高等數(shù)學(xué)函數(shù)

1、1函數(shù)本節(jié)內(nèi)容：一、鄰域二、函數(shù)的概念三、基本初等函數(shù)四、復(fù)合函數(shù)五、初等函數(shù)一、鄰域.定義1:設(shè)aR,R,則一點a的鄰域aU(a,)的中心，U(a,)的半徑.定義2:一點a的去心鄰域二、函數(shù)的概念f定義在D上的函數(shù)；D定義域；x自變量；y因變量；f(Xo)X0處的函數(shù)值；Wy|yf(x),xD值域.注意:函數(shù)的兩個要素一一定義域和對應(yīng)法則.補(bǔ)例1求下列函數(shù)的定義域.yln入基本初等函數(shù)基本初等函數(shù)指下列5類:哥函數(shù)yx（是常數(shù)）指數(shù)函數(shù)yax（a是常數(shù)，a0,a1）對數(shù)函數(shù)ylogax（a是常數(shù)，a0,a1）三角函數(shù)ysinx,ycosx,ytanx,ycotx,ysecx,ycscx反三角

2、函數(shù)yarcsinx,yarccosx,yarctanx,yarccotx（一）冪函數(shù).冪函數(shù)的定義:.冪函數(shù)的圖形與性質(zhì)(a)取不同值，騫函數(shù)的定義域與值域均可能不同；(b)對任意，函數(shù)圖形都過點(1,1);當(dāng)時，圖形過點(0,0)和(1,1);(c)當(dāng)0時，哥函數(shù)在。)為單調(diào)遞增函數(shù);而而0時，騫函數(shù)在(0,)為單調(diào)遞減函數(shù);(d)騫函數(shù)為無界函數(shù).騫函數(shù)的運(yùn)算性質(zhì)：(a)aa(b)aa(c)(a)(d)(ab)1指數(shù)函數(shù)的定義：2.指數(shù)函數(shù)的圖形與性質(zhì)：2.指數(shù)函數(shù)的圖形與性質(zhì)：(a)定義域為R,值域為R ;(b)a不論取何值，函數(shù)圖形都過點(0,1);當(dāng)a1時，指數(shù)函數(shù)為單調(diào)遞增函數(shù)而

3、0a1時，指數(shù)函數(shù)為單調(diào)遞減函J(d)指數(shù)函數(shù)為無界函數(shù)；(e)指數(shù)函數(shù)是非奇非偶函數(shù)3.指數(shù)函數(shù)的運(yùn)算性質(zhì)：與騫函數(shù)的運(yùn)算性質(zhì)相似(三)對數(shù)函數(shù).對數(shù)函數(shù)的定義：其中a底數(shù).一種特殊對數(shù)：ylnx.，略.對數(shù)函數(shù)的圖形與性質(zhì)：，略.(a)定義域為R,值域為R;(b)a不論取何值，函數(shù)圖形都過點(1,0);當(dāng)a1時，對數(shù)函數(shù)為單調(diào)遞增函數(shù)而0a1時，對數(shù)函數(shù)為單調(diào)遞減函(d)對數(shù)函數(shù)為無界函數(shù)；(e)對數(shù)函數(shù)是非奇非偶函數(shù).對數(shù)函數(shù)的運(yùn)算性質(zhì)(b)lOgaU vlOgaUv(d)lOg(b)lOgaU vlOgaUv(d)lOgaXlOgaUlOgaV；VlOgaU；lnx.lna(四)三角函

4、數(shù)ysinx,ycosx:ysinx正弦函數(shù)；ycosx余弦函數(shù).(b)ysinx,ycosx均為非單調(diào)函數(shù)；ysinx,ycosx均為有界函數(shù)；ysinx為奇函數(shù)，ycosx為偶函數(shù);(e)ysinx,ycosx均為周期函數(shù).ytanx,ycotx:ytanx正切函數(shù)；ycotx余切函數(shù).(a)ytanx定義域為1R(k2),ycotx定乂域為Rk,值域均為R；ytanx,ycotx均為非單調(diào)函數(shù);ytanx,ycotx均為無界函數(shù)；ytanx,ycotx均為奇函數(shù)；(e)ytanx,ycotx均為周期函數(shù).ysecx,ycscx：1ysecx正割函數(shù)；cosx1ycscx余割函數(shù).sinx

5、(五)反三角函數(shù)yarcsinx,yarccosx:yarcsinx反正弦函數(shù)；yarccosx反余弦函數(shù).yarcsinx,yarccosx的圖形與性質(zhì):圖1-7(a)yarcsinx,yarccosx定義域均為yarcsinx的值域為/2,/2,yarccosx的值域為0,;yarcsinx,yarccosx均為單調(diào)函數(shù);yarcsinx,yarccosx均為有界函數(shù);yarcsinx為奇函數(shù),yarccosx為非奇非偶函數(shù).2.yarctanx,yarccotx：yarctanx反正切函數(shù)；yarccotx反余切函數(shù).yarctanx,yarccotx的圖形與性質(zhì):圖1-8(a)yarc

6、tanx,yarccotx的定義域均為R,yarctanx的值域為(/2,/2),yarccotx的值域為(0,);(b)yarctanx,yarccotx均為單調(diào)函數(shù)；(c)yarctanx,yarccotx均為有界函數(shù);(d)yarctanx為奇函數(shù),yarccotx為非奇非偶函數(shù).四、復(fù)合函數(shù)設(shè)ylnu,utanx,則yln(tanx).定義:設(shè)有函數(shù)yf(u),uDiu(x),xD2,則且(D2)Di函數(shù)yf(x)稱為由yf(u)及u(x)復(fù)合而成的復(fù)合函數(shù)，其中u稱為中間變量.寫出下列復(fù)合函數(shù)的復(fù)合過程，并求其定義域.22(1)yarctan(x);(2)ysin(x);(3)y(sinx)2.五、初等函數(shù)1.定義：由常數(shù)及基本初等函數(shù)經(jīng)過有限次四則運(yùn)算及有限次的復(fù)合步驟所構(gòu)成并且可以用一個式子表示的函數(shù)叫做初等