)..................................................................123

一、復(fù)數(shù)的相關(guān)概念.........................................................123

1、復(fù)數(shù)的引入..........................................................123

2、復(fù)數(shù)的模............................................................123

3、共桅復(fù)數(shù)............................................................124

4、好數(shù)的輻角..........................................................124

二、復(fù)數(shù)的運(yùn)算.............................................................124

1、復(fù)數(shù)的加減..........................................................124

2、復(fù)數(shù)的乘法..........................................................125

3、復(fù)數(shù)的除法..........................................................125

※三、復(fù)數(shù)知識(shí)進(jìn)階.........................................................125

1、i的乘方法則.........................................................125

2、棣莫佛定理..........................................................126

第一講集合

一、集合的含義與表