高數(shù)知識(shí)點(diǎn)總結(jié)

高數(shù)知識(shí)點(diǎn)總結(jié)高數(shù)知識(shí)點(diǎn)總結(jié)篇一:高等數(shù)學(xué)知識(shí)點(diǎn)歸納第一講:一.數(shù)列函數(shù):1.類型:極限與連續(xù)(1)數(shù)列:*an?f(n);*an?1?f(an)(2)初等函數(shù):(3)分段函數(shù):*F(x)???f1(x)x?x0?f(x)x?x0;*F(x)??;*,,?ax?x0?f2(x)x?x0(4)復(fù)合(含f)函數(shù):y?f(u),u??(x)(5)隱式(方程):F(x,y)?0(6)參式(數(shù)一,二):??x?x(t)?y?y(t)(7)變限積分函數(shù):F(x)??xaf(x,t)dt(8)級(jí)數(shù)和函數(shù)(數(shù)一,三):S(x)?2.特征(幾何):?ax,x??nnn?0?(1)單調(diào)性與有界性(判別);(f(x)單調(diào)??x0,(x?x0)(f(x)?f(x0))定號(hào))(2)奇偶性與周期性(應(yīng)用).3.反函數(shù)與直接函數(shù):y?f(x)?x?f二.極限性質(zhì):1.類型:*liman;*limf(x)(??含x???);*limf(x)(含x?x0?)n??x???1(y)?y?f?1(x)x?x02.無(wú)窮小與無(wú)窮大(注:無(wú)窮量):3.未定型:0??,,1,?????,0??,00,?00?4.性質(zhì):*有界性,*保號(hào)性,*歸并性三.常用結(jié)論:ann?1,a(a?0)?1,(a?b?c?maxa(b,,c,)?a?0??0n!nn??1n1n1nn1xnlnnxxx?1,lix?0?0,(x?0)??,lim,lim?x???x???x?0xexxxlnx?0lim,e??x?0?n?0x???,???x???四.必備公式:1.等價(jià)無(wú)窮小:當(dāng)u(x)?0時(shí),ux(?)ux(;)tanu(x)?u(x);1?csu(x)?sin12u(x);2eu(x)?1?u(x);ln(1?u(x))?u(x);(1?u(x))??1??u(x);unx(?)ux;(arctanu(x)?u(x)arcsi2.泰勒公式:12x?(x2);2!122(2)ln(1?x)?x?x?(x);2134(3)sinx???x?x?(x);3!12145(4)csx?1?x?x?(x);2!4!?(??1)2?x?(x2).(5)(1?x)?1??x?2!(1)e?1?x?x五.常規(guī)方法:前提:(1)準(zhǔn)確判斷,1.抓大棄小(0??1,1,?M(其它如:???,0??,00,?0);(2)變量代換(如:?t)0?x?),?2.無(wú)窮小與有界量乘積(??M)(注:sin?1?1,x??)x3.1處理(其它如:0,?)4.左右極限(包括x???):11x(1)(x?0);(2)e(x??);ex(x?0);(3)分段函數(shù):x,[x],maxf(x)x005.無(wú)窮小等價(jià)替換(因式中的無(wú)窮小)(注:非零因子)6.洛必達(dá)法則(1)先”處理”,后法則(0xlnxxlnx最后方法);(注意對(duì)比:lim與lim)x?11?xx?01?x0v(x)(2)冪指型處理:u(x)(3)含變限積分;?ev(x)lnu(x)(如:e1x?1?e?e(e1x1x11?x?1x?1))(4)不能用與不便用7.泰勒公式(皮亞諾余項(xiàng)):處理和式中的無(wú)窮小8.極限函數(shù):f(x)?limF(x,n)(?分段函數(shù))n??六.非常手段1.收斂準(zhǔn)則:(1)an?f(n)?limf(x)x???(2)雙邊夾:*bn?an?cn?,*bn,cn?a?(3)單邊擠:an?1?f(an)*a2?a1?*an?M?*f(x)?0??f?fx0()?x?0?x1112nlif[?)f(??)?f(?)]fxd(3.積分和:,x)?0n??nnnn2.導(dǎo)數(shù)定義(洛必達(dá)?):li4.中值定理:lim[f(x?a)?f(x)]?alimf(?)x???x???5.級(jí)數(shù)和(數(shù)一三):?2nn!(1)?an收斂?liman?0,(如limn)(2)lim(a1?a2???an)??an,n??n??nn??n?1n?1??(3){an}與?(an?1n?an?1)同斂散七.常見(jiàn)應(yīng)用:1.無(wú)窮小比較(等價(jià),階):*f(x)?kx,(x?0)?(1)f(0)?f(0)???f(2)(n?1)n(0)?0,??f(n)(0)?a?f(x)?anax??(xn)?xnn!n!?xf(t)dt??ktndtx2.漸近線(含斜):f(x),b?lim[f(x)?ax]?f(x)?ax?b??x??x??x1(2)f(x)?ax?b??,(?0)x(1)a?lim3.連續(xù)性:(1)間斷點(diǎn)判別(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)1.連通性:f([a,b])?[m,M](注:?0???1,“平均”值:?f(a)?(1??)f(b)?f(x0))2.介值定理:(附:達(dá)布定理)(1)零點(diǎn)存在定理:f(a)f(b)?0?f(x0)?0(根的個(gè)數(shù));(2)f(x)?0?(?xaf(x)dx)?0.第二講:導(dǎo)數(shù)及應(yīng)用(一元)(含中值定理)一.基本概念:1.差商與導(dǎo)數(shù):f(x)?lim?x?0f(x??x)?f(x)f(x)?f(x0);f(x0)?limx?x0?xx?x0(1)f(0)?limx?0f(x)?f(0)f(x)?A(f連續(xù))?f(0)?0,f(0)?A)(注:limx?0xx(2)左右導(dǎo):f?(x0),f?(x0);(3)可導(dǎo)與連續(xù);(在x?0處,x連續(xù)不可導(dǎo);xx可導(dǎo))2.微分與導(dǎo)數(shù):?f?f(x??x)?f(x)?f(x)?x?(?x)?df?f(x)dx(1)可微?可導(dǎo);(2)比較?f,df與0的大小比較(圖示);二.求導(dǎo)準(zhǔn)備:1.基本初等函數(shù)求導(dǎo)公式;(注:(f(x)))2.法則:(1)四則運(yùn)算;(2)復(fù)合法則;(3)反函數(shù)三.各類求導(dǎo)(方法步驟):dx1?dyyf(x?h)?f(x?h)h1.定義導(dǎo):(1)f(a)與f(x)x?a;(2)分段函數(shù)左右導(dǎo);(3)limh?0(注:f(x)???F(x)x?x0,求:f(x0),f(x)及f(x)的連續(xù)性),x?x0?a2.初等導(dǎo)(公式加法則):(1)u?f[g(x)],求:u(x0)(圖形題);(2)F(x)??xaf(t)dt,求:F(x)(注:(?f(x,t)dt),(?f(x,t)dt),(?f(t)dt))aaaxbb?f1(x)x?x0,(3)y??,求f?(x0),f?(x0)及f(x0)(待定系數(shù))?f2(x)x?x0dyd2y,3.隱式(f(x,y)?0)導(dǎo):dxdx2(1)存在定理;(2)微分法(一階微分的形式不變性).(3)對(duì)數(shù)求導(dǎo)法.?x?x(t)dyd2y,24.參式導(dǎo)(數(shù)一,二):?,求:dxdx?y?y(t)5.高階導(dǎo)f(n)(x)公式:(e)ax(n)1(n)b??nn!;)??ae;(a?bx(a?bx)n?1nax(n)(sinax)?ansin(ax??2?n);(csax)(n)?ancs(ax??2?n)1(n?1)2(n?2)(uv)(n)?u(n)v?Cnuv?Cnuv??注:f(n)f(n)(0)(0)??與泰勒展式:f(x)?a0?a1x?a2x2???anx???an?n!n四.各類應(yīng)用:1.斜率與切線(法線);(區(qū)別:y?f(x)上點(diǎn)M0和過(guò)點(diǎn)M0的切線)2.物理:(相對(duì))變化率?速度;3.曲率(數(shù)一二):??曲率半徑,曲率中心,曲率圓)4.邊際與彈性(數(shù)三):(附:需求,收益,成本,利潤(rùn))五.單調(diào)性與極值(必求導(dǎo))1.判別(駐點(diǎn)f(x0)?0):(1)f(x)?0?f(x)?;f(x)?0?f(x)?;(2)分段函數(shù)的單調(diào)性(3)f(x)?0?零點(diǎn)唯一;f(x)?0???駐點(diǎn)唯一(必為極值,最值).2.極值點(diǎn):(1)表格(f(x)變號(hào));(由limx?x0f(x)f(x)f(x)?0,lim?0,lim2?0?x?0的特點(diǎn))x?x0x?x0xxx??(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);(2)實(shí)例:由f(x)??(x)f(x)?g(x)確定點(diǎn)“x?x0”的特點(diǎn).(3)閉域上最值(應(yīng)用例:與定積分幾何應(yīng)用相結(jié)合,求最優(yōu))篇二:高等數(shù)學(xué)知識(shí)點(diǎn)總結(jié)高等數(shù)學(xué)知識(shí)點(diǎn)總結(jié)導(dǎo)數(shù)公式:2(tanx)??secx(ct??anx)???cscx(secx)??secx?tanx(cscx)???cscx?ctx(a)??alna(lgaxx2(arcsinx)??(arccsx)???(??arctanx)??1?x21?x121?x2x)??1xlna(arcctx)???11?x2基本積分表:三角函數(shù)的有理式積分:?tan?sec?a?x?a?xdx??lncsx?C?ctxdx?lnsinx?Cxdx?lnsecx?tanx?C?c??s?sindx2xx???sec?csc2xdx?tanx?Cxdx??ctx?C??dx22?cscxdx?lncscx?ctx?Cdx2?secx??x?tanxdx?secx?Cxdx??cscx?Cx?xdx?adx?xdx22???1a1arctanlnlnxa?C?C?C?cscx?ct?adx?ax?ax?aa?xa?xxalna?C222a12a?shxdx?chxdx??2?chx?C?shx?C?ln(x?x?a)?C2222a?x2?arcsin?Cdxx?a22?2In??sin02nxdx??csnxdx?2n?1naaa2In?2x?a??)?Cx?axa?C2222???sinx?2u1?ux?adx?x?adx?a?xdx?22222x2x2x2x?a?x?a?a?x?22??22222ln(x?lnx?arcsin22?C2，csx?21?u1?u2，u?tan2x2，dx?2du1?u2一些初等函數(shù):兩個(gè)重要極限:e?e2e?e2shxchx2x?xx?x雙曲正弦:shx?雙曲余弦:chx?雙曲正切:thx?arshx?ln(x?archx??ln(x?arthx?12ln1?x1?xlimsinx(1?x1xx?0?1)xlime?ee?exx?x?xx???e?x?1)x?1)2三角函數(shù)公式:?誘導(dǎo)公式:?和差角公式:?和差化積公式:sin(???)?sin?cs??cs?sin?cs(???)?cs?cs??sin?sin?tan(???)?ct(???)?tan??tan?1?tan??tan?ct??ct??1ct??ct?sin??sin??2sinsin??sin??2cs???2cssin???2???2???2cs??cs??2cscs??cs??2sin???2cssin???2???2???2?倍角公式:sin2??2sin?cs?cs2??2cs??1?1?2sin??cs??sin?ct2??tan2??ct??12ct?2tan?1?tan?222222sin3??3sin??4sin?cs3??4cs??3cs?tan3??3tan??tan?1?3tan?2333?半角公式:sintan?2?????cs?21?cs?1?cs?asin??A1?cs?sin?bsinB?csct?2??1?cs?2?21?cs?sin?2?2??csin?1?cs??2??1?cs?1?cs?2?sin?1?cs??正弦定理:?sinC?2R?余弦定理:??c?a?b?2abcsC?反三角函數(shù)性質(zhì):arcsinx??2?arccsxarctanx??2?arcctx高階導(dǎo)數(shù)公式——??萊布尼茲(Leibniz)公式:n(uv)?u(n)??Ck?0knu(n?k)v(k)(n)v?nu(n?1)v??n(n?1)2!??u(n?2)v?????n(n?1)?(n?k?1)k!u(n?k)v(k)???uv(n)中值定理與導(dǎo)數(shù)應(yīng)用:拉格朗日中值定理:柯西中值定理:f(b)?f(a)?f?(?)(b?a)?f?(?)F?(?)拉格朗日中值定理。f(b)?f(a)F(b)?F(a)當(dāng)F(x)?x時(shí)，柯西中值定理就是曲率:弧微分公式:平均曲率:K?ds????s?y?dx,其中y??tg???:從M點(diǎn)到M?點(diǎn)，切線斜率的傾角變???sd?dsy??(1?y?)232化量;?s:MM?弧長(zhǎng)。M點(diǎn)的曲率:直線:K?0;K?lim?s?0??.半徑為a的圓:K?1a.定積分的近似計(jì)算:b矩形法:?f(x)?abb?an(y0?y1???yn???1)梯形法:?f(x)?abb?a1[(y0?yn)?y1???yn?1]n2b?a3n[(y0?yn)?2(y2?y4???yn?2)?4(y1?y3???yn?1)]拋物線法:?f(x)?a定積分應(yīng)用相關(guān)公式:功:?F?s水壓力:F?p?A引力:F?km1m2r2,k為引力系數(shù)函數(shù)的平均值:y?1b?ab?b?aa1??bf(x)dx均方根:?af(t)dt2空間解析幾何和向量代數(shù):空間2點(diǎn)的距離:向量在軸上的投影:d?M1M2?(x2?x1)?(y2?y1)?(z2?z1)222PrjuAB?cs?,?是AB與u軸的夾角。????Prju(a1?a2)?Prja1?Prja2????a?b?a?bcs??axbx?ayby?azbz,是一個(gè)數(shù)量?jī)上蛄恐g的夾角:cs??k,axbx?ayby?azbzax?ay?az?bx?by?bz222222i???c?a?b?axbxjayb??y???az,c?a?bsin?.例:線速度:bzaybycyazbzcz???v??r.ax??????向量的混合積:[abc]?(a?b)?c?bxcx代表平行六面體的體積。????a?b?ccs?,?為銳角時(shí)，平面的方程:1、點(diǎn)法式:?A(x?x0)?B(y?y0)?C(z?z0)?0，其中n?{A,B,C},M0(x0,y0,z0)Ax?By?Cz?D?0xa?yb?zc?1d?Ax0?By0?Cz0?DA?B?C空間直線的方程:2222、一般方程:3、截距世方程:平面外任意一點(diǎn)到該平面的距離:???x?x0?mtx?x0y?y0z?z0?????t,其中s?{m,n,p};參數(shù)方程:?y?y0?ntmnp?z?z?pt0?2222二次曲面:1、橢球面:2、拋物面:3、雙曲面:單葉雙曲面:雙葉雙曲面:xaxa222??2xa222??yb?2zc?1xy2p2q?z(,p,q同號(hào))??ybyb2222??zczc2222?1?(馬鞍面)1多元函數(shù)微分法及應(yīng)用全微分:dz??z?xdx??z?ydydu??u?xdx??u?ydy??u?zdz全微分的近似計(jì)算:多元復(fù)合函數(shù)的求導(dǎo)法?z?dz?fx(x,y)?x?fy(x,y)?y:dz?z?u?z?vz?f[u(t),v(t)]????dt?u?t?v?t?z?z?u?z?vz?f[u(x,y)??,v(x,y)]?????x?u?x?v?x當(dāng)u?u(x,y)，v?v(x,y)時(shí)，du??u?xdx??u?ydydv??v?xdx??v?ydy隱函數(shù)的求導(dǎo)公式:FFFdydy??dy隱函數(shù)F(x,y)?0??x2?(?x)，(?x)?dxFy?xFy?yFydxdxFyFx?z?z隱函數(shù)F(x,y,z)?0?????xFz?yFz2篇三:高數(shù)知識(shí)點(diǎn)總結(jié)(1)專接本高數(shù)知識(shí)點(diǎn)總結(jié)(上冊(cè))——北雁友情提供函數(shù):極限與連續(xù)性:數(shù)列的極限:夢(mèng)想這東西和經(jīng)典一樣，永遠(yuǎn)不會(huì)因?yàn)闀r(shí)間而褪色，反而更顯珍??貴～夢(mèng)想這東西和經(jīng)典一樣，永遠(yuǎn)不會(huì)因?yàn)闀r(shí)間而褪色，反而更顯珍貴～夢(mèng)想這東西和經(jīng)典一樣，永遠(yuǎn)不會(huì)因?yàn)闀r(shí)間而褪色，反而更顯珍貴～夢(mèng)想這東西和經(jīng)典一樣，永遠(yuǎn)不會(huì)因?yàn)闀r(shí)間而褪色，反而更顯珍貴～夢(mèng)想這東西和經(jīng)典一樣，永遠(yuǎn)不會(huì)因?yàn)闀r(shí)間而褪色，反而更顯珍貴～篇四:高數(shù)上冊(cè)知識(shí)點(diǎn)總結(jié)高數(shù)重點(diǎn)知識(shí)總結(jié)1、基本初等函數(shù):反函數(shù)(y=arctanx)，對(duì)數(shù)函數(shù)(y=lnx)，冪函數(shù)(y=x)，指數(shù)函數(shù)(y?ax)，三角函數(shù)(y=sinx)，常數(shù)函數(shù)(y=c)2、分段函數(shù)不是初等函數(shù)。x2?xx?lim?13、無(wú)窮小:高階+低階=低階例如:limx?0x?0xxsinx4、兩個(gè)重要極限:(1)lim?1x?0x(2)lim?1?x?ex?0??1x?1?lim?1???ex???x?g(x)x經(jīng)驗(yàn)公式:當(dāng)x?x0,f(x)?0,g(x)??，lim?1?f(x)?x?x0?ex?x0limf(x)g(x)例如:lim?1?3x?ex?01xx?0??3x?lim???x??e?35、可導(dǎo)必定連續(xù)，連續(xù)未必可導(dǎo)。例如:y?|x|連續(xù)但不可導(dǎo)。6、導(dǎo)數(shù)的定義:lim?x?0f(x??x)?f(x)?f(x)?xx?x0limf(x)?f(x0)?f?x0?x?x07、復(fù)合函數(shù)求導(dǎo):df?g(x)??f?g(x)??g(x)dx例如:y?x?x,y?2x?2x?12x?x4x2?xx1?18、隱函數(shù)求導(dǎo):(1)直接求導(dǎo)法;(2)方程兩邊同時(shí)微分，再求出dy/dxx2?y2?1例如:解:法(1),左右兩邊同時(shí)求導(dǎo),2x?2yy?0?y??xydyx法(2),左右兩邊同時(shí)微分,2xdx?2ydy???dxy9、由參數(shù)方程所確定的函數(shù)求導(dǎo):若??y?g(t)dydy/dtg(t)??，則，其二階導(dǎo)數(shù):dxdx/dth(t)?x?h(t)d(dy/dx)d?g(t)/h(t)?dyd?dy/dx????2dxdxdx/dth(t)210、微分的近似計(jì)算:f(x0??x)?f(x0)??x?f(x0)例如:計(jì)算sin31?11、函數(shù)間斷點(diǎn)的類型:(1)第一類:可去間斷點(diǎn)和跳躍間斷點(diǎn);例如:??y?sinx(x=0是x函數(shù)可去間斷點(diǎn))，y?sgn(x)(x=0是函數(shù)的跳躍間斷點(diǎn))(2)第二類:振蕩間斷點(diǎn)和無(wú)窮間斷點(diǎn);例如:f(x)?sin??(x=0是函數(shù)的振蕩間斷點(diǎn))，y?斷點(diǎn))12、漸近線:水平漸近線:y?limf(x)?cx???1??x?1(x=0是函數(shù)的無(wú)窮間xlimf(x)??，則x?a是鉛直漸近線.鉛直漸近線:若，x?a斜漸近線:設(shè)斜漸近線為y?ax?b,即求a?limx??f(x),b?lim?f(x)?ax?x??xx3?x2?x?1例如:求函數(shù)y?的漸近線x2?113、駐點(diǎn):令函數(shù)y=f(x)，若f(x0)=0，稱x0是駐點(diǎn)。14、極值點(diǎn):令函數(shù)y=f(x)，給定x0的一個(gè)小鄰域u(x0,δ),對(duì)于任意x?u(x0,δ)，都有f(x)?f(x0)，稱x0是f(x)的極小值點(diǎn);否則，稱x0是f(x)的極大值點(diǎn)。極小值點(diǎn)與極大值點(diǎn)統(tǒng)稱極值點(diǎn)。15、拐點(diǎn):連續(xù)曲線弧上的上凹弧與下凹弧的分界點(diǎn)，稱??為曲線弧的拐點(diǎn)。16、拐點(diǎn)的判定定理:令函數(shù)y=f(x)，若f(x0)=0，且xx0,f(x)0;xx0時(shí)，f(x)0或xx0,f(x)0;xx0時(shí)，f(x)0，稱點(diǎn)(x0，f(x0))為f(x)的拐點(diǎn)。17、極值點(diǎn)的必要條件:令函數(shù)y=f(x)，在點(diǎn)x0處可導(dǎo)，且x0是極值點(diǎn)，則f(x0)=0。18、改變單調(diào)性的點(diǎn):f(x0)?0，f(x0)不存在，間斷點(diǎn)(換句話說(shuō)，極值點(diǎn)可能是駐點(diǎn)，也可能是不可導(dǎo)點(diǎn))19、改變凹凸性的點(diǎn):f(x0)?0，f(x0)不存在(換句話說(shuō)，拐點(diǎn)可能是二階導(dǎo)數(shù)等于零的點(diǎn)，也可能是二階??導(dǎo)數(shù)不存在的點(diǎn))20、可導(dǎo)函數(shù)f(x)的極值點(diǎn)必定是駐點(diǎn)，但函數(shù)的駐點(diǎn)不一定是極值點(diǎn)。21、中值定理:(1)羅爾定理:f(x)在[a,b]上連續(xù)，(a,b)內(nèi)可導(dǎo)，則至少存在一點(diǎn)?，使得f(?)?0(2)拉格朗日中值定理:f(x)在[a,b]上連續(xù)，(a,b)內(nèi)可導(dǎo)，則至少存在一點(diǎn)?，使得f(b)?f(a)???(b?a)f(?)(3)積分中值定理:f(x)在區(qū)間[a,b]上可積，至少存在一點(diǎn)?，使得b?f(x)dx?(b?a)f(?)a22、常用的等價(jià)無(wú)窮小代換:x~sinx~arcsinx~arctanx~tanx~ex?1~2(?x?1)~ln(1?x)1?csx~12x2111tanx?sinx~x3,x?sinx~x??3,tanx?x~x326323、對(duì)數(shù)求導(dǎo)法:例如，y?xx，解:lny?xlnx?1y?lnx?1?y?xx?lnx?1?y24、洛必達(dá)法則:適用于“0?”型，“”型，“0??”型等。當(dāng)0?x?x0,f(x)?0/?,g(x)?0/?，f(x),g(x)皆存在，且g(x)?0，則f(x)f(x)ex?sinx?10ex?csx0ex?sinx1lim?lim例如，limlimlim?2x?x0g(x)x?x0g(x)x?0x?0x?0x2x2225、無(wú)窮大:高階+低階=高階例如，26、不定積分的求法(1)公式法(2)第一類換元法(湊微分法)(3)第二類換元法:哪里復(fù)雜換哪里，常用的換元:1)三角換元:23?x?1??2x?3?lim?x???2x5x2?2x?lim?4x???2x53a2???x2，可令x?asint;x2?a2，可令x?atant;x2?a2，可令x?asect2)當(dāng)有理分式函數(shù)中分母的階較高時(shí)，常采用倒代換x?1t27、分部積分法:udv?uv?vdu，選取u的規(guī)則“反對(duì)冪指三”，剩下的作v。分部積x3分出現(xiàn)循環(huán)形式的情況，例如:ecsxdx,secxdx????28、有理函數(shù)的積分:例如:3x?22(x?1)?x11dx????2dx??x(x?1)3?x(x?1)3?x(x?1)2?x?13dx11x?1?xx?1?x1dx???需要進(jìn)行拆分，令?x(x?1)2x(x?1)2x(x?1)2x(x?1)(x?1)2其中，前部分?111??2xx?1(x?1)29、定積分的定義:?f(?)?x?f(x)dx?lim?a???0iii?1bn30、定積分的性質(zhì):b(1)當(dāng)a=b時(shí)，?f(x)dx?0;aba(2)當(dāng)ab時(shí)，?f(x)dx???f(x)dxaba?aa(3)當(dāng)f(x)是奇函數(shù)，?f(x)dx?0,a?0a(4)當(dāng)f(x)是偶函數(shù)，b?a?f(x)dx?2?f(x)dxcb(5)可加性:?f(x)dx??f(x)dx??f(x)dxaacxxd31、變上限積分:?(x)??f(t)dt??(x)?f(t)dt?f(x)?dxaad推廣:dxu(x)?f(t)dt?f?u(x)?u(x)ab32、定積分的計(jì)算(牛頓—萊布尼茨公式):bb?f(x)dx?F(b)?F(a)a33、定積分的分部積分法:udv??uv??vdu例如:xlnxdx?aba?a???bb???34、反常積分:(1)無(wú)窮限的反常積分:?f(x)dx?lim?f(x)dxaabbt?a?(2)無(wú)界函數(shù)的反常積分:35、平面圖形的面積:(1)A??f(x)dx?lim?f(x)dxatd??f(x)?f(x)?dx(2)A????(y)??(y)?dy2121ac2(2)繞y軸旋轉(zhuǎn)，????f(x)d??xV???(y)dy??2acbdb36、旋轉(zhuǎn)體的體積:(1)繞x軸旋轉(zhuǎn)，V??篇五:高等數(shù)學(xué)知識(shí)點(diǎn)歸納第一講:一.數(shù)列函數(shù):1.類型:極限與連續(xù)(1)數(shù)列:*an?f(n);*an?1?f(an)(2)初等函數(shù):(3)分段函數(shù):*F(x)???f1(x)x?x0?f(x)x?x0,,;*F(x)??;*?ax?x0?f2(x)x?x0(4)復(fù)合(含f)函數(shù):y?f(u),u??(x)(5)隱式(方程):F(x,y)?0(6)參式(數(shù)一,二):??x?x(t)?y?y(t)(7)變限積分函數(shù):F(x)??xaf(x,t)dt(8)級(jí)數(shù)和函數(shù)(數(shù)一,三):S(x)?2.特征(幾何):?ax,x??nnn?0?(1)單調(diào)性與有界性(判別);(f(x)單調(diào)??x0,(x?x0)(f(x)?f(x0))定號(hào))(2)奇偶性與周期性(應(yīng)用).3.反函數(shù)與直接函數(shù):y?f(x)?x?f二.極限性質(zhì):?1.類型:*liman;*limf(x)(含x???);*limf(x)(含x?x0)n??x???1(y)?y?f?1(x)x?x02.無(wú)窮小與無(wú)窮大(注:無(wú)窮量):3.未定型:0??,,1,???,0??,00,?00?4.性質(zhì):*有界性,*保號(hào)性,*歸并性三.常用結(jié)論:ann?1,a(a?0)?1,(a?b?c)?maxa(b,,??c,)?a?0??0n!nn1n1n1nnxnlnnx1xx?1,lix?,0lim?0,(x?0)??,lim?x???x???x?0exxxlnx?lim?x?0n,0e??x?0x???,???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);1?csu(x)?eu(x)12u(x);2?1?u(x);ln(1?u(x))?u(x);(1?u(x))??1??u(x);arcsiunx(?)ux;(arctanu(x)?u(x)2.泰勒公式:12x?(x2);2!122(2)ln(1?x)?x?x?(x);2134(3)sinx?x?x?(x);3!12145(4)csx?1?x?x?(x);2!4!?(??1)2?(5)(1?x)?1??x?x?(x2).2!??(1)ex?1?x?五.常規(guī)方法:前提:(1)準(zhǔn)確判斷1.抓大棄小,2.無(wú)窮小與有界量乘積(??M)(注:sin?3.1處理(其它如:0,?)0??1,,1,?M(其它如:???,0??,00,?0);(2)變量代換(如:?t)0?x??1?1,x??)x4.左右極限(包括x???):11x(1)(x?0);(2)e(x??);ex(x?0);(3)分段函數(shù):x,[x],maxf(x)x5.無(wú)窮小等價(jià)替換(因式中的無(wú)窮小)(注:非零因子)6.洛必達(dá)法則(1)先”處理”,后法則(0xlnxxlnx最后方法);(注意對(duì)比:lim與lim)x?1x?001?x1?xv(x)(2)冪指型處理:u(x)?ev(x)lnu(x)(如:e1x?1?e?e(e1x1x11?x?1x?1))(3)含變限積分;(4)不能用與不便用7.泰勒公式(皮亞諾余項(xiàng)):處理和式中的無(wú)窮小8.極限函數(shù):f(x)?l??imF(x,n)(?分段函數(shù))n??六.非常手段1.收斂準(zhǔn)則:(1)an?f(n)?limf(x)x???(2)雙邊夾:*bn?an?cn?,*bn,cn?a?(3)單邊擠:an?1?f(an)*a2?a1?*an?M?*f(x)?0??f?fx0()?x?0?x1112n3.積分和:lif,x)[?)f(??)?f(??)]fxd(0n??nnnn2.導(dǎo)數(shù)定義(洛必達(dá)?):li4.中值定理:lim[f(x?a)?f(x)]?alimf(?)x???x???5.級(jí)數(shù)和(數(shù)一三):?2nn!(1)?an收斂?liman?0,(如limn)(2)lim(a1?a2???an)??an,n??n??n??nn?1n?1?(3){an}與?(an?1?n?an?1)同斂散七.常見(jiàn)應(yīng)用:1.無(wú)窮小比較(等價(jià),階):*f(x)?kx,(x?0)?(1)f(0)?f(0)???f(2)(n???1)n(0)?0,f(n)(0)?a?f(x)?anax??(xn)?xnn!n!?xf(t)dt??ktndtx2.漸近線(含斜):f(x),b?lim[f(x)?ax]?f(x)?ax?b??x??x??x1(2)f(x)?ax?b??,(?0)??x(1)a?lim3.連續(xù)性:(1)間斷點(diǎn)判別(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)1.連通性:f([a,b])?[m,M](注:?0???1,“平均”??值:?f(a)?(1??)f(b)?f(x0))2.介值定理:(附:達(dá)布定理)(1)零點(diǎn)存在定理:f(a)f(b)?0?f(x0)?0(根的個(gè)數(shù));(2)f(x)?0?(?xaf(x)dx)?0.第二講:導(dǎo)數(shù)及應(yīng)用(一元)(含中值定理)一.基本概念:1.差商與導(dǎo)數(shù):f(x)?lim?x?0f(x)?f(x0)f(x??x)?f(x);f(x0)?limx?x0x?x0?x(1)f(0)?limx?0f(x)?f(0)f(x)(注:lim?A(f連續(xù))?