基本初等函數(shù)與初等函數(shù)課件

1、 基本初等函數(shù)基本初等函數(shù) 初等初等函數(shù)函數(shù)第二節(jié)第二節(jié) 基本初等函數(shù)與初等函數(shù)基本初等函數(shù)與初等函數(shù)一、基本初等函數(shù)一、基本初等函數(shù)(1) 常函數(shù)常函數(shù) y = c(2) 冪函數(shù)冪函數(shù) y =x (3) 指數(shù)函數(shù)指數(shù)函數(shù) xxy aaay e(0,1) 或或 ayx aayxlog(0,1)ln 或或(4) 對(duì)數(shù)函數(shù)對(duì)數(shù)函數(shù) (5) 三角函數(shù)三角函數(shù) yx yx yxsin,cos,tan, yx yx yxcot ,sec ,csc (6) 反三角函數(shù)反三角函數(shù) yarcx yarcx yarcx yarcxsin,cos,tan,cot 應(yīng)熟練掌握其表達(dá)式、定義域、值域、幾何特性、常應(yīng)熟

2、練掌握其表達(dá)式、定義域、值域、幾何特性、常見(jiàn)公式、圖象及性質(zhì)見(jiàn)公式、圖象及性質(zhì).1、冪函數(shù)、冪函數(shù)yx() 是是常常數(shù)數(shù) oxy)1 , 1(112xy xy yx1 xy 2、指數(shù)函數(shù)、指數(shù)函數(shù))1, 0( aaayx)1 , 0( xey xay)1( xya a(1) 3、對(duì)數(shù)函數(shù)、對(duì)數(shù)函數(shù)xyln )1, 0(log aaxyaayxlog xya1log )1( a)0 , 1( 4、三角函數(shù)、三角函數(shù)正弦函數(shù)正弦函數(shù)xysin xysin xycos xycos 余弦函數(shù)余弦函數(shù)正切函數(shù)正切函數(shù)xytan xytan xycot 余切函數(shù)余切函數(shù)xycot 正割函數(shù)正割函數(shù)1sec

3、cosyxxxysec 1cscsinyxx余割函數(shù)余割函數(shù)xycsc 5、反三角函數(shù)、反三角函數(shù)xyarcsin xyarcsin 反反正正弦弦函函數(shù)數(shù)xyarccos xyarccos 反余弦函數(shù)反余弦函數(shù)xyarctan xyarctan 反反正正切切函函數(shù)數(shù) 常函數(shù)常函數(shù), 冪函數(shù)冪函數(shù), 指數(shù)函數(shù)指數(shù)函數(shù), 對(duì)數(shù)函數(shù)對(duì)數(shù)函數(shù), 三角函三角函數(shù)和反三角函數(shù)統(tǒng)稱為數(shù)和反三角函數(shù)統(tǒng)稱為基本初等函數(shù)基本初等函數(shù).xycot 反反余余切切函函數(shù)數(shù)arcxycot arc二、初等函數(shù)二、初等函數(shù) 由常數(shù)和基本初等函數(shù)經(jīng)過(guò)有限次四則運(yùn)算和有限次的由常數(shù)和基本初等函數(shù)經(jīng)過(guò)有限次四則運(yùn)算和有限次的函數(shù)

4、復(fù)合步驟所構(gòu)成并可用函數(shù)復(fù)合步驟所構(gòu)成并可用一個(gè)式子表示一個(gè)式子表示的函數(shù)的函數(shù),稱為稱為初等初等函數(shù)函數(shù).注注 分段函數(shù)不為初等函數(shù)分段函數(shù)不為初等函數(shù).都是初等函數(shù)都是初等函數(shù).xyxxyx2cos2arcsin log 3 ,(sin ) 例例1 分段函數(shù)雖然不為初等函數(shù)分段函數(shù)雖然不為初等函數(shù), 但是由于分段函數(shù)在其但是由于分段函數(shù)在其定義域的子區(qū)間內(nèi)都是初等函數(shù)定義域的子區(qū)間內(nèi)都是初等函數(shù), 所以所以仍可通過(guò)初等函仍可通過(guò)初等函數(shù)來(lái)研究分段函數(shù)數(shù)來(lái)研究分段函數(shù).復(fù)合函數(shù)的分解復(fù)合函數(shù)的分解函數(shù)復(fù)合的逆運(yùn)算函數(shù)復(fù)合的逆運(yùn)算 將一個(gè)復(fù)合函數(shù)分解成若干簡(jiǎn)單函數(shù)將一個(gè)復(fù)合函數(shù)分解成若干簡(jiǎn)單函

5、數(shù)(基本初等函數(shù)或基本初等函數(shù)或由基本初等函數(shù)與常數(shù)的四則運(yùn)算所得到的函數(shù)由基本初等函數(shù)與常數(shù)的四則運(yùn)算所得到的函數(shù))是是由外由外到里到里, 從左到右從左到右, 逐層分解逐層分解, 使每個(gè)層次都是簡(jiǎn)單函數(shù)使每個(gè)層次都是簡(jiǎn)單函數(shù).例例2 將下列函數(shù)分解為簡(jiǎn)單函數(shù)并求其定義域：將下列函數(shù)分解為簡(jiǎn)單函數(shù)并求其定義域：222(1) cos(2) ln(11)1(3) log (56)ayxyxyxx yu ux2(1) cos,(,) yu uv vx2(2) ln,1,1,(,) 21(3) ,log,56,ayuv vxxu 55555555 (,)(,2)(3,)(,)2222 解解引入中間變量

6、引入中間變量將將 函數(shù)分解為簡(jiǎn)單函數(shù)函數(shù)分解為簡(jiǎn)單函數(shù).xxxye22coslnsin1 解解wyewu v ux vt tzxzx22,cos ,ln ,sin,1 2sinhxxeex 雙雙曲曲正正弦弦xycosh xysinh ),(: D奇函數(shù)奇函數(shù).2coshxxeex 雙曲余弦雙曲余弦),(: D偶函數(shù)偶函數(shù).1、雙曲函數(shù)、雙曲函數(shù)xey21 xey 21三、雙曲函數(shù)與反雙曲函數(shù)三、雙曲函數(shù)與反雙曲函數(shù)(簡(jiǎn)介簡(jiǎn)介)xxxxeeeexxx coshsinhtanh雙雙曲曲正正切切奇函數(shù)奇函數(shù),),(:D有界函數(shù)有界函數(shù),雙曲函數(shù)常用公式雙曲函數(shù)常用公式;sinhcoshcoshsin

7、h)sinh(yxyxyx ;sinhsinhcoshcosh)cosh(yxyxyx ;1sinhcosh22 xx;coshsinh22sinhxxx .sinhcosh2cosh22xxx 2、反雙曲函數(shù)、反雙曲函數(shù)奇函數(shù)奇函數(shù),),(:D.),(內(nèi)單調(diào)增加內(nèi)單調(diào)增加在在;sinh xy 反雙曲正弦反雙曲正弦ar).1ln(sinh2 xxxyarsinhar xy.), 1內(nèi)內(nèi)單單調(diào)調(diào)增增加加在在 ), 1 :D y反反雙雙曲曲余余弦弦coshar).1ln(cosh2 xxxyarxcosharx y.11ln21xx )1 , 1(: D奇函數(shù)奇函數(shù),.)1 , 1(內(nèi)單調(diào)增加內(nèi)單調(diào)增加在在 y反雙曲正切反雙曲正切tanharxytanh arxtanharx y函數(shù)表示