成人高考专升本高等数学二复习教程

第一讲函数、连续与极限arctanxx2x3)arctanxx2x31limlim6等价小量与洛必达）x0x06x(x)6f(x)x3x2x0x0limsin6xxf(x)lim6cos6xf(x)xy'3x2x0x3x06x2yxy''6x6x3y''xy'''6x0x03y''(0)0y''(0)66f(x)y'y''22x0洛必达）x22xx0x0lim(2x)x12x重要极限）x1x1ab3)xxxb求lim(2x0ab33xxt(),lnt[ln(ab)ln2]xxx2x33limlntlim(alnablnb)ln(ab)xxabx2变量替换）xx0x0t(ab)3/21lim(cosx)2x)x011t(cosx)lnt2,lntln(cosx)x)x2)x2xx01te变量替换）1/22x02ft)dtx'(x)f(0)f'(0)0lim1设f0x2xft)dtx00洛必达与微积分性质）ln(cos),0xx2x(x)f在aa,x0alimln(cosx)/x1/22连续性的概念）x0ex1xlim3洛必达）1xxx011ctgx()洛必达或）xxx0xxet21洛必达与微积分性质）01ex2x0第二讲导数、微分及其应用xarctantyy(x2ytye52t|1yy(x由ln(xy)xysinx23x0时y'ycosxyyy(x2xydy|(ln2dxx0e在（，（e/2,/2)cosxe,(x,y)|ey'|1/2ye/2/2ye/2x5f(6),f'(6)或f,f'fsinx)3fsinx)sinxlimx0ft)f3ft)f]sinxtttt04f'8f'2y2(x6)f(x)对一切x满足xf''(x)2[f'(x)]1e，y2x若f'(x)0(x0)(x,y)0000x0ex01xx代入，f''(x)0x000exx000x3y(x2x()y'0x0x3y''0拐点xx：铅垂；yx：斜(xey/2xx2xe/2x驻点x与x1y'解：，1x2渐：ye(x2)与yx2d0xt)x2x21(xt)2(2n(2nsin(xt)(xt)(xt)(226n11(xt)4n1sin(xt)dt(xt)(xt)(237n13(4nn11x4n1(4nnsin(xt)xx(x237n30ddx01x2(2nsin(xt)dtxx(sinx2x226n(2n0xx求ftudu()dux222x0(x)xx)在x处的n阶导数f(0)2n()x2x3xn2x2x)x(x(2n1o(x)2n23n2x4x5xn=x3(n1o(xn)23n2!f(0)((n)n1n2x(设，1111x)x21x)lnx)x，1222g(x)x)lnx)x,g(0)0222x)x)2g'(x),g''(x),g'''(x)g'(0)g''(0)0x(时g''(x单调下降，g''(x)g'(x单调下降g'(x)g(x单调下降，g(x)；得证。11h(x),xh'(x)x)xf(x[f(0,1f，，使'''()3f'(0)0f11f(x)f(0)f'(0)xf''(0)xf'''()x23(0,x),x[其中110f(f(0)f''(0)f'''()261将111ff(0)f''(0)f'''()262'''()f'''()6f121，，'''()f['''()'''(3ff212124abe2baba)22ee2fb)f(a):f'()balnblna2ln22令f(x)lnx,2balnt1lntln2()令t,'()t0)()e2tt2e24baba)22e2f(x)ln1x求y''(0)31x22xett(处切线为y2x102yett11yxln(e)(x0)的渐进线方程为yxxe(xlnx(x时222(x2g(x)(xlnx(x,g'(xg''(xg'''(x)22x3gg'，g''20x(g'''g''2x(g'0g''0g0x),g'''g''2x),g'0第三讲不定积分与定积分dxdxx2arcsinC2x(4x)4(2)2xexe2eexC2x22x22x2xx)设fx)f(x)xex)f(x)exexee))xe)e)Cxxxx1exx11x1bb1x|()24221xx1xf(x)x21(x)(x)1f(xt)dtA(x)并讨论'(x)f且x0x0在x0xf(y)0(0)(0)y(x)fx(x)f(y)xA'()x'(0)'(0)/2'(0)A022xx0dx0d(xt)f(xt)dtx)x22222220dx2f(y)d(y)(x2)20a'(x)f(x)x设f(x)f(x)02，2又f(x)与f(x)时S绕xdf(x)aa1(x)f(x)xf(x)2c4a2220a10f(x)x(4xV'(y2dx)'0a522yx1x绕x2ydsyx/2绕x02x1绕x5y6156lnxxln2xxxCx2x5x26xarcsinxdxx第四讲向量代数、多元函数微分与空间解析几何(x)(sin)fzfeyzze2xzxf(x)f''f0f(u)ceceuu1212zzyf(xy)y(xyxyy(x),zz(x由zxf(xy),F(x,y,z)决定dz/dx4.求xyzax/xy/yz/zada000x22y3z212,2)22z(x,y)x26102180xyyyzz设z22求zz(x,y)xy2zzf(xy,)g(),求yxyxyzxzzf(xy,g(求yxyu,uy,,求x22x第五讲多元函数的积分by2(x)f(x,y)dydxf(x,y)dxdyayx)r2()(,)2dfrrdrD1r)by2(x)z2(x,y)f(x,y,z)dzdxdyayx)zx,y)f(x,y,z)dxdydzz2dz2()d2()(,,)frzrdrzrzrz)z1z)V2()r2()f(r,,)rsindrdd2)r)zf(x,y)A1z'z'22dxdyxyDL:yy(x)f(x,y(x))1y'dxb2xaxxt)f(x,y)dlL:fxtytx((),())''ydt22yyt)ttLL:rr(f(rcos,sin)r)r'r2d2与f(x,y,z(x,y))1z'f(x,y,z)dSz'dxdy22xyS:zz(x,y)DxyEdV通量，散度）Gauss:EdSSVStokes:FdrF)dS旋度）LS22yzI(xy),22绕zx0d8(xy)822zr2I2230x2y22z00022dxdy,D为yaaxa与(I224axy222D12yxIa()222yxxyxf(x,y)，f(x,y)dxdy,D:xy2x求22D(eybxy))e(()yIxxLL(2ay2axx从O沿y至AO2I(ba)dxdy(bx)dx(2)a2aba23220L11Dxdyydx,L为以0)为中心，R(为半径的圆周正向。I4xy22L2xrcosL1:，yrsin0L1L1L1，xf(x)dydzxyf(x)dzdxe2xzdxdy0f(x)在S求f(x)1f(x)。x0(f(x)xf'(x)xf(x)e)dV0FdSFdV2xs11exy(yey(e2xxxxx第六讲常微分方程f(x),y''f(x,y')(y'p(x)),y''f(y,y')(y'p(y))y(n)y''q02pq0yce1ce21xx2121齐次)y(ccx)ex12112(x)xciyecx112or()Qxexy2nf()P(ex()Qxxexy非齐次)n122nandQxx()y2ex122nf()ex(p()cosxp()sin)xij(()cos()siniyeqxxrx(非齐xx2nn(()cos()sin(max(,)iyxeqxxxrxxnij2nn)(3x2xyy)dx(x2xy)dy0求222通解。（xy2xyxc)23u2.yy''cos2'sin3cosxyxyxexx2xxexu''ue,yc2cx（）x5x1213.设yy(x)(x,y)(处1y'2yy(x)11y''y'10y|x)|1y122422xy(x)y((0),求oxyy(4）ye1x21求y''4ye2xyce2xce22xxe2x）411(yxy)0(x0),y0y(x求）222211yxe2)求y''2ye0,y(0)y'(0)12x2x44第七讲无穷级数p与[l,l]的[0,l]第八讲线性代数用初等变换求矩阵的秩与逆n掌握线性相关、线性无关的判别掌握用初等行变换求解线性方程组的方法掌握用正交变换、配方法化二次型为标准形的方法第九讲概率统计初步会计算古典型概率与几何型概率掌握概率的加减、乘、全概率与贝叶斯公式数掌握常用分布函数的数字特征，会求随机变量的数学期望2tF第十讲总结1[(x)(x2a)...(x(na)x1aa(几何级nnnn2n)2lim(x)e/2)1/xx0xlim(2x)tan2x1lim(3x)2x16xx(xa)na(xa)nnn1lim(xa)2xa1cos2x,x0x2lim()fxf(x)x0x0costdtx(x0)0xabxabxa)()()xbyxxy)0xxettyy(x)yett2xylny14xy(2xyy'02221e,ulnv,vx求y'y2u33xt1I(x)xtt1）202x21|x1|21x)dx123/201xx)dttt2114x14x2x2zf(,e)zz','yxyzzz(x,y)由F(x,y)0z''zyxxy求u(x,y)xy2xy在22usinxln(xy)uyyzxf()xz'2yz'nx2xy求f(x,y)4x4yxy在D:xy182222(ab)brotaarotbII(4xy)dxdy,D:xy2y22D(xy)dxdy,D:x