高-数强化-第三四讲

考研数学高等数学主讲：张宇（15第一讲极限 第二讲一元函数微积分 第三讲多元函数微分 第四讲二重积 张宇：名师，博士，著名考研数学辅导，教育部“国家精品课程建设骨（15第一讲极限 第二讲一元函数微积分 第三讲多元函数微分 第四讲二重积 第一讲4.4.1.1）limf(xA00，当0xx0时，恒有f(xAx（xx，xx，xx，x，xx 2）limxa0N0，当nNxannx若limf(x)f(xx0f(xx0中有无定义点，则limf(x不如sin(x 1xxcosxN③取f(xxn的范围2. 若limf(x)A，则A唯一

ln(1ex1ln(1ex

(x2(x21)若limf(x)A，则M0，0，使得当0 x0时，f(x)Mxf(x(x1)sin3，若limf(xA0（或xf(x0（或推论：若x中，f(x)0（或，若limf(x)，则limf(x)0（或 【例】设limf(xf(x0)1f(xxx处（ (xx0 1cosxcos2xcoslim （2）limx0ex1[t(e 11x21xlimlnxln(1limxlnx0（00lim[4x2xln(21)2ln2 （6）（6）lim(2xtanx2)sin（7）lim(xsinx)sinx(11xlimx2f(x)

limsin2xf(x)

， x0xsin xsin（1）lim(111)n 1lim(12n3n)nsinn x

1

ln(1x)x (11)(12 (1n)，求limx

n

1f(xx（II）设xn满足lnxn

11，证明limxn存在，并求此极限 1p(xabxcx2dx3x0时p(xtanxx3p(x2】若ln(0intdt与cxk为等价无穷小量，求cktxxx(xxxx(x1)ln

x(x24f(x xsinx(x2x x第二讲一元函数微积1.f1.f(xx0f(xx0f(x0存在x2sin1xxx2】设0f(x在[,f(0)1，且limln(12x2xf(x)0f(xx0f(0)3f(xx0处连续，且limf(xf(xf(0) 4f(xx0处连续，且limf(2xf(xf(0) 5f(xx0处连续，且limf(axf(x)baba f(0

a1 yf①yf(x0x)f(x0

③

0y

f(xx 故一点可导AxdyAxy(x0)xf(uyf(x2xx1处取x0.1的线性主部为0.1，则f(1) xIF(xf(xF(xf(xI上的一个原函数f(x)dxF(x)x0IF(x0f(x0F(xf(xI上的原函数xf(xIF(xf(t)dt(axI必可导F(xf(xxIx1F(x

x

xx x【例2】f(x)

x①f(x)

xxx2xsin12cos1 x②f(x)1③f(x)

xxx2xcos1 ④f(x)

x f(x连续F(x)f(t)dtf(x可积F(x)f(t)dt xF(xaf(t)dtx【例】设函数yfx在区间1,3上的图形为ffO-0-123xx则函数Fx0ftdt的图形为 xFFF11-0123x-0123x-- FF11-0123x-0123x- 1）f(xf.2）f(xf.x3）f(xF(xa0ff(t)dt(a f f(xF(xaa

f(t)dt(axF(x)0f(t)dt是 x （C）x0为间断点的奇函 （D）x0为间断点的偶函2f(x是连续的奇函数，a0y.yx yx2 f(u)du f(u)du；③ xf(u)du a0 0a xfa1）f(x以Tf(x也是以T为周期的2）f(x以TF(xf(t)dt以TTx0f(x)dx0T【定理】若可积函数f(x)以T为周期，则0f(x)dx f(x)dx，af(x在(0,内可导，则（f(x在Xf(x在Xf(x在Xf(x在Xf(x在(0,f(x在(0,f(x在(0,f(x在(0,1】lim(n1n2n3nn2 n2 n2nnn21n12】求lim(bn bsinnbx5.变限积 x

f(t)dt，

f(t)dt，

2(x

fb 1(xbb属于定积分af(x)dx f(t)dt)f[2(x)]2(x)f[1(x)]1【例】设f(x)是连续函数，则(xtf(x2t2)dt)x 06.①定义 fbaf(x)dxab x x 1 p1 0x p11】设0

1ln02】设k0

x(lnx)k的敛散性xsinxx

f

，满足何种关系时，f(xx处连续

x2yf(xy3xy2x2y60f(x的极值12xy(n0【例5】设y x12xy(nn2ln(xln(x1x2)1x2【例1】 3I1x)dx(xx14Iex1x25x2

(2x1)34(2x1)34x10①x(x0,x0)，f(x)0；x(x0,x0)，f(x)0②x(x0,x0)，f(x)0；x(x0,x0)，f(x)0

x0为极小值x0为极大值点yy(xx2y2y1yy(2)0yx的极值 f(x) f(n1)(x)设f(x)在x0处n阶可导，且f(n)(0x)0 f(nx0x n为偶数时，若f(nx0x 【例】设yy(x)满足y(4)3y5yecosx，其中y(2)y(2)y(2)y(2)0，x220拐点与凹f(xx0f(x变号x0f(x0为曲线上的拐点f(x)f(x) f(n1)(x)设f(x)在x0处n阶可导，且f(n)(0x) 1yx1)(x2)2x3)3(x4)4的一个拐点为（ （B）(2, 2f(xg(x)f(0)(1xf(1)x，则在[01]上（（A）f(x0f(xf(x)0f(x)f(x0f(xf(x0f(x30渐近线——求解程limy(x)（xxxxxx为铅直渐近线；反之亦反x limy(xA(yAlimy(xlimy(x)a(0)lim[y(xaxb(yaxb为斜渐近线 4x2xln(21)的渐近线 条x在[abf(x)0x0f(x不存在x1③端点abf(x0、f(x1)、f(a)、f(b，比较大M,在(ab③

f(x)，

ff(xex2sinx21012b12b

y1(x)y2(x) 20旋转体的

r2()r2()x轴旋转

bf2aby轴旋转Vy2axf(x)bb30平均值 yay(x)dxbab(dx)2(dx)2bb

1[f(x)]2[r()]2[r[r()]2[r()]2S[(t)]2(t)]2dt，其中xy50旋转体的侧面 S2f 1[f(x)]2ba注：40、 均为数一、二考查内容，数三不要1【例】设曲线ye 周所得旋转体体积V2.物理应用（数一二 x0x【例】设某产品的总成本函数为C(x1003xx2p100xx21f(x在[aa](a0)f(0)0（I）f(x) faa 2f(x在[01](0,1f(0)0f(1)1证明：存在不同的123(0,1)f(1f(2f(33 f f(a)f( 0f( f(x)0f(x)；f(x)0ff(nx0至多有kf(n1)(x0至多有k1个根【例】证明lnxex1cos2xdx0有且仅有两个根0 g(x)在[a,b]上连续，且f ，0g(x) g(t)dtxa，xxa[a,b]abg(t

f(x)g(x)dx 第三讲多元函数微概念——51.（1）limf(xyy(x,y)(x0,y0f(x,(x2y2)1f(xyx2 (x,y)(0,，求limf(xy(x,y)(0, 2】求lim22xyy2xy.2 【例3】求 x2y2 x2 (x2y24】求limsin(x2yy4

x5f(xyx2 x2y2，求limf(xx2y22.

f(xyf(x0y0f(xy在(x0y0处连续y

xsin1

(0,f(xy)

ysin (x,y) (x,y)(0,y

f(x,y)f(x0,y0)，称为间断，但多元时，不间断类型 z(x,y)f(x (x0,y0 (x0,y0f(x,y)limf(x,y0)f(x0,y00 x0f(x,y)limf(x0,y)f(x0,y00 y y0(xyDz

fx(x,y)0f(xyxx1x2，则f(x1y0f(x2y0f(xy)0f(x,y)0f(xyf(xy，则（（A）x1x2，y1（C）x1x2，y1

x1x2，y1（D）x1x2，y1

x2【例】设f(x,y) 求f(0,0)，f(0,x2 zf(x0xy0yf(x0y0 Ax 线性增量Bfy(x0,y0z(Axlim 0f(xy在(xy处可微.(x)2(x)2zAxByo((x)2y)2①f(xx,yy)f(x,y)AxByo((x)2(y)2 ②f(x,y)f(x,y)A(xx)B(yy)o((xx)2(y (x0,y0 dzf(x,y)dx(x0,y0 dzf(x,y)dxf(x,y)dy，dzzdxz zf(x,y满足limf(xy2xy2.5.zf(x,yfx(x0y0fy(x0y0fx(xyfy(xlimf(xyf(xylimf(x,y)f(x,yyx yy x2x2(yf(xy在(x0y0处的偏导数连续.xy(x2y2【例】设f(x,y) x2 (x,y)(0,0)，求f(0,0)，f

(0,0) (x,y)(0,zf(u,v,w)，vv(x,uu(w v wzzvz1zfexcosyx2yf2.2zf(xy,f(x,y))，求x1.1.zf(x,y在点(xy)一阶偏导数存在f'(xy0,f'(xy0 记''(x,y)0极值A0''(xyB，则BAC0非极 2 ''(x0,y0)(x,y,z)问题提法：求目标函数uf(x,yz(x,y,z)F(x,y,z,,)f(x,y,z)(x,y,z)(x,y,FxFFz解上述方程组P(x,y,z)(i1, )u(P)

i2.1f(xykx22kxyy2在点(0,0)处取得极小值，求k的取值范围zx2【例2】求ux2y2z2在约束条件 xyz3】求uxy2yzx2y2z210下的最大值与最小值第四讲二重积2f(x,y)d f(x,y)f(x, f(x,y)f(x,

f(x,y)d

2f(x,y)d

f(x,y)f(x, f(x,y)f(x,2f(x,y)d f(x,y)f(x,Df