大学高等数学知识点(全)

大学高等数学知识点整理公式，用法合集极限与连续一.数列函数:(1)数:*a f(n); *a f(a)n n1 n(2)初等函数:f(x)xx

f(x)

xx(3)分段函:*F(x)1 ,

0;*F(x) ,

0;*f(x)xx2 0

a xx0(4)复(含f)函: yf(u),u(x)(5)隐(方): F(x,y)0xx(t)

yy(t)变限积分函数:F(x)a

f(x,t)dtS(x)n0

axn,xnf(x)单调x0

,(xx0

)(f(x)f(x0

))定号)反函数与直接函: yf(x)xf1(y)yf1(x)二.极限性质:类型:*lim

;*limf(xx);*

f(x)(含xx)n n

x

xx0 0

0 0,

,1,,0,00,0性质:有界性,保号性,1 1 nn1 , an(a0)1 , (anbncn)nmax(a,b,c) ,1 1 ana00n!n lnnx1(x0), limxx1, limx 0, lim

0,x x0 xex0 x

x xlimx0

xlnn

x0, ex

x四.必备公式:u(x);等价无穷小:当u(x)u(x);u(x);sinu(x);u(x);u(x)euu(x);u(x)

tanu(x)ln(1u(x))

1cosu(x) ;1u2(x)2u(x1u2(x)2u(x);u(x);u(u(x);u(x);1

arctanu(x)ex

1x

x2o(x2);2!1(2)ln(1x)x x2o(x2);21(3)sinxx x3o(x4);13!1 1(4)cosx1 x2 x4o(x5);2! 4!(5)(1x)1x(1)x2o(x2).2!五.常规方法:前提:(1)1t)x

0, ,M0000);(2)0抓大弃小),1x无穷小与有界量乘积(M)注:sin1x3.1处理(其它如:00,0)x):1

1,x)(1)(x0); (2)ex(x); ex(x0); (3)分段函数:x ,[x],xmaxf(x)洛必达法则0最后方法);(注意对比:limxlnx与limxlnx)0 1 1

1x1

x01

1x幂指型处理:u(x)vx

evx)lnux)(如:ex1ex

ex(ex1x1))极限函: f(x)limF(x,n)(分段函)n六.非常手段an

f(n)limf(x)x双边:*b an n

c?,*b,cn n

a?单边挤:

n1

f(an

) *a2

a? *a1 n

M? *f'(x)0?导数定洛必?):

ff'(x)x0 x 01 1 2 3.积分:lim [f( )f( ) f( )]1 1 2 nn n n n 0

f(x)dx,中值定: lim[f(xa)f(x)]alimf)x xaa)n(1)

收敛lima

0,如lim2nn!) (2)lim(aa

a ,n1(3){a

n}与(

na

n

n nn

n 1 2

nn1n n1

n1七.常见应用:*f(x)

kxn,(x0)?f(f(n1)axnn!(1)f(0)f'(0)

(0)0,f(n)(0)af(x) xnn!

(xn)(2)0

f(t)dtxktndtalimf(x),blim[f(x)ax]f(x)

axbx

x(2)f(x)axb,(1x

0)连续性: (1)间断点判别(个数); (2)分段函数连续性(附:极限函数,f'(x)连性)八.[a,b]上连续函数性质1.连通性: f([a,b])[m,M] (注:01,“平均”值:f(a)(1)f(b)f(x))0零点存在定理:f(af(b0f(x0

)0(根的个数);(2)f(x)0(xa

f(x)dx)'0.第二讲:导数及应用(一元)(含中值定理)一.基本概念:差商与导: f'(x)

f(xx)f(x)

; f'(x

)lim

f(x)f(x)0x0 x

0 xx xx000(1)f'(0)

f(x)f(0) f(x)注:lim f连续)f(0)0,f'(0)A)x0左右: f'(x

x), f'(x);

x0 x 0 0可导与连;在x0处,x连续不可;xx可导)微分与导: ff(xx)f(x)f'(x)xo(x)dff'(x)dx可微可; 比较f,df与"0"的大小比(图);二.求导准备:1.基本初等函数求导公式;(注:(f(x))')dx 12.法则:(1)四则运算;(2)复合法则;(3)反函数dy y'三.各类求导(方法步骤):1.定义导: (1)f'(a)与f'(x)f(xh)f(xh)

xa

; (2)分段函数左右导; (3)limh0

hF(x) xx(:f(x) ,

0,求:f'(x

),f'(x)f'(xa xx 002.初等导(公式加法则):(1)uf[g(x)]u'(x0

)(图形题);

F(x)a

f(t)dt ,

求 : F'(x) ( 注 :(a

f(x,t)dt)',(a

f(x,t)dt)',(a

f(t)dt)')f(x) xx(3)y1 ,

0f(x

),f'(x

)及f'(x

) f(x) xx2

0 0 03.f(xy)0

dy,d2ydx dx2(3xx(t) dy d2y参式(数,): ,求y y(t)fn(x

,dx dx2(eax)(n)

aneax; (

)(n

bnn! ;(sinax)(n)ansin(ax

abx (abx)n1n); (cosax)(n)ancos(ax2

n)2(uv)(n

u(n)vC1u(n1)v'C2u(n2)v"n nax nan nf(n)(0)n!注:f(n)(0)与泰勒展:f(xax nan nf(n)(0)n!0 1 2 2四.各类应用:斜率与切(法); 区:yf(x)上点M0f"(x)(1f"(x)(1f'2(x))3

和过点M0

的切线)曲率(数一二):

(曲率半径,曲率中心,曲率圆)f'(x0

)0):;(1)f'(x)0f(x) ; f'(x)0f(x);分段函数的单调性f'(x)0零点唯; f"(x)0驻点唯(必为极,最).f'(x变号);(由

f'(x)

0,lim

f'(x)

x0,limx

f''(x)

0x0的特点)f'(x0

)0)

xx x0

xx xx x20 0注(1)ff',fff'(x(xf(x)g(x)xx0

”的特点.f(x)0)(1)区:单变量与双变? *x[a,b]与x[a,),x(,)?(2)类:*f'0,f(a)0; *f'0,f(b)0*f"0,f(a),f(b)0; *f"(x)0,f'(x0

)0,f(x0

)0注意:单调性端点值极值凹凸性.(如:f(x)Mf函数的零点个数:单调介值六.凹凸与拐点(必求导!):

max

(x)M)1.yf"(x0

)0)2.应:泰勒估; (2)f'单调; 凹.1.结论:F(bF(aFf()02.辅助函数构造实例:(1)f()F(x)xa

f(t)dt(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.fn0f(x有n1个零点fn1)(x有2个零点4.特例:证明f(n)aF(x)f(xP(xn1P(x待定)5.注:含1 2

n n时,分家!(柯西定理)6.附(达布定理):f(x在[abcf'(a),f'(b[ab]fc八.拉格朗日中值定理1.结论:f(bf(a)f)(ba;((a(b0)fff)xf,f',f1 1.结:f(x)f(x)f'(x)(xx) f"(x)(xx)2 f)(x1

)3;0 0 0 2! 0 0 3! 02.应用:在已知f(a)或f(b)值时进行积分估计十.积分中值定理(附:广义):[注:有定积分(不含变限)条件时使用]第三讲:一元积分学一.基本概念:F(x:(1)F'(x)f(x); (2)f(x)dxdF(x); (3)f(x)dxF(x)c注(1)F(x)a

f(t)dt(连续不一定可导);(2)x(xt)f(t)dt

f(t)dtf(x) f(xa a(1)

f(x)dx)'f(x); d(

f(x)dx)f(x)dx(2)f'(x)dxf(x)c; df(x)f(x)c二.不定积分常规方法熟悉基本积分公式(k1

f(x)k2

g(x))dxk1

f(x)dxk2

g(x)dx1sin2

xcos2x)dx

1 1 dxx d(ax b), xdx dx2, dlna 2 x

dx2dx1x2x dxd 1xx1x2

(1lnx)dxd(xlnx)axb常用(三角代换,根式代换,倒代换):xsinaxb作用与引化简): x21xt

t,

1t, texex1含需求导的被积函数(如lnx,arctanxa

f(t)dt);“反对幂三指xneaxdx, xnlnxdx,特: xf(x)dx (*已知f(x)的原函数为F(x);*已知f'(x)F(x))6.特例:(1)

asinxbcosx1 1

dx;(2)p(x)ekxdxp(x)sinaxdx快速法;

(3) v(x)un(x)

asinxbcosx三.定积分:(2*a

axx2dx(a0)

a2; *

b(xa

)dx00附:a

8f(x)dxMba), a

af(x)g(x)dxMa

2g(x)dx)2:变限积分(x)a

f(t)dt的处理(重点)(1)f可积连续,f连续可导(2) (a

f(t)dt)' f(x) ; (x(xt)f(t)dt)'a a

f(t)dt ;xf(x)dt(xa)f(x)a(3F(x)xa

f(t)dt参与的求导,极限,极值,积分(方程)问题NL公式:a

f(x)dxF(bF(a)F(x在[ab注:(1)分段积分,对称性(奇偶),周期性(2)有理式,三角式,根式(3)含a

f(t)dt的方程.

bf(x)dxa

f(u(t))u'(t)dt0

f(x)dx0

f(ax)dx(xat),a f(a f(x)dxa f(x)dx(xt)a[f(x)(x)]dx 1 dx)aa01sinx(3)In

2sinnxdx0

n1n

4I ,n2(4)

f(sinx)dx220 022

f(cosx)dx;

f(sinx)dx220 02

f(sinx)dx,(5)xf(sinx)dx0 2 0

f(sinx)dx,分部积分准备时“凑常数”f'(xf(x)xa附:三角函数系的正交性:

时,求a

f(x)dx2sinnxdx2cosnxdx2sinnxcosmxdx00 0 02sinnxsinmxdx2cosnxcosmxdx(nm)00 02sin2nxdx2cos2nxdx0四.反常积分:1.(1)a

0f(x)dx,a

f(x)dx,

f(x)dx f(x

a

f(x)dx:(f(x)在xa,xb,xc(acb)处为无穷间断)计算:积分法NL公式4.(1)

1dx; (2)11dx1 xp 0xp五.应用:(柱体侧面积除外)(1)Sb[f(x)g(x)d; (2)S

f1(y)dy;a cS

1r2)d; (4S

b2f(x)1f'2(x)dx2 aV

b[f2(x)g2(x)dx; )

d[f1(y)2dybxf(x)dx(3)V

xxx0

a与Vyy0

y c a(dx)2(dx)2(dy)2(1)yf(x),x[a,b] sa

1f'2(x)dxxx(t)(2)y y(t)

,t[t,t1

] st1

x'2(t)y'2(t)dt(3)rr),,]: s

r2)r'2)d(1)f[a,b] 1 baa

f(x)dx;(2)f[0)

lim

xf(t)dt0

T,(f以T为周期:f 0

f(t)dt)x x T第四讲:微分方程一.基本概念xx(tyDy"(如欧拉方程)令uu(xyyy(xuy1.形:(1)y'f(x,y); (2)M(x,y)dxN(x,y)dy0; (3)y(a)b变量分离型:yf(x)gy)解法:dyg(y)

f(x)dxG(y)F(x)C“偏”微分方程:zf(xy;x一阶线性(重点):y'p(xyq(x)M(x)exp(x)dx

y

1 [

xM(x)q(x)dxy]解(积分因子): (2)变化:x'p(y)xq(y);

M(x) x 00(3)推广:伯努利(数一)y'p(x)yq(x)y齐次方程:yy)x解法:u

yuxu'(u),x

du(u)u

dxxdy a

x

yc

1dx 1

xb2

yc1211N M1全微分方(数):M(x,y)dxN(x,y)dy0且 x ydUMdxNdyUC一阶差分方程(数三):y

ay

0 y

cax三.二阶降阶方程

x1

x bxp(x) x

xnQ(x)bx1.y"f(x):yF(x)c1

xc22.yf(xyyp(xydpf(x,p)dx3.yfyyypyypdpfy,p)dya(xyb(xy'c(xyf(x)齐次解:y0

c1

y(x)c1

y(x)2非齐次特解:y(x)c1

y(x)c1

y(x)y*(x)2常系数方程:ayby'cyf(x)特征方程与特征根:a2c0非齐次特解形式确:待定系; 附:f(x)keax的算子)欧 拉 方 程 ( 数 一 ):

ax2ybxy'cyf(x) , 令xetx2y"D(D1)y,xy'Dy五.应用(注意初始条件):注:切线和法线的截距a

f(x)dxF(x),F(a)0的方程含双变量条件f(xy)的方程Fmadvdt

d2xdt2Q路径无关得方(数): x

Py幂级数求和; (2)方程的幂级数解法:ya0

axa1

x2 ,a0

y(0),a1

y'(0)

第五讲:多元微分与二重积分一.二元微分学概念ff(x0limf, f

limxf,

flim y

xf

x xy df,

y fdf

(判别可微性)x,y 0y), fx,y 0y), ff(x x,y), ff(x,y y)x00y0 0注:(0,0)点处的偏导数与全微分的极限定义:f(x,0)f(0,0) f(0,y)f(0,0)f(0,0)x x0

, f(0,0)limx y y0 y

xy (0,0)(1)f(x,y)x2y2

:(0,0)点处可导不连续; 0 ,(0,0)x2y2x2y2(2)f(x,y) 0 ,二.偏导数与全微分的计算:

:(0,0)点处连续可导不可微;显函数一,二阶偏导:zf(xy):(1)xy型; (2)zx

(x,y0 0

; 复合函数的一,二阶偏导(重点):zf[u(x,yv(x,y)]熟练掌握记号f', f', f", f", f

的准确使用1 2 11 12 22F(x,y,z)0(1)形:*F(x,y,z)0; *G(x,y,z)0

(存在定理)Fdx

dyFdz0 导)注:(xy0 0

)z0

的及时代入

x y z三.二元极值(定义?);(1目标函数与约束条:zf(x,y)(x,y)0, 或:多条)(2)求解步骤:L(xyf(xy(xy(1)zf(x,y)MD{(x,y)(x,y)0}(2)实例:距离问题四.二重积分计算:(1)d,D对称性(熟练掌握):*D域轴对称;*f奇偶对称;*字母轮换对称;*重心坐标;D;2“分块”积分:*DD;21

*f(x,y)分片定; *f(x,y)奇偶D(2极坐标使(转): f(x2y2)附:Dxa)2yb)2

R2; D

x2y2

1;a2 b2双纽线(x2y2)2a2(x2y2) D:xy1单变量:f(xfy)利用求积分:要求:题型

(kxky)dxdy,且已知DS1 2

与重心D(x,y)f(M)d:D):1.“尺寸”:(1)dSDD

; 第六讲:无穷级数(数一,三)一.级数概念1.定义:(1){an

},(2)Sn

aa aa;n

(3)limSn n

(如n1

n)(n1)!)(1)lim

;(2)qn(或 1);(3“伸缩级:(

a收敛}收敛.

n n an

n1 n n性质:)收敛的必要条件:lim

0;n n(2)加括号后发散,则原级数必发散(交错级数的讨论);(3)s s,a 0s ss s;2n n 2n1 n二.正项级数1.正项级数:(1)定义:a 0; (2)特征:Sn n界)

; (3)收敛S M(有n2.标准级数:(1)

1, (2)lnkn, (3)np n

1nlnkn2aba2b2alnbblna)

k 1(估计),如

f(x)dx;

P(n)n np

0 Q(n)nun比值与根值:*nun

un1 *lim

(应用:幂级数收敛半径计算)nu nn(1n1a(an n

0)1.“审”前考察:(1)an

0? (2)an

0?; 注:若liman11,则u 发散na nn2.(1)(1n11

; (2)(1n11

; (3)(1n1 1n np lnpn前提:a 发散; (2)条件:an n

0; (3)结论:(1n1a 条件,,a n收敛.加括号后发,则原级数必发; (2)s s,a 0s ss s.2n n 2n1 n注意事项:对比

a;(1nan

;an

;an

之间的敛散关系(1)axn, (2)a(xx)n, (3)a(xx)2nn n 0 n 0结论:xx*敛Rx*x0

; xx*散Rx*x0注:当xx*条件收敛时Rxx* n a n n注(1)

naxn

nx n

ax同收敛半径n(2)axn与a(xx)2n

之间的转换n n 0,R1ex1x1x2 x,R12! 3!,,R(exex)1 x2 x42 2! 4!,,R(exex)x x3 x52 3! 5!,,R,Rsinxx x3 x5 cosx1 x2 x4 ;1 3! 5! 2! 4!1 1xx2 ,x(1,1); 1xx2 ,x(1,1)1x 1x1 1ln(1x)x x2 x3 ,x(1,1]2 31 1ln(1x)x x2 x3 ,x[1,1)2 31 1arctanxx x3 x5 ,x[1,1]3 5分:f(x)g(x)h(x)(注:中心移) 特

1ax2bx

,xx)0考察导函数:g(x)考察原函数:g(x)

f'(x)f(x)xg(x)dxf(0)0xf(x)dxf(x)g'(x)0幂级数求和法(注:先求收敛域,(1)S(x),(2)S'(x)

,(注意首项变化)(3)S(x)()',S(xS(x的微分方程a

axn

S(x)a

S(1).n n n方程的幂级数解法(1)复:A(1p)n; (2)现: A(1p)nT)0傅氏级三角级): S(x)a02Dirichlet(1f(xS(x)(2)S(x)1[f(x)f(x)]2

ann1

cosnxbn

sinnx1a 11

f(x)cosnxdx

a 0

f(x)dx,

n 11b

,n1,2,3,f(x)sinnxdx n4.题型:(注:f(x)S(xx?)

(1)T且f(x) ,x(,](分段表)(2)x(,x[0,2](3)x[0,正弦或余弦*(4)x[0,](T)*5.T2l06.附产: f(x)S(x)a02

ann1

cosnxbn

sinnxa 1S(x) 0 acosnxbsinnx [f(x

)f(x

)]0 2 n1

0 n 0 2 0 0第七讲:向量,偏导应用与方向导(数一)一.向量基本运算kakb;(平行ba)1 21a;(单位向方向余) a0

a (cos,cos,cos))a3.ab;(投:(b) a

aa

abab0;夹角:(a,b)abab)ab)ab(a,b)abab)ab)二.平面与直线1.平面M0

(x,y,z0 0

)n(A,B,C)方 程 ( 点 法 式 )::A(xx0

)B(yy0

)C(zz0

)0AxByCzD0其它:xy

1;三点式a b cLM0

(x,y,z0 0

)s(m,n,p)L

xx0

y

zz0m n pAxB

y

zD0一般方程(交面式):A

1 xB

1yC

1zD 02 2 2 2其它: *二点式; *参数式;(附:线段AB 的参数表示:xa(aa)t 1 2 1yb(bb)t,t[0,1])z 1 2 1c(cc)t1 2 1平面束方程::Ax

y

zD

(

x

y

zD

)0AxAxByCz D0 0 0A2B2C2M0

(x,y0

)到平面的距离d三.曲面与空间曲线(准备)曲面(1)形式:F(x,y,z)0 或zf(x,y); 注:柱面f(x,y)0)(2)法向

,Fnn(F,F

)(cos,cos,cos) (或n(zx

,z

1))y

xx(t)

F(x,y,z)0

:yy(t),或 ;

zz(t)

G(x,y,z)0s{x'(t),sns{x'(t),snn)1 2应用(3四.常用二次曲面1.圆柱面:x2y2

R22.球面:x2y2z2R2R2(x2y2)变:x2y2R2(x2y2)x2y2z2x2yx2y2

2az, (xx0

)2(yy0

)2(zz)20

R2x2y2变:x2y2z2, zx2y2抛物面:zx2y2,变:x2y2z, za(x2y2)双曲面:x2y2

z21马鞍面:zx2y2zxy五.偏导几何应用曲面(1)法向:F(x,y,z)0(2)切平面与法线:

,Fnn(F,F

, 注:zf(xy

nn(f,fx y曲线(1)切向:xx(t),yy(t),zz(t)

ss(x',y',(2)切线与法平面snn3.综合:snnG0 1 2六.方向导与梯度(重点)l定义(条件):lmn,p(cos,cos,cos)计(充分条:可): uul x

cosuyz

cosuz

cos附:zf(x,y),l0{cos,sin}l

f cosfx

sin

f cos22f sincosf sin22l2 2G:G:zf(x,y)Ggradz(f,f);x yuf(x,y,z)Ggradu,u,u)x y zGGl0;ulGG(M)0第八讲:三重积分与线面积分(数一)一.三重积( fdV )对称重点)::关于坐标; 关于变; 关于重心投影法:Dxy

{(x,y)x2y2R2}z1

(x,y)zz2

(x,y)(3)截面:D(z){(x,y)x2y2R2(z)}azb(4)其它:长方体,四面体,椭球f(1)单变量f(z),(2)f(x2y2),(3)f(x2y2z2),(4)faxbyczd“积”: *dv; 利用对称(重)Iba

fdxdy(细腰或中空,f(z,f(x2y2)D(z)投影法(直柱体):I

dxdyz2(x,yz(x,y)

fdzD 1xyI2dnd

f()2d,0 0 0faxbyczd):IaxbyczdGauss 公式二.第一类线积分(fds)LdsL; 对称; 代入“L”表达式Lxx(t)计算公: y y(t)

t[a,b]fdsaL

f(x(t),y(t)) x'2(t)y'2(t)dt

(axbyc)ds(axbyc)L;AAdsAdrL应用范围

L Lzx,ydsL三.第一类面积( fdS)(1)dS;(代入F(x,yz)0)(2)对称性(如:字母轮换,重心)(3)分片(1)zz(x,y),(x,y)D If(x,y,z(x,y)) 1z2z2dxdyDxyDxyAndSAdS(2)与第二类互换: 四:第二类曲线积分(1):

P(x,y)dxQ(x,y)dy LLxx(t)直接计算:y y(t)

,t:tt1

It2[Px'(t)Qyt1常(1)水平线与垂直; (2)x2y21PdxPdxQdyL(1)

(QP)dxdy;x y(2)

D : *P

Q换路径; *P

Q围路径L(AB

y y y y(3)L

(Q Px

但D内有奇点)L L*(变形)

Qy yPdxQdydu

uBA

(道路变形原理)L(AB)P(xy)dxQ(xy)dyf待定):微分方程.L应用功环流):IFdr (有向,F(P,Q,R),drds(dx,dy,dz))五.第二类曲面积分:

PdydzQdzdxRdxdy,或R(x,y,z)dxdy (其中含侧)

R(x,yz)dxdy,其中zz(x,y)注:垂直侧面,双层分隔合一投影(多项,单层):n(zx

,zy

,1)PdydzQdzdxRdxdy[P(z)Q(z)R]dxdyx y 化第一(不投影): n(cos,cos,cos)PdydzQdzdxRdxdy(PcosQcosRcos)dS Gauss P Q R散度计算:divA

xyzGauss 公式:封闭外侧,内无奇点PdydzQdzdxRdxdydivAdv (含奇点)注: 补充“盖”平:; (含奇点) 0AdS 有向nAP,QRdSndS(dydzdzdxdxdy)六:第二类曲线积分(2):

P(x,y,z)dxQ(x,y,z)dyR(x,y,z)dzFdr参数式曲线Fdr注(1)当rotA0时,可任选路; (环流):IRA(x,y Stokes公RA(x,y

,z)(P,Q,R)n{F,Fn{F,FG 0

,F}或x y z{G,Gx

,G};Adr(ndSStokesafdS

PdydzQdzdxRdxdy;(bR(x,yz)dxdy;(c 高数重点知识总结1(y=arctanx(y=lnx(y=xyax)，三角函数(y=sinx)，常数函数(y=c)2、分段函数不是初等函数。3、无穷小：高低阶低阶 例如：

x2xlimx1x0 x x0x4、两个重要极限：

limsinx1x0 x

(2)lim1x1exx0x

lim1x

1x ex经验公式：当

xx0

,f(x)0,g(x)

lim1f(x)g(xxx

limf(x)g(x)x

0lim3xlim1

e

xe3x0

x x0 5、可导必定连续，连续未必可导。例如：y|x|连续但不可导。6

f(xx)f(x)

f'(x)

lim

f(x)f(x)0

f'xx0

x x

xx 007dfg(x)f'g(x)g'(x)dxx x x

,y' 1112 x2 x x2 x14 x2x xx2y21解：法左右两边同时求导2x2yy'0y'xy法(2),左右两边同时微分,2xdx2ydy

dyxdx yyg(t) dy9、由参数方程所确定的函数求导：若 ，则

dy/dt

g'(t)，其二阶导数：xh(t) d(dy/dx) dg'(t)/h'(t)

dx dx/dt h'(t)d2yd

dy/

dt dtdx2 dx dx/dt h'(t)10、微分的近似计算：f(x0

x)f(x0

)xf'(x0

)例如：计算sin3111ysinx（x=0x函数可去间断点，ysgn(x)（x=0是函数的跳跃间断点(2间断点；例如：f(x)sin1（x=0是函数的振荡间断点，y1（x=0是函数的无穷间 x x断点）12、渐近线：水平渐近线：ylimf(x)cx铅直渐近线：limf(x)，则xa是铅直渐近线.xa斜渐近线：设斜渐近线为yaxb,即求alim

f(x),blimf(x)axx3x2x1例如：求函数y 的渐近线x21

x x

x13、驻点：令函数y=f(x)，若f'(x0)=0，称x0是驻点。14y=f(xx0u(x0,δx∈u(x0,δ)，都f(x)≥f(x0x0f(x)的极小值点；否则，称x0f(x极大值点统称极值点。15、拐点：连续曲线弧上的上凹弧与下凹弧的分界点，称为曲线弧的拐点。16、拐点的判定定理：令函数y=f(x)，若f"(x0)=0，且x<x0,f"(x)>0；x>x0时，f"(x)<0或x<x0,f"(x)<0；x>x0时，f"(x)>0，称点(x0，f(x0))为f