大學(xué)高等數(shù)學(xué)知識(shí)點(diǎn)【掌握這些高數(shù)就不怕了】

1、大學(xué)高等數(shù)學(xué)知識(shí)點(diǎn)整理公式，用法合集極限與連續(xù).數(shù)列函數(shù):種類(lèi):(1)數(shù)列:*anf(n);*an1f(an)初等函數(shù):(3)分段函數(shù):*F(x)f1(x)xx0f(x)xx0,;*F(x),;*f2(x)xx0axx0(4)復(fù)合(含f)函數(shù):yf(u),u(x)隱式(方程):F(x,y)0 x(t)參式(數(shù)一,二):y(t)變限積分函數(shù):F(x)x(7)f(x,t)dta(8)級(jí)數(shù)和函數(shù)(數(shù)一,三):S(x)anxn,xn0特點(diǎn)(幾何):(1)單一性與有界性(鑒別);(f(x)單一x0,(xx0)(f(x)f(x0)定號(hào))奇偶性與周期性(應(yīng)用).3.反函數(shù)與直接函數(shù):yf(x)xf1(y)y

2、f1(x)二.極限性質(zhì):1.種類(lèi):*liman;*limf(x)(含x);*limf(x)(含xx0)nxxx0無(wú)窮小與無(wú)窮大(注:無(wú)窮量):3.未定型:0,1,0,00,00性質(zhì):*有界性,*保號(hào)性,*合并性.常用結(jié)論:111annn1,an(a0)1,(anbncn)nmax(a,b,c),a00n!11(xxxlimxnlimlnnx0),lim1,x0,0,xx0 xexxlimxlnnx,0ex0,xx0 x四.必備公式:1.等價(jià)無(wú)窮小:當(dāng)u(x)0時(shí),sinux()ux(;)tanu(x)u(x);1cosu(x)1u2(x);2eu(x)1u(x);ln(1u(x)u(x);(1

3、u(x)1u(x);arcsiunx()ux;(arctanu(x)u(x)泰勒公式:(1)ex1x1x2o(x2);2!1x2(2)ln(1x)xo(x2);2(3)sinxx1x3o(x4);3!(4)cosx11x21x4o(x5);2!4!(5)(1x)1x(1)x2o(x2).2!五.常例方法:前提:(1)正確判斷0,1,M(其余如:,0,00,0);(2)變量代換(如:1t)0 x抓大棄小(),2.無(wú)窮小與有界量乘積(M)(注:sin11,x)x1辦理(其余如:00,0)4.左右極限(包括x):11(1)(x0);(2)ex(x);ex(x0);(3)分段函數(shù):x,x,maxf(x

4、)x無(wú)窮小等價(jià)替換(因式中的無(wú)窮小)(注:非零因子)洛必達(dá)法例先”辦理”,后法例(0最后方法);(注意比較:limxlnx與limxlnx)0 x11xx01x211111(2)冪指型辦理:u(x)v(x)ev(x)lnu(x)(如:ex1exex(ex1x1)含變限積分;不能用與不便用泰勒公式(皮亞諾余項(xiàng)):辦理和式中的無(wú)窮小8.極限函數(shù):f(x)limF(x,n)(分段函數(shù))n六.特別手段1.收斂準(zhǔn)則:(1)anf(n)limf(x)x(2)雙邊夾:*bnancn?,*bn,cna?(3)單邊擠:an1f(an)*a2a1?2.導(dǎo)數(shù)定義(洛必達(dá)?):limf(x0)0fxx3.積分和:li

5、m1f(1)f(2)f(n)nnnnn4.中值定理:limf(xa)f(x)alimxx*anM?*f(x)0?1f(x)dx,0f()級(jí)數(shù)和(數(shù)一三):(1)an收斂liman0,(如lim2nn!)(2)lim(a1a2an)an,n1nnnnn1n(3)an與(anan1)同斂散n1七.常有應(yīng)用:無(wú)窮小比較(等價(jià),階):*f(x)(1)f(0)f(0)f(n1)(0)xx(2)f(t)dtktndt00漸近線(xiàn)(含斜):(1)alimf(x),blimf(x)xxx1(2)f(x)axb0),(xkxn,(x0)?0,f(n)(0)af(x)axn(xn)axnn!n!axf(x)axb3

6、.連續(xù)性:(1)中止點(diǎn)鑒別(個(gè)數(shù));(2)分段函數(shù)連續(xù)性(附:極限函數(shù),f(x)連續(xù)性)八.a,b上連續(xù)函數(shù)性質(zhì)31.連通性:f(a,b)m,M(注:0,“平均”值:f(a)(1)f(b)f(x0)1介值定理:(附:達(dá)布定理)(1)零點(diǎn)存在定理:f(a)f(b)0f(x0)0(根的個(gè)數(shù));(2)f(x)0(x0.f(x)dx)a第二講:導(dǎo)數(shù)及應(yīng)用(一元)(含中值定理)一.基本觀點(diǎn):1.差商與導(dǎo)數(shù):f(x)limf(xx)f(x);f(x0)limf(x)f(x0)x0 xxx0 xx0(1)f(0)limf(x)f(0)(注:limf(x)A(f連續(xù))f(0)0,f(0)A)x0 xx0 x左

7、右導(dǎo):f(x0),f(x0);(3)可導(dǎo)與連續(xù);(在x0處,x連續(xù)不可導(dǎo);xx可導(dǎo))2.微分與導(dǎo)數(shù):ff(xx)f(x)f(x)xo(x)dff(x)dx(1)可微可導(dǎo);(2)比較f,df與0的大小比較(圖示);.求導(dǎo)準(zhǔn)備:基本初等函數(shù)求導(dǎo)公式;(注:(f(x)2.法例:(1)四則運(yùn)算;(2)復(fù)合法例;dx1(3)反函數(shù)ydy三.各類(lèi)求導(dǎo)(方法步驟):1.定義導(dǎo):(1)f(a)與f(x)xa;(2)分段函數(shù)左右導(dǎo);(3)limf(xh)f(xh)h0h(注:f(x)F(x)xx0,求:f(x0),f(x)及f(x)的連續(xù)性),xx0初等導(dǎo)(公式加法例):ufg(x),求:u(x0)(圖形題);

8、F(x)x(注:(xbb(2)f(t)dt,求:F(x)af(x,t)dt),(f(x,t)dt),(f(t)dt)aaaf1(x)xx0及(3)y,f(x0),f(x0)f(x0)(待定系數(shù))f2(x)求xx043.隱式(f(x,y)0)導(dǎo):dy,d2ydxdx2存在定理;微分法(一階微分的形式不變性).對(duì)數(shù)求導(dǎo)法.xx(t)24.dy,dy參式導(dǎo)(數(shù)一,二):,求:yy(t)dxdx25.高階導(dǎo)f(n)(x)公式:ax(n)nax(a1(n)bnn!(e)ae;bx)(abx)n1;(sinax)(n)ansin(axn);(cosax)(n)ancos(axn)22(uv)(n)un(v

9、)Cnu1n(v1)Cnu2nv(2)注:f(n)(0)與泰勒展式:f(x)a0a1xa2x2anxnanf(n)(0)n!四.各類(lèi)應(yīng)用:1.斜率與切線(xiàn)(法線(xiàn));(區(qū)別:yf(x)上點(diǎn)M0和過(guò)點(diǎn)M0的切線(xiàn))2.物理:(相對(duì))變化率速度;3.曲率(數(shù)一二):f(x)(曲率半徑,曲率中心,曲率圓)(1f2(x)3邊際與彈性(數(shù)三):(附:需求,利潤(rùn),成本,利潤(rùn)).單一性與極值(必求導(dǎo))鑒別(駐點(diǎn)f(x0)0):(1)f(x)0f(x);f(x)0f(x);分段函數(shù)的單一性(3)f(x)0零點(diǎn)唯一;f(x)0駐點(diǎn)唯一(必為極值,最值).極值點(diǎn):(1)表格(f(x)變號(hào));(由limf(x)0,lim

10、f(x)f(x)0,lim20 x0的特點(diǎn))xx0 xxx0 xxx0 x(2)二階導(dǎo)(f(x0)0)注(1)f與f,f的匹配(f圖形中包含的信息);5(2)實(shí)例:由f(x)(x)f(x)g(x)確定點(diǎn)“xx0”的特點(diǎn).(3)閉域上最值(應(yīng)用例:與定積分幾何應(yīng)用相聯(lián)合,求最優(yōu))不等式證明(f(x)0)(1)區(qū)別:*單變量與雙變量?*xa,b與xa,),x(,)?(2)種類(lèi):*f0,f(a)0;*f0,f(b)0*f0,f(a),f(b)0;*f(x)0,f(x0)0,f(x0)0(3)注意:單一性端點(diǎn)值極值凹凸性.(如:f(x)Mfmax(x)M)函數(shù)的零點(diǎn)個(gè)數(shù):單一介值.凹凸與拐點(diǎn)(必求導(dǎo)!

11、):y表格;(f(x0)0)2.應(yīng)用:(1)泰勒估計(jì);(2)f單一;(3)凹凸.七.羅爾定理與協(xié)助函數(shù):(注:最值點(diǎn)必為駐點(diǎn))1.結(jié)論:F(b)F(a)F()f()02.協(xié)助函數(shù)結(jié)構(gòu)實(shí)例:(1)f()F(x)xf(t)dta(2)f()g()f()g()0F(x)f(x)g(x)(3)f()g()f()g()0F(x)f(x)g(x)(4)f()()f()0F(x)e(x)dxf(x);3.f(n)()0f(x)有n1個(gè)零點(diǎn)f(n1)(x)有2個(gè)零點(diǎn)4.特例:證明f(n)()a的常例方法令f(x)Pn(x)有n1個(gè)零點(diǎn)(Pn(x)待定):F(x)注:含1,2時(shí),分家!(柯西定理)6.附(達(dá)布定

12、理):f(x)在a,b可導(dǎo),cf(a),f(b),a,b,使:f()c八.拉格朗日中值定理1.結(jié)論:f(b)f(a)f()(ba);(a)(b),()0)62.估計(jì):ff()x九.泰勒公式(連結(jié)f,f,f之間的橋梁)1.結(jié)論:f(x)f(x0)f(x0)(xx0)1f(x0)(xx0)21f()(xx0)3;2!3!應(yīng)用:在已知f(a)或f(b)值時(shí)進(jìn)行積分估計(jì).積分中值定理(附:廣義):注:有定積分(不含變限)條件時(shí)使用第三講:一元積分學(xué).基本觀點(diǎn):原函數(shù)F(x):(1)F(x)f(x);(2)f(x)dxdF(x);(3)f(x)dxF(x)cx(1)F(x)f(t)dt(連續(xù)不一定可導(dǎo))

13、;axx(2)(xt)f(t)dtf(t)dtf(x)(f(x)連續(xù))aa2.不定積分性質(zhì):(1)(f(x)dx)f(x);d(f(x)dx)f(x)dx(2)f(x)dxf(x)c;df(x)f(x)c.不定積分常例方法熟悉基本積分公式基本方法:拆(線(xiàn)性性)(k1f(x)kg(x)dx1k(f)xdxk(g)xdx22湊微法(基礎(chǔ)):要求巧,簡(jiǎn),活(1sin2xcos2x)如:dx1d(axb),xdx1dx2,dxdlnx,dx2dxa2xxxdxd1x2,(1lnx)dxd(xlnx)x2變量代換:(1)常用(三角代換,根式代換,倒代換):xsint,axbt,1t,ex1tx(2)作用

14、與引伸(化簡(jiǎn)):x21xt7分部積分(巧用):(1)x含需求導(dǎo)的被積函數(shù)(如lnx,arctanx,f(t)dt);a(2)“反對(duì)冪三指”:xneaxdx,xnlnxdx,(3)特別:xf(x)dx(*已知f(x)的原函數(shù)為F(x);*已知f(x)F(x)6.特例:(1)a1sinxb1cosxdx;(2)p(x)ekxdx,p(x)sinaxdx迅速法;(3)v(x)dxasinxbcosxun(x).定積分:觀點(diǎn)性質(zhì):積分和式(可積的必要條件:有界,充分條件:連續(xù))幾何意義(面積,對(duì)稱(chēng)性,周期性,積分中值)*ax2dx(a0)a2;bax*(x08abM(ba),bb(3)附:f(x)dx

15、f(x)g(x)dxMaaa(4)定積分與變限積分,反常積分的區(qū)別聯(lián)系與重視b)dx02g(x)dx)2:變限積分(x)xf(t)dt的辦理(重點(diǎn))a(1)f可積連續(xù),f連續(xù)可導(dǎo)(2)xf(x);xxx(f(t)dt)(xt)f(t)dt)f(t)dt;f(x)dt(xa)f(x)aaaax由函數(shù)F(x)f(t)dt參與的求導(dǎo),極限,極值,積分(方程)問(wèn)題aNL公式:bF(b)F(a)(F(x)在a,b上必須連續(xù)!)3.f(x)dxa注:(1)分段積分,對(duì)稱(chēng)性(奇偶),周期性(2)有理式,三角式,根式b(3)含f(t)dt的方程.abf(u(t)u(t)dt4.變量代換:f(x)dxaaf(x

16、)dxax)dx(xat),(1)f(a00aaa41(2)f(x)dxaf(x)dx(xt)f(x)f(x)dx(如:dx)a041sinx(3)In2sinnxdxn1In2,0n8(4)2f(sinx)dx2f(cosx)dx;f(sinx)dx22f(sinx)dx,0000(5)xf(sinx)dx2f(sinx)dx,00分部積分準(zhǔn)備時(shí)“湊常數(shù)”(2)已知f(x)或f(x)xb時(shí),求f(x)dxaa6.附:三角函數(shù)系的正交性:222sinnxcosmxdx0sinnxdxcosnxdx00022m)0sinnxsinmxdxcosnxcosmxdx(n0022nxdx2sincos

17、2nxdx00四.反常積分:1.種類(lèi):(1)f(x)dx,af(x)dx(f(x)連續(xù))f(x)dx,ab(f(x)在xa,xb,xc(acb)處為無(wú)窮中止)(2)f(x)dx:a斂散;3.計(jì)算:積分法NL公式極限(可換元與分部)4.特例:(1)1(2)11xpdx;xpdx10.應(yīng)用:(柱體側(cè)面積除外)面積,(1)Sbf(x)g(x)dx;(2)Sd1(y)dy;afcS1r2()d;(4)側(cè)面積:Sb(3)2f(x)1f2(x)dx2a體積:(1)Vxb2(x)g2(x)dx;(2)Vyd1(y)2dy2bffxf(x)dxacaVxx0與Vyy03.弧長(zhǎng):ds(dx)2(dy)2(1)y

18、f(x),xa,bb12(x)dxsfa(2)xx(t)tt1,t2t2x2(t)y2(t)dt,syy(t)t19(3)rr(),:sr2()r2()d物理(數(shù)一,二)功,引力,水壓力,質(zhì)心,平均值(中值定理):(1)fa,b1bbaf(x)dx;axTf(t)dtf(t)dt(2)f0)lim00,(f以T為周期:f)xxT第四講:微分方程.基本觀點(diǎn)知識(shí):通解,初值問(wèn)題與特解(注:應(yīng)用題中的隱含條件)變換方程:令xx(t)yDy(如歐拉方程)(2)令uu(x,y)yy(x,u)y(如伯努利方程)成立方程(應(yīng)用題)的能力.一階方程:1.形式:(1)yf(x,y);(2)M(x,y)dxN(x

19、,y)dy0;(3)y(a)b變量分別型:yf(x)g(y)(1)解法:dyf(x)dxG(y)F(x)Cg(y)“偏”微分方程:一階線(xiàn)性(重點(diǎn)):zf(x,y);xyp(x)yq(x)xp(x)dx1x(1)解法(積分因子法):M(x)ex0yM(x)q(x)dxy0M(x)x0(2)變化:xp(y)xq(y);(3)推廣:伯努利(數(shù)一)yp(x)yq(x)y4.齊次方程:y(y)x(1)解法:uyxududxu(u),xx(u)u10dya1xb1yc1(2)特例:a2xb2yc2dx5.全微分方程(數(shù)一):M(x,y)dxN(x,y)dy0且NMxydUMdxNdyUC6.一階差分方程(

20、數(shù)三):yx1ayx0yxcaxbxp(x)yx*xnQ(x)bx.二階降階方程yf(x):yF(x)c1xc22.yf(x,y):令yp(x)ydpf(x,p)dx3.yf(y,y):令yp(y)ypdpf(y,p)dy四.高階線(xiàn)性方程:a(x)yb(x)yc(x)yf(x)通解結(jié)構(gòu):(1)齊次解:y0(x)c1y1(x)c2y2(x)(2)非齊次特解:y(x)c1y1(x)c2y2(x)y*(x)常系數(shù)方程:aybycyf(x)特點(diǎn)方程與特點(diǎn)根:a2bc0(2)非齊次特解形式確定:待定系數(shù);(附:f(x)keax的算子法)由已知解反求方程.3.歐拉方程(數(shù)一):ax2ybxycyf(x),

21、令xetx2yD(D1)y,xyDy.應(yīng)用(注意初始條件):幾何應(yīng)用(斜率,弧長(zhǎng),曲率,面積,體積);:切線(xiàn)和法線(xiàn)的截距積分等式變方程(含變限積分);x可設(shè)f(x)dxF(x),F(a)0a導(dǎo)數(shù)定義立方程:含雙變量條件f(xy)的方程114.變化率(速度)5.dvd2xFmadt2dtQP6.路徑無(wú)關(guān)得方程(數(shù)一):xy級(jí)數(shù)與方程:(1)冪級(jí)數(shù)求和;(2)方程的冪級(jí)數(shù)解法:ya0a1xa2x2,a0y(0),a1y(0)彈性問(wèn)題(數(shù)三)第五講:多元微分與二重積分.二元微分學(xué)觀點(diǎn)極限,連續(xù),單變量連續(xù),偏導(dǎo),全微分,偏導(dǎo)連續(xù)(必要條件與充分條件),(1)ff(x0 x,y0y),xff(x0 x

22、,y0),yff(x0,y0y)(2)limf,fxlimxf,fylimyfxy(3)fxxfyydf,limfdf(鑒別可微性)x)2(y)2(注:(0,0)點(diǎn)處的偏導(dǎo)數(shù)與全微分的極限定義:fx(0,0)limf(x,0)f(0,0),fy(0,0)limf(0,y)f(0,0)x0 xy0y特例:xy(0,0)(1)f(x,y)x2y2(0,0)點(diǎn)處可導(dǎo)不連續(xù);:0,(0,0)xy(0,0)f(x,y)x2y2(0,0)點(diǎn)處連續(xù)可導(dǎo)不可微;(2):0,(0,0)二.偏導(dǎo)數(shù)與全微分的計(jì)算:1.顯函數(shù)一,二階偏導(dǎo):zf(x,y)注:(1)xy型;(2)zx(x,y);(3)含變限積分002.

23、復(fù)合函數(shù)的一,二階偏導(dǎo)(重點(diǎn)):zfu(x,y),v(x,y)12嫻熟掌握記號(hào)f1,f2,f11,f12,f22的正確使用隱函數(shù)(由方程或方程組確定):(1)形式:*F(x,y,z)0;*F(x,y,z)0G(x,y,z)0(存在定理)微分法(嫻熟掌握一階微分的形式不變性):FxdxFydyFzdz0(要求:二階導(dǎo))注:(x0,y0)與z0的實(shí)時(shí)代入會(huì)變換方程.二元極值(定義?);二元極值(顯式或隱式):必要條件(駐點(diǎn));充分條件(鑒別)條件極值(拉格朗日乘數(shù)法)(注:應(yīng)用)(1)目標(biāo)函數(shù)與拘束條件:zf(x,y)(x,y)0,(或:多條件)(2)求解步驟:L(x,y,)f(x,y)(x,y)

24、,求駐點(diǎn)即可.有界閉域上最值(重點(diǎn)).(1)zf(x,y)MD(x,y)(x,y)0實(shí)例:距離問(wèn)題.二重積分計(jì)算:觀點(diǎn)與性質(zhì)(“積”前工作):d,D(2)對(duì)稱(chēng)性(嫻熟掌握):*D域軸對(duì)稱(chēng);*f奇偶對(duì)稱(chēng);*字母輪換對(duì)稱(chēng);*重心坐標(biāo);(3)“分塊”積分:*DD1D2;*f(x,y)分片定義;*f(x,y)奇偶計(jì)算(化二次積分):直角坐標(biāo)與極坐標(biāo)選擇(變換):以“D”為主;交換積分序次(嫻熟掌握).3.極坐標(biāo)使用(變換):f(x2y2)附:D:(xa)2(yb)2R2;D:x2y2221;ab雙紐線(xiàn)(x2y2)2a2(x2y2)D:xy14.特例:13(1)單變量:f(x)或f(y)(2)利用重心求

25、積分:要求:題型(k1xk2y)dxdy,且已知D的面積SD與重心(x,y)D無(wú)界域上的反常二重積分(數(shù)三)五:一類(lèi)積分的應(yīng)用(f(M)d:D;L;):1.“尺寸”:(1)dSD;(2)曲面面積(除柱體側(cè)面);D質(zhì)量,重心(形心),轉(zhuǎn)動(dòng)慣量;為三重積分,格林公式,曲面投影作準(zhǔn)備.第六講:無(wú)窮級(jí)數(shù)(數(shù)一,三)一.級(jí)數(shù)觀點(diǎn)1.定義:(1)an,(2)Sna1a2an;(3)limSn(如n)nn1(n1)!注:(1)liman;(2)qn(或1(3)“伸縮”級(jí)數(shù):(an1an)收斂an收斂.an);n2.性質(zhì):(1)收斂的必要條件:liman0;n加括號(hào)后發(fā)散,則原級(jí)數(shù)必發(fā)散(交錯(cuò)級(jí)數(shù)的議論);(

26、3)s2ns,an0s2n1ssns;二.正項(xiàng)級(jí)數(shù)1.正項(xiàng)級(jí)數(shù):(1)定義:an0;(2)特點(diǎn):Sn;(3)收斂SnM(有界)2.標(biāo)準(zhǔn)級(jí)數(shù):(1)1,(2)lnkn1npn,(3)knnln3.審斂方法:(注:2aba2b2,alnbblna)k1P(n)(1)比較法(原理):an(估計(jì)),nf(x)dx;np如0Q(n)(2)比值與根值:*limun1*limnun(應(yīng)用:冪級(jí)數(shù)收斂半徑計(jì)算)unnn三.交錯(cuò)級(jí)數(shù)(含一般項(xiàng)):(1)n1an(an0)1.“審”前考察:(1)an0?(2)an0?;(3)絕對(duì)(條件)收斂?14注:若liman11,則un發(fā)散nan2.標(biāo)準(zhǔn)級(jí)數(shù):(1)n11n1

27、1n11(1);(2)(1)np;(3)(1)pnnln萊布尼茲審斂法(收斂?)(1)前提:an發(fā)散;(2)條件:an,an0;(3)結(jié)論:(1)n1an條件收斂.補(bǔ)充方法:(1)加括號(hào)后發(fā)散,則原級(jí)數(shù)必發(fā)散;(2)s2ns,an0s2n1ssns.5.注意事項(xiàng):比較a;(1)na;a;a2之間的斂散關(guān)系nnnn四.冪級(jí)數(shù):1.常有形式:(1)anxn,(2)an(xx0)n,(3)an(xx0)2n阿貝爾定理:(1)結(jié)論:xx*斂Rx*x0;xx*散Rx*x0(2)注:當(dāng)xx*條件收斂時(shí)Rxx*收斂半徑,區(qū)間,收斂域(求和前的準(zhǔn)備)注(1)nanxn,anxn與anxn同收斂半徑n(2)an

28、xn與an(xx0)2n之間的變換冪級(jí)數(shù)展開(kāi)法:前提:熟記公式(雙向,注明斂域)ex1x1x21x3,R2!3!1(exex)11x21x4,R22!4!1(exex)x1x31x5,R23!5!sinxx1x31x5,Rcosx11x21x4,R;13!5!12!4!x1xx2,x(1,;1)1xx2,x(1,1)11xln(1x)x1x21x3,x(1,123ln(1x)x1x21x3,x1,1)2315arctanxx1x31x5,x1,1351(2)分解:f(x)g(x)h(x)(注:中心移動(dòng))(特別:,xx0)2bxaxc(3)考察導(dǎo)函數(shù):g(x)f(x)f(x)xg(x)dxf(0

29、)0考察原函數(shù):g(x)xf(x)g(x)(4)f(x)dx0冪級(jí)數(shù)求和法(注:*先求收斂域,*變量替換):(1)S(x),(2)S(x),(注意首項(xiàng)變化)(3)S(x)(),S(x)S(x)的微分方程(5)應(yīng)用:ananxnS(x)anS(1).方程的冪級(jí)數(shù)解法經(jīng)濟(jì)應(yīng)用(數(shù)三):復(fù)利:A(1p)n;(2)現(xiàn)值:A(1p)n.傅里葉級(jí)數(shù)(數(shù)一):(T2)1.a0ancosnxbnsinnx傅氏級(jí)數(shù)(三角級(jí)數(shù)):S(x)2n12.Dirichlet充分條件(收斂定理):(1)由f(x)S(x)(和函數(shù))(2)S(x)1f(x)f(x)2an11f(x)cosnxdx3.系數(shù)公式:f(x)dx,n

30、1,2,3,a0bn1f(x)sinnxdx4.題型:(注:f(x)S(x),x?)(1)T2且f(x),x(,(分段表示)16(2)x(,或x0,2x0,正弦或余弦*(4)x0,(T)*5.T2l6.附產(chǎn)品:f(x)S(x)a0ancosnxbnsinnx2n1S(x0)a0ancosnx0bnsinnx01f(x0)f(x0)2n12第七講:向量,偏導(dǎo)應(yīng)用與方向?qū)?數(shù)一)一.向量基本運(yùn)算1.k1ak2b;(平行ba)2.a;(單位向量(方向余弦)a01a(cos,cos,c)os)a3.ab;ab;垂直:abab0;夾角:(a,b)ab(投影:(b)a)aab4.ab;(法向:naba,b

31、;面積:Sab)二.平面與直線(xiàn)1.平面(1)特點(diǎn)(基本量):M0(x0,y0,z0)n(A,B,C)(2)方程(點(diǎn)法式):A(xx0)B(yy0)C(zz0)0AxByCzD0(3)其余:*截距式xyz1;*三點(diǎn)式abc2.直線(xiàn)L(1)特點(diǎn)(基本量):M0(x0,y0,z0)s(m,n,p)(2)方程(點(diǎn)向式):L:xx0yy0zz0mnp(3)一般方程(交面式):A1xB1yC1zD10A2xB2yC2zD2017xa1(a2a1)t(4)其余:*二點(diǎn)式;*參數(shù)式;(附:線(xiàn)段AB的參數(shù)表示:yb1(b2b1)t,t0,1)zc1(c2c1)t實(shí)用方法:(1)平面束方程:A1xB1yC1zD1

32、(A2xB2yC2zD2)0(2)距離公式:如點(diǎn)M0(x0,y0)到平面的距離Ax0By0Cz0DdB2C2A2對(duì)稱(chēng)問(wèn)題;投影問(wèn)題.曲面與空間曲線(xiàn)(準(zhǔn)備)曲面(1)形式:F(x,y,z)0或zf(x,y);(注:柱面f(x,y)0)(2)法向n(Fx,Fy,Fz)(cos,cos,cos)(或n(zx,zy1)曲線(xiàn)xx(t)0(1):yF(x,y,z)形式y(tǒng)(t),或;zG(x,y,z)0z(t)(2)切向:sx(t),y(t),z(t)(或sn1n2)應(yīng)用交線(xiàn),投影柱面與投影曲線(xiàn);旋轉(zhuǎn)面計(jì)算:參式曲線(xiàn)繞坐標(biāo)軸旋轉(zhuǎn);錐面計(jì)算.常用二次曲面圓柱面:x2y2R22.球面:x2y2z2R2變形:x2

33、y2R2z2,zR2(x2y2),x2y2z22az,(xx0)2(yy0)2(zz0)2R2183.錐面:zx2y2變形:x2y2z2,zax2y2拋物面:zx2y2,變形:x2y2z,za(x2y2)5.雙曲面:x2y2z216.馬鞍面:zx2y2,或zxy.偏導(dǎo)幾何應(yīng)用曲面(1)法向:F(x,y,z)0n(Fx,Fy,Fz),注:zf(x,y)n(fx,fy1)切平面與法線(xiàn):曲線(xiàn)(1)切向:xx(t),yy(t),zz(t)s(x,y,z)切線(xiàn)與法平面F0sn1n23.綜合:,G0六.方向?qū)c梯度(重點(diǎn))1.方向?qū)?l方向斜率):(1)定義(條件):l(m,n,p)(cos,cos,co

34、s)(2)計(jì)算(充分條件:可微):uuxcosuycosuzcosl附:zf(x,y),l0cos,sinzfxcosfysinl(3)附:2ffxxcos22fxysincosfyysin2l22.梯度(取得最大斜率值的方向)G:19計(jì)算:(a)zf(xy,)Ggradzfx(fy,;(b)uf(x,y,z)Ggradxu(y,uz,uu結(jié)論(a)uGl0;l(b)取lG為最大變化率方向;(c)G(M0)為最大方向?qū)?shù)值.第八講:三重積分與線(xiàn)面積分(數(shù)一).三重積分(fdV)域的特點(diǎn)(不波及復(fù)雜空間域):對(duì)稱(chēng)性(重點(diǎn)):含:對(duì)于坐標(biāo)面;對(duì)于變量;對(duì)于重心(2)投影法:Dxy(x,y)x2y2

35、R2z1(x,y)zz2(x,y)(3)截面法:D(z)(x,y)x2y2R2(z)azb其余:長(zhǎng)方體,四面體,橢球f的特點(diǎn):(1)單變量f(z),(2)f(x2y2),(3)f(x2y2z2),(4)faxbyczd選擇最適合方法:“積”前:*dv;*利用對(duì)稱(chēng)性(重點(diǎn))截面法(旋轉(zhuǎn)體):投影法(直柱體):Ibfdxdy(細(xì)腰或中空,f(z),f(x2y2)dzaD(z)Idxdyz2(x,y)fdzDxyz1(x,y)(4)2sindR)2d,球坐標(biāo)(球或錐體):Idf(000(5)重心法(faxbyczd):I(axbyczd)V應(yīng)用問(wèn)題:同第一類(lèi)積分:質(zhì)量,質(zhì)心,轉(zhuǎn)動(dòng)慣量,引力Gauss公式20二.第一類(lèi)線(xiàn)積分(fds)L“積”前準(zhǔn)備:(1)dsL;(2)對(duì)稱(chēng)性;(3)代入“L”表達(dá)式L