大一高数期末复习重点

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᩽▲᩽▲ᙠᑣᨵ᩽▲.sinx.rಗlk=l⌕᩽▲,Ilim(l+J_)x=elXfoovᨵ▲Ḅ!"ឋ$‘&ᨵ'(Ḅ!;ᜧI"+▤-▤.▤,/0,k▤1

1ᵨx—40,eX-1xax-1Inasinx~xarcsinx~xtanx~xarctanx~xln(l+x)~x(1+x)a-1ax1-COST~__tanx—sinx~-2

2প456789;(2).◀ᨬ+<=>(3)?ᑖ;(4).AᐳC7D>(5)ᑭᵨGHឋ$园(6)IᔠKᦪM᩽▲9ᑣ群笳NᑭᵨOஹQ᩽▲MᑖRKᦪ᩽▲;同空(8)ᑭᵨᜳTUᳮ>刻(9)ᑭᵨ⌕᩽▲;苒(10)ᑭᵨ/0Wᣚ;(11)ᑭᵨKᦪḄឋ$(Wᐭ9);(12)ᑭᵨ\]9ᑣ.

3_HV1+tanxvl+sinxmoe^tanx—e^sinxtanx-sinx=lim------/-----------------tanxsinx1஺(V1+tanX+V1+Sinx)(e-e)11.=limtanx-sinx_limtanx—sinxtanxsinxtanx-sinx2x^0e-sinxe(e-l)`xf0,etanx-sinx-1~tanx-sinx,]ᦑbD=limtanx-sinx~2xfoesinx(etanx-sinx-1)1,■•=limtanx-sinx

42x->oesinx(tanx-sinx)2

5d⌕Ḅᓫf᩽▲(஻>1)lim஻"=0,lima*=8,%—0
Xfo+1R1Rlimarctan
=

limarctan_=_.1o-x2io+x2
◤⌕hiḄ᩽▲Xf—8jkx-»+ooDj^

6Hm/(x)=/(x)0x-»x"ḄḄḄr0ெOஹQ
mn"p5q,rsq1mnoḄᑖ,tmn"qu₩q1wᨬᜧᨬxUᳮwyzmKᦪḄឋ$jᨵ'ឋw
x
ᳮ.6oUᳮ

7_M/(%)=
tḄmno}ᢣᐸpl-elx`=0"=1,KᦪU,!KᦪḄmno.1%=0,ᵫlim/(x)=lim-------=ᔩ10xf0-1,-e1-Xᡠ*=0!KᦪḄtmo,!q.x=1,ᵫlim/(x)=lim_____=0xfr—11+001lim/(x)=lim______=1Xfl+1-xf-001-eᡠx=1!KᦪḄ
mno,!rsq.

8_M/(#=£ᓝ:ஹḄmno}ᑨᐸq.k+xT)=-1,%=1/=0!mno-llim"=Isinl,x='-11x|(x>4)(x-1)2x=_l
p5mnox=l,lim/(x)=00,x—>1X=1tmnox=0,limf(x)=-1,limf(x)=1.x->0+x->0-%=0
rsmno

92X—11_My=~[——+sm(x-l)sin------Ḅmno,1x-12X+1}ᑨnᐸq.px=0,x=l!pḄmno.(1)ᙠx=0ᜐlimy=_1+sin2(-l),limy=1+sin2(-1)x->0-x->0+7ᙠx=0ᜐḄOQ᩽▲¢ᙠ£¤¥/ᡠ=0KᦪḄ
mno!rsmno.

10ফᙠ=1ᜐ,limy=lim[Tsin(x-1)sin]=X1§—^1Xߟ^1-Xx—132+1ᓽᙠ1=1ᜐKᦪḄOQ᩽▲¢ᙠ¥/ᡠ=1!KᦪḄ
mno!p5mno.

11a(l-cosx)X<•2_©Kᦪ/(x)=1,x=ᙠx=0ª«|஻=2,b=e.®¯f(0)=lim"(1-cQs")=a2x->o-x2f(0+)=limIn(b+x2)=lnbx-»0+஻11A2—=1=111)1
COS%—x22

12_°±/(%)=7"u0,x=0ᙠx=0ᜐḄឋp²ឋ.²r<0_᝞´/(%)=µ'ᜐᜐp²·¸[5(1-)X>09()(A)a=b=l;(B)a=-2,b=-l;(C)a=l^b=0;(D)஻=0,)=1.

13t²ᦪ¹ᑖO²ᦪ('(%),Q²ᦪ4'(%)U3ஹ²ᦪᙠḄᐙ⌕ᩩ¼²ᦪீ¾¿iᑗÁÂ᳛᝕=1'(1)I0ŲឋឋḄᐵÇp².¹ᑖ!M¹ᑖdy=fr(x)dx0"Ip²=p¹

14ᢥUM²IM²ᦪÊ9IᔠKᦪM²«◚KᦪÍᦪÊÎM²dᦪ9M²ᑖRKᦪᙠᑖRoMÏ1+▤²ᦪ(sinx,cosx^e,---)ெ1-x

15fX=(p(t)ÍᦪÊÎீM²ᦪdj-d--j___d_r_____♦_'_(_)dxdx'পd(᝞)d(%(ὡ2d(")dxe'(ÑdJ=dx=df=dldx2dxdxdxddt

16Ò¹ᑖÓxUᳮÔᐸÕᵨ"wÖ×UᳮÓxUᳮீIØᘤÚUᳮÛÜÓxUᳮIÝÞUᳮ"ÝÞßDàáâ᪍ßD1\]9ª«1
äH°,,8-8/8/çUq᩽▲1000ÓxUᳮḄÕᵨwèé¤/Dbw°±ÊÎ᪷Ḅᙠᦪ

17/r’KᦪḄᓫìឋ"ᑭᵨ²ᦪᑨn1n函KᦪḄ᩽x1íᖆᙠḄ⌕ᩩ¼^n数—IU性---------1᩽xᙠḄᐙᑖᩩ¼态KᦪḄïðឋ"ñoïðឋᑨ91KᦪḄᨬᜧᨬxKᦪḄòóÁ"ôᚖö1

18÷Peanoqø⚗ḄÝÞßD©/(x)ᙠúxḄzmᑁᨵ஻▤0²ᦪ,ᑣdxᨵ/(X)=/(x)+/\x)(x-x)+^\x-x)20000")(%)4-----F,("—x0)"+o[(x-x0)"].

19ᵨᦪḄ᪍352n+lsinx=x-—+--------+——+o(x2n+2)3:5!(2n+1)!

20(1)：，(six)o+six+：・+z+x+JxH+I+XII)I

21\]9ᑣ஺
2üýq000þ:0.oo,00-00,0°,1°°00°99ᑣlim/(%)_limf'(%)g(x)/(%)•(1)᩽▲ᙠ,ᵨᑣ
(2)ᙠ᩽▲
⌕ᵨᑣ
(3)ᙠᵨ,⌕〉Ḅᓄ"
(4)ᙠᵨ,#$ᐸ&᩽▲'()ᔠ.

22-ᳮ(01ᐙᑖᩩ5)7஻x)ᙠ:;U(Xo)ᑁ,(a)x0;GXு/,ᨵf(x)<0,ᑣf(x)ᙠ/ᜐK᩽ᜧM.(b)x*o,ᨵf(x)>0,ᑣ/(>ᙠ/ᜐK᩽OM.(c)Qf(x)ᙠ:;U(%)ᑁST(U
ᑣf(x)ᙠX஺ᜐW᩽M.

23-ᳮ(0Xᐙᑖᩩ5)7f(x)ᙠYᜐᐹᨵX▤\ᦪ
^,(%)=0,`(a)b0,ᑣ(«)dX)ᙠ*0ᜐKe᩽ᜧM
3)r(x0)>0,f(x)ᙠX஺ᜐKe᩽OM.

24᩽MḄgh:a.\ᦪ
'(*)jb.kl('f{x}=0Ḅ᪷)no(x)pᙠḄl.C.qrf(x)ᙠ'ᡠᨵltḄuvT
ᡈ`(X)ᙠxlḄST
ᑨz᩽Ml.d..᩽M.

25{|}Ḅভ{|}Qᦪdlimf(x)=a,X—>00(-00,4-00)ᑣᦪ/(x)Ḅ}ᨵ{|}y=a.(b)ᚖ{|}Qᦪf(x)limf(x)=00,x->x(xojxii)oᑣᦪ/(x)Ḅ}ᨵᚖ{|}X=x.o

26⚪=2x<1[.7y/(x)=11+x2
ᦪᙠ[ax+bx>lx=lᜐ\
-,b.a2.1imp-2lii(l+—PxXfoo|_XJ3.᩽▲sinx-x(l+x)i(2)lim(e%—1—%(1)lime

27

28⚪1.ᵫឋ
ᨵHm/(x)=lim/(x)=/(l)Xf1-x-»l+.a+b=l(1)ᵫ\ឋ
ᨵlim/(%)-/প=limপx-»rx—111+X—12-1y-1=Hml+%21nXfl+X-lX-12-11------J4x/.a=lim1+x_lim=-1T-X-lar(1+X2)2

29஻=—1ᵫ(1)=2.r1"|l-xln(l+~)2.limix-x2ln(l+)|_lim-----------------x->ool“Jx-»oonX]XY1ᑣ=lim_______X->0t1-=lim^-ln(1+^=lim1+/2r->otr-»o2t=lim1+/1_1--------72t(l+t)2

303.পeஹinx+excosx-1-2xX11-------------------3x2=lime"(sinx+cosx)+ex(cosx-sinx)-256x=limcosx-la஺3x_Hm-exsinx+excosx-o3~3

312:ex(sinx-x)ex-l-xl=lim-------z-------+------2——xcosx-1ex-llime*lim------—+lim--------2xfOx->03xx-»o623

32

33ᑖ'ᭆ¢£ᦪ
p-¤ᑖJ/x
dx
ឋ¥£a\
¦ᑖ§¨ᐵª;}ឋ«ឋ¬1®ᣚᐗ£±¦ᑖ¬3.ᣚᐗ¤ᑖ,¤|0X®ᣚᐗ£²³´ᣚ᎔´ᣚ¬ᑖ·¤ᑖ1f¹º»Ḅ¤ᑖᨵᳮ¸ᦪḄ¤ᑖ1¼ᓄ½ᨵᳮᦪḄ¤ᑖ

34

35J.3-i-ij-1Y-dx=—\dx2Jx4+1+C(xwO)

36¾Jmax(l,|x|dx.tx<7f(x)=max{l,|x|},ᑣdx)=<1,-11/(%)ᙠ(-00,+00)
ᑣ¿ᙠᦪF(x),—.x<-121F(x)=jx+C,-11I2ÀWxÁᜐᜐ
ᨵ

37lim(x+C2)=.(1-x2+c\ᓽ_]+G=__+G,Drx->T+x-»-l-22lim(x2+C)=lim(x+C),ᓽ+C=1+C,X->1+2x->l~2232ὶÄÅG=c,e஺2=Æ+஺
c.=i+c.----+(JX<-121ᦑJmax{l|x}dx=1

38!ᑖᦈ¤ᑖḄ-ÊËÌ#Êឋ¥£}ឋ
ͨÎឋ
ÏÐឋ¥¬-¤ᑖÑM-ᳮ«(஺-a)

39oᣚᐔrJ/£x¬dx%=0Q¬jIbbbெᑖ·¤ᑖ஻1Jvdw`ʤᑖoÛÜÝᑖ

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