数学分析试题

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ᦪ§1ᦪᭆᑡ:MIIMTW+z+…ᓃಘ+ᕜ+…+IX)+x+•••X+XI>IX|-(IXjI+IxIH---Fix").2;J23..a+b\a-bmax(a,b)=a+b\a-b\min(஻,b)4.789:;Ḅ=ᩩ?ᑖAB஺Db,EFGHḄᜳ:B0,J.K9:;Ḅ☢s(0),O.ᐸQRS.5.ᙠVWBrḄᳫᑁ[ᐭ]ᑁ^ᙊ`aJbᙊ`Ḅcd⊤BᐸfḄᦪaO.KᦪḄQRS.6.hiᐳklmnᐰpB20km,ἠstQ᝞vwᙶ5kmy(ᒹ{5km)ὅᦈ~1ᐗ5kmᙠ15kmy(ᒹ{15km)ὅᦈ~2ᐗᐸᦈ~2ᐗ5:.Jbἠs⊤BmḄᦪaOᦪḄ;.7.]ᾭᘤ]9:.EHrḄᓄtB/(/),9:ᑖAᨵᐵ/(0)=0,/(10)=20,f(20)=0,.f(t)(0

110.")=*■ᙠ(-8,+8)ᨵµ.11.ᵨ·Q¸¹º»ᦪ¼µaOf(x)=—ᙠ(0,1)¼µ.X12.J=ᏔᦪḄwd¬Ꮤᦪa=᜻ᦪḄwd¬Ꮤᦪa]᜻ᦪD]ᏔᦪḄwd¬᜻ᦪ.13.½f(x)BQRᙠ(-00,+00)ᑁḄ¾¿ᦪa/(x)ÀᑖÁᡂ᜻ᦪDᏔᦪGD.14.ᵨ·Q¸¹º»vᙠ(-co,+oo)Ã(1)f(x)¬᜻ᦪ(2)/(x)¬ᓫÅÃᓣᦪ(3)/(x)¼ÇÈ(4)/(x)¼Ãµ.§2ÉᔠᦪËÌᦪ1.½f(x)=1X,./(/(x))=X.1+X2..ᑡᦪḄÌᦪÍᐸQRSv(1)y=—(x+—),10.**)[-x2,x>0..Éᔠᦪ/(g(x))ag(7(x)).5.½f(x)=-^=,.(f஺/஺…஺/)(x).J"'~~'6.½/(x)=ll+xl-ll-xl,J.(/஺(஺…஺/)(x).7.½/(%)=—5—a.f(fM),/(/(/(x))),/(-5—).1-xf[x}§3Øᦪ2

21.ᑡᦪᑖAÙÚᦪḄQRSDÛSa᜻Ꮤឋaᕜ®ឋaᨵµឋaOᦪḄ;v(1)y=lxl(2)y=x-[x](3)y=tan1x1(4)y=y]x(2-x)(5)y=sin2x(6)y=|sinx|+1cosx\.2.78ᦪy=Ḅ;aᦪ=|/(^)|ay=f(-x)ay=-f(-x)23Ḅ;aOÝ%,%,%Ḅ;ËyḄ;Ḅᐵ.3.78ᦪ஻x),g(x)Ḅ;aJᦪy=g[/(x)+g(x)±|/(x)-gu)|]Ḅ;aOÝyḄ;Ë/(x)ஹg(x);Ḅᐵ.4.ᑡᦪḄ;v(1)y=xsinx(2)y=sin—.X5.âãᦪ1,x>0,y=sgnx=<0,x=0,-1,x<0,JᑖAsgnx,sgn(2x),sgn(j-2)Ḅ;.6.ᑡᦪḄ;:Y(1)y=sgncosx(2)y=[x]-2—9᩽▲ËᦪḄçèឋ§1᩽▲é⚪Ḅë§2ᦪᑡḄ᩽▲1.ᵨQRᑡᦪᑡḄ᩽▲BÇv/ì1.«+■(1)lim—~;I00n~+1ersin”(2)hm------n—>oofiJTবlim
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3(4)“T8]|lim(Jn+1-)(5)H—>00lim”;(6)"->81n..n.lim----(Q>1);n->ooa”..n\
lim---n->001+2+3+…+”(9)lim2“TOOn(10)lim($+a~n),a>1.“T8஻2.ᵨQR:3n~+n3(1)lim——-----=is2n-12õNn?+niফhm---------=1“T8஻n-1஻BᏔᦪ,n(3)limxn=\fᐸö”T8H+lnB᜻ᦪ;n3n=3k.3஻+l(4)limxn=3,ᐸö஻=+1(k=l,2,…)a”—>8n2+1.஻n=3k+2.3-y/n+n3.ᵨQR:(1),ᑣ¾]ø᦮ᦪk,ᨵlimJ”->8ফliman=a,ᑣIim|a”|=|al.ÌG¬ᔲᡂú?n->«>"ߟ>8বliman—a,,ᑣýᙠN,ÿ஻>Nᨵa«>b;n—>oo(4)liman=a,4>0,!]lim=\[a.n—>oo“TOOY4.᩽▲Ḅᦋᡂ☢ᔲ!(ᐸ$“&”()*+⊤-.ᙠ”.)পVQO,37V>0,n>Nᔢᨵ\x„~a\<£79Q)Ve>0,37V>0,n>Nᨵ\xn-a\<£7(2)Ve>0,3N>0,9஻ுNᨵ<(M>?ᦪ).4

45.Bx“y“EᦈGHᔲIBᓃEஹBy„/LᦈG!6.Nx^aoo8.Tᑡ᩽▲:lim-…+')7প…1223஻(஻+l)lim(4+—++—!-y);ফ஻T8n-(஻+l)2(2n)211_বlim(+•••+n—\ln2+1y/n2+2J/+஻lim(-+-4+-2n-1ভ••+-----v7…2222"(5)lim(1-cosn7஻q>8(6)lim“T8iim(VTVTVT---V7)7(8)lim[(n4-1)M-nn],08i122n-\im(9)>oo242n135…(2஻-1)(10)lim{246lim|7(11)(12)limyjnInn.9.U~B4EB"E$qᦈGᦪᑡqᦣᦪᑡᑣB஺,,±"E5

5ᦣᦪᑡ7B஺LE]30)ᔲLᦣᦪᑡ!>!10.NS=(q1)",U~B0Eᦣ.11.6%,89…4>ᦪU~limWa7+a<+…+"=max(qd…q”).12.Nlima“=a,U~“T81-[”J(1)lim----=a7஻q>8஻(2)a>0,6>0,IIlim=1.n—>ooY13.ᑭᵨᓫᨵᳮU~limx„.ᙠT(1)x=>J~2,x=y12x_,=2,3,…7l2n1(2)Xj=\Tc>0,x=7c+x»-i»”2,3,…7n(3)4=J(c>0)7n\Y(4)%=L=1+"஻/஻=1,2,….1+K14.¥=a>0,%=Z?>0(aoon?»-><»n15.U~6a஻>0,lim-^-=/>1,lim=0.“TaCU71"Toort+I16.Nlima”=a,U~n—>oo(1)lim%+%+••,+""=“7(Ḅ⌮®⚪ᡂ°ᔲ!)“T8஻(2)6%>0,ᑣlim4q4…=a-”T8Y17.±ᵨ²⚪Ḅ³´U~ᑡᔜ⚪,111[+_+_(1)lim—-----=07n—>oo஻(2)lim#7=1(a>0)7஻T8(3)lim4=176

6(4)=071+J2+-^3+…+\fn~(5)11m---------------------=17஻T8஻(6)lim=a(bn>0),ᑣlim=a."f8b/1->«Y18.ᵨU~ᑡᦪᑡ>¶·ᜧ¹:(1){7Q){஻}(3){Inn};(4)1+W…+L23n19.U~Bx.E>¶·ᜧ¹B»,E>ᨵ¼¹ᑣBx“½>E>¶·ᜧ¹.20.(1)¾¶·ᜧ¹ḄḄ᩽▲᝞À!ÁÂÃᔜÄHឋ!(2)Âö·ᜧ¹¶·¹ḄஹÆஹᖪḄ᩽▲ḄÈ;(3)Âö·ᜧ¹¶·¹ḄÉÊHËḄᔜᑭᘂ.21.ᑭᵨ+=e,Tᑡ᩽▲:஻-8I]n(4)lim“Too§3ÎᦪḄ᩽▲1.ᵨ᩽▲U~ᑡ᩽▲:r—1(3)lim-=—=27TVX-1(4)lim-----------=05x-3(5)limJx2+5=37x->27

7x(x-l)J(6)limXTlx2-l2..xÏlim—---=oo7x—3——x—\(8)lim--=17*fgx+2ÓÔ.X2(9)hm-----=oo7XT00x+\x2-5(10)limV^=l.XT8£qJ2.ᵨ᩽▲Ḅ]ᑣ_`aᑣTᑡ᩽▲..x4(1)hm---------;JO2x"-X-1..x2-iQ)hm---------;—2x"-x-l(X-1)3+(1-3X)বlimKTO×x2-Wভlim——■==—7—IVxJl+x_2rমlim--------713x-3..x2-5x+6(6)lim---------13x2-8x+15rM-1Ïlim---(n,m>᦮ᦪ);limÙÚ(8)14VT-23.N/(x)>09U~:limf(x)=A,ᑣlim=VT,ᐸ$᦮ᦪ஻22..t->RXT%4.U~:lim/(x)=A,ᑣUml/(x)l=IAI,ÝÞßàá.X->XQX->X05.TâᑡÎᦪãᡠ-åḄæç᩽▲0,x>1,(1).f(x)=<1,X=l,ᙠX=1;X2+2,x<1,,1xsinq,x>0,ফ/3)=/Xᙠx=07l+x2,x<0,xl1ব/(x)=-ᙠx=07xl+x291(4)ᙠx=—,n᦮ᦪ:n8

82",x>0,(5)/«=<0,x=0,ᙠx=0.\+x2,x<0,7.ᵨ¼¹éᣚTᑡ᩽▲:(1)limx[~]7I஺’X(2)limxaInx(a>0)7x->(rInr(3)lim—(a>0)7XT+oo££(4)limxx.X—>+»8.Nf(x)ᙠ(a,+oo)²ᓫ²ᓣlimx=+oo,lim/(%஻)=4,TU:limf(x)=An(A>¶·).9.Nf(x)ᙠìᔠX²ᑣf(x)ᙠX²¶Ḅᐙ⌕ᩩñ.ᙠx.wX,஻=1,2,…òlim"(x)I=+8.”q>810.ᑭᵨó⌕᩽▲T᩽▲:sin2xপlimATOx×sinx1ফlim------i஺(sinx)tan3x(3)lim---------sin5x9

92sinx-sin2x(4)limA-*0X3cos5x-cos3xমlim.10tanx-sinx(6)limATOarctanxÏlimKTOxsin4xôlimKTO\lx+1-1V1-COSX2õlimKTO1-COSXCOS(Harccosx),ᙠᜧøஹ(10)lim——---------(஻>᜻ᦪ)7x->0xtanx-1(11)limT71X---4..sinmx/1.Lúûü(12)hm-....(ý,஻>᦮ᦪ)7xTksinnxcosx(13)hm-----,Yr---2(14)limxsin-7A'->-KDX(15)lim[cosV«+1-cos\Tn];X->+a)(16)limsin(ÿJ+1)(஻᦮ᦪ)limfl--(17)X(18)lim(l+஻x)x(஻᦮ᦪ);x->0(19)lim(1+tanx)CMxA->0(20)3x+2(21)lim(■31(22)lim(sinx),anx2(23)limXT610

10(24)lim.-1)11.!"limcosl%&ᙠ.KTOX12.!"limD(x)%&ᙠ(ᐸ*XT.%.(ஹ[1,Xᨵᳮᦪ,X=[0,x4ᳮᦪ.13.5᩽▲XXXhmcos—cos---cos—.:242஻14.ᵨ<=!">(1)?lim/(x)=+co,limg(x)=4,ᑣlim"(x)+g(x)]E+8x->“x->a(2)?lim/(x)=+oo,lim^(x)=A(>0)9ᑣlim"(x)g(x)]=+8.x->aATa15.?limf(x)=A,limg(x)=B,!">lim[f(x)g(x)]=AB.16.!"limf(x)=AḄᐙ⌕ᩩOP>QRSᦪᑡx,->+oo(஻->oo),ᨵ-A(஻-8).17.!"lim/(x)=+ooḄᐙ⌕ᩩOP>QRSᦪᑡX”f/(஻W°°),ᨵA(77->00).18.YZᦪ/(x)ᙠ(0,+oo)[\]^_f(2x)=/(x),`limf(x)=A,!":/(x)=A,xe(0,+oo).§4ZᦪḄcdឋ1.ᵨ<=!"fᑡZᦪᙠ<=gᑁcd>(1)y=\/~x(2)y=—;x(3)y=lxl(4)y=sin—.x2.ᢣkfᑡZᦪḄlmnop"ᐸqr>11

11(1)/(x)=X+—;XX(2)/(x)=-J+X)2'(3)/(x)=cos2-X(4)/(x)=[x]+[-x]ஹsinx(5)(6)/(x)=sgn|xl(7)/(x)=sgn(cosx)1(8)fW=Inxx,1x1<1,(9)((x)=<1,|xl>l;cos—,1xy1,(10)f(x)=•2|x-l|,IXl>1sin7TX,ᡀᨵᳮᦪ,(11)/U)="0,x4ᳮᦪX,Xᨵᳮᦪ((12)/(x)=<-X,X4ᳮᦪ.3.|x=0}fᑡZᦪ4<=(~<=/(0)Ḅ(/(x)ᙠx=0cd:(2)=X(3)/'(x)=sinxsin—xI(4)((x)=(l+xF.4.Y/(%)PcdZᦪ(!"QRSc>0,Zᦪ-c,f(x)<-cg(x)=(/(Al/(x)|cPcdḄ.5.?/(x)ᙠ/ncd(|/(x)|f\x)Pᔲ:ᙠX஺ncd᝞S6.?Zᦪf(x)x=0ncd(g(x)ᙠx=0n%cd(EZᦪḄஹᙠX஺nPᔲcd?/(X)g(x)ᙠX஺n%cd(EZᦪḄஹᙠX஺nPᔲ%cd12

127.!"?cdZᦪᙠᨵᳮnḄZᦪ0,ᑣZᦪឤ0.8.?/(x)ᙠKa,bLcd(ឤ(ᢥ<=!"—ᙠKa,bLcd.9.?f(x)g(x)ᙠa,bLcd,~!"max(/(x),g(x))min(/(x),g(x))ᙠKa,bLcd.10.!">Yyl(a,b)[ᓫZᦪ(?xe(a,b)f(x)Ḅlmn(ᑣPo/(x)ḄWqlmn.11-?/(%)ᙠKa,bL,a?f(x)ᙠa,bLcd(`%&ᙠxwa,b£,/(x)=0,ᑣf(x)ᙠKa,bLឤᡈឤ¥.14.Y/(X)Ka,bL[Ḅ⌴§Zᦪ(gKf(a),f(b)L,!"f(x)ᙠKa.bL[cd.15.Y/(x)ᙠa,+oo)[cd(`0V/(x)4x(x*0),?a,>0,4+i="a")("=1,2,…).5!:(1)lima&ᙠn—>cc(2)Ylim%=/,ᑣ/(/)=/(3)᝞ª«ᩩOᦋ04/(x)0),ᑣ/=0.16.5fᑡ᩽▲>l+.t(2)lim(arctanx)cos—1(3)lim(cosx)x2XTOe'cosx+5(4)hm-------------.J஺1+x+ln(l-x)17.!"^_x3+px+q=o(>0)ᨵ`±ᨵW²³᪷.p§54µ¶·4µᜧ·Ḅ¹º1.|x-0}(»x᪗½5fᑡ4µ¶·Ḅ▤:(1)sin2x-2sinx(2)J--(1-x)1+x13

13বJ'lxlWE;(4)V1+tanx-Vl-sinx;(5)In(1+x)(6)V5X*2-4?ÀVl+7-1(8)e'-1.2.|xf±oo}(»x᪗½5fᑡ4µᜧ·Ḅ▤:(1)X2+X6<ফ4x2+6x4-x5ksing(3)ভx3+1মX2+2X-3(6)x2arctan—.x3.|x-0}(fᑡÆÇᡂÉᔩপo(x2)=o(x)ফ0(x2)=o(x)বxo(x2)=o(x3)(4)(5)(6)o(x)=O(x2).4.~!fᑡᔜ⚪>3(1)xsin\/~x=0(x^)(x7()+)ফ2x3+2x2=0(x3)(x—>co)বo(g(x))±o(g*))=o(g(x))(x-%);(4)o(x"')+o(x")=o(x")(X->0),/H>M>0মo(xm)o(xn)=o(xm+n)(X->0),/H>/I>0.5.!"fᑡᔜÇ>পtanxx(x—>0)14

14(2)arcsinxx(x—>0)(3)arctanxx(r—0)1ஹ(4)1-cosx—x"(x—>0)2(5)cx—1x(x—>0)(6)(l+x)a-1ax(x-0),ᐸ*awO.6.ÑᵨÆÒ4µ¶·5᩽▲>c12arctanW(1)lim-------zooX-COSX—1(2)hm---------XT°1-cosx(4)lim^—.XT°xsinx7.Yf(x)g(x)(x->x),!">0fM-g(x)=o(f(x))ᡈ/(x)-g(x)=o(g(x)).8.Yx-^a}(/1(x)f(x)ÓÆÒ4µ¶(&(x)g2஺)PÆÒ4µᜧ(`2\imf(x)g(x)&ᙠ(5!22limf(x)g(x)=limf(x)g(x).KTaxIKT"22Ö×ØᖪØᑖ§1ØᖪᭆÜÝᐸÞß1.5àᱥây=fᙠ4(1,1)n3(-2,4)nḄᑗâ^_äâ^_.1)2.?5=-/gå(5(1)ᙠ1=1,1=1+ælḄçᙳéê(Y4=1,0.1,0.01)(2)ᙠf=1Ḅë}éê.3.~ì<íây=InxᙠîïnḄᑗâçðñfᑡòâ>(1)y=ó-1(2)y=2x-3.15

15x2,x>34.Y/a)=ax+b,x<3,~ì<Ḅ(/(x)ᙠx=3ᜐõö.5.5fᑡíâᙠᢣ0,ফ/(x)=]x<0;"isin—xw07.YZᦪ/(x)={xx'(m᦮ᦪ).0,x=0~>(DmÆñS}(/(x)ᙠx=0cd(2)mÆñS}(/(x)ᙠx=0õö;(3)mÆñS}(/'(x)ᙠx=0cd.^(x)sin—,xwO.8.Yg(O)=g'(O)=O,/(x)=Jx0,x=Q.5.0).9.!">?å(X஺)&ᙠ(ᑣ]æ/(/+¶)W/(*஺W.)=/'(X).0-2Ax10.Y/(X)P<=ᙠ(-00,+00)[ḄZᦪ(`QRûX[/2€(-8,+8),ᨵ/(ý+x)=/(x)/(x).212?f'(0)=1,!"Rûxe(-00,+00),ᨵf\x)=f(x).16

1611.Yf(x)PᏔZᦪ,`ÿ(0)ᙠ,/(0)=0.12.f(x)᜻ᦪ/'(%)=3,/'(/).13.ᵨ!ḄᏔᦪḄ!ᦪ᜻ᦪ!Ḅ᜻ᦪḄ!ᦪᏔᦪ.14.%ᑡᦪḄ!ᦪ(1)y=x2sinx+(2)y=xcosx+3x2+(3)y=xtanx-7x4-6+(4)y=exsinx-7cosx4-5x2+(5)y-4>/x+--2x3+x7(6)y=3x+Syfx4——+x'14-X2(7)1-X2](8)y1+X+x~X(9)(l-x)(2-x)(12)y——~r=+\[x+3yjx(13)y=x3Inx--xn+n…ஹcosx11(14)y-——In—+xx(15)y=(x4--)Inx+x…ஹxcosx-lnx(16)y=------------+x+117

171(17)y=--------+X+COSXxsinx+cosx(18)y=—:----------+xsinx-cosxxex(19)y=------+sinx(20)y=xsinxlnx.15.%ᑡCᔠᦪḄ!ᦪ:(1)y=(x3-4)3+(2)y=x(a2+x2)>Ja2-x2+(5)y=ln(lnx)+a+x(6)y=—In2a-x(7)y=ln(x++x2)+x(8)y=Intan—+2(9)y=cos(cosVx)+(10)y=cos3x-cos3x+(11)y^-^=Ve'3x'F(12)y=arcsin(sinxcosx)+(13)y=arctan——H1-x2(14)y=e~x+2x+IIl(x+2)(x+3)15)y=in--------——;Vx+118

182(16)y=e2xsin3x+—+/ஹsincoxJஹLMFஹ(17)y=---------(k,6yKLᦪ)+1+x(18)y=xyla2-x2+^:+77^7(19)y=sinz,xcosnx+Jl+X-yj\-X(20)y=In,---.--.yjl+X+Jl-X16.ᵨPᦪ!Q%ᑡᦪḄ!ᦪ(1)y=xj---+Vl+x/ஹ//1+Xফ>'=i-A1------T+1—xVl+x+x(3)y=(x+y/1+x2)M+(4)y=xx,(x>0)+(5)y=xInr,(x>0)+(6)y=(l+x)x,(xு0)+(7)y=xtanr,(x>0)+(8)WX"Z"(஻>()).17./(x)Px!Ḅᦪ\dxপ>=/஺2);(2)y=f(ex)efM<বJ=/(/(/«)).18.p(x)c/(X)Px!Ḅᦪd:dx(1)y=e(x)+஻2(x)+19

19(2)y=arctan0")(஻(x)w0)+W(x)(3)y=YZ஺)(஻(x)>0,/(x)w0)+ভ)=log>*“(x)(஻(x)>0,^(x)>0,9(x)w1).19.%ᑡᦪḄ!ᦪ(1)y=eax(cos/?x+sinbx)+12(2)y=xarctanx-—ln(l+x)+Jl—-2—12x(3)y=arctan----------+arctan-----+7x\-x(4)y=arctan(tan2x)+y=(»•&(o(a,/?>0)+bxa/[xc7a•x/Jஹ(6)y=—\a-x+—arcsin—(a>0)+Inx+yJa2+x(«>0)+y=ln(arccos(9)y=x1'"+ax+apaு0q+(x+l)212x-l(10)y=—In+arctan6x?—x+1J3§2rᑖᭆuvᐸxy1.%ᑡᦪᙠᢣ{Ḅrᑖ(1)y-axn+a_x"~'+',,+ax+a,dy(0),dyপ;nnil0ஹ71(2)y=secx+tanx,dy(0),dy(—)cdy(/r)+4]x(3)y=—arctan—,dy(0)dy(a)+20

20(4)y=—4--^-,dy(O.l),dy(O.Ol).xx2.%ᑡᦪḄrᑖஹx(zI)(2)y=x

21x-x<(3)y=yjx-+-

22x---j=+(4)y=arcsinVl-x2(5)y=e"2+(6)y=Intan(—+—).243.஻/xḄrᦪdy/ஹ஻(1)y=arctan+v(2)y=ln7w2+v2+(3)y=lnsin(w+v)+(4).=/1=J/+.24.%ᑡᦪḄrᑖdy(1)y=sin2t,t=ln(3x+1)+(2)y=ln(3/4-l),r=sin2x+(3)y=e3u,u=—Infj=x3-2x+5+(4)y=arctanu,u=(lnr)2,r=14-x2-cotx.§3◚ᦪᦪrᑖQ1.%ᑡ◚ᦪḄ!ᦪᒹ:dx21

2322(1)—7+77=1(஻/Kᦪ)+ab(2)y2=2px(pKᦪ)+(3)x2+xy+y2=a2(QKᦪ);(4)/+W3=஺+(5)y=x+—siny+(6)x3+y3=a3(aKᦪ);(7)y=cos(x+y)+(8)y=x+arctany+(9)y=1-ln(x+y)+ey+(10)arctan—=In^x2+y2.x'2.%ᑡᦪḄ!ᦪx=---1+(1)\y=cos-tx-e2tcos17/•2[y=esinttভx=6z(lntan—+cosr)y=asint3.ᦪy=y(x)ᙠᢣ{Ḅ!ᦪ:(1)y=cosx+^siny,(y,0)+(2)yex+

24y=1,(0,1)+y=1-cosr,222

25+V2ᙠ=J,Jᜐ.y=1,234.•ᙊ┵ᘤK10m,☢Ḅ⚔ᙊK4m.(1)ᐭḄVP☢஻Ḅᓄ᳛+(2)VPᘤ¢☢ᙊRḄᓄ᳛.5.ᑐ=acos"y=asin”.(1)y'(f)+(2)¦§Ḅᑗ§©ᙶ᪗¬ᡠ¢Ḅ®Kᦪ.6.¦§¯,°{ḄQ§ᑮ²{Ḅ³´ឤ¶·a.y=a(sinf-fcost)§4▤rᖪ▤rᑖ1.%ᑡᦪᙠᢣ{Ḅ▤!ᦪ(1)/(X)=3X3+4X2-5X-9,/"প/"'পJভপ;X(2)/(%)=-==,^/"(0),/"(D,/"(-1).,/I•2.%ᑡᦪḄ▤!ᦪ:(1)y=xlnx,y"+(2)y=e",y"'+(3)”¾»;(5)y=x56cosx,y0°)+(6)y=x3-~~-,»3.%ᑡᦪḄn▤!ᦪ(1)y=¼+(2)y=Inx.4.%ᑡᦪḄn▤!ᦪ:23

26](1)x(l-2x)(2)y=sin2x+1(3)y=---------•X2-2x-8(4)y=—+xix+2(5)y=In----+1-x(6)y=2AInx.5.Ḅᔜ▤!ᦪᙠ,y”vq».(1)y=/(x2)+(2)y=/(-)+X(3)y=3)+(4)y=/(Inx)+(5)y—.6.À/(x)=k',f")(0)=0.0,x=07.%ᑡᦪḄX▤rᑖপy=3+7x(2)y=xarctanx+(3)y=f(w)=e",஻=(p(x)—x2.8.%ᑡᦪḄH▤rᑖ(1)஻(x)=lnx#(x)=",J3(wv),J3(—)vIu(2)஻(x)=¯2/(%)=cos2x,/(-'[().9.ÅᑡᦪḄX▤!ᦪ:24

27x=2t-ry=3t-t3x=acost(2)

28d2xÈᦪx=x(y)ËÌ-=1,ÏÐÑZy=y(x).dyÒÓÔrᑖÕÖᳮvØᵨ§1rᑖÕÖᳮ1.(1)X3—3X+C=0(cᦪ)ᙠÙÚ[0,1]ᑁÜÝᨵßÜàḄá᪷+(2)xn+px+q=0("Kä᦮ᦪKáᦪ)æ஻KᏔᦪçèᨵßá᪷+æ஻K᜻ᦪçèᨵHá᪷஺2./(x)=x"'(l-Kä᦮ᦪxe[0,l],ᑣᙠJw(0,l),êm_Jn1-J3.ØᵨëìᨽîÕÖᳮ%ᑡܶï(1)|sinx-siny|<|x-y|,x,ye(-oo,+oo)+(2)|x|<|tanx\,xe(-/ñ),¶òᡂôæõæx=0+(3)e*>1+x,xw0;(4)y~X0.l+x24.ᦪᙠ{aᐹᨵ÷øḄX▤!ᦪ/(./0/(.-/Q-2/(.)lim++ú-0û2J'/5.lim/(x)=a,°ü7>0,ᨵXT+00Um[/(x+r)-/(x)]=Ta.X->+ao6.ᦪ/(x)ᙠ[a,ᑗ!ᐸÕa20,ᙠJw(a,6),êþ2ÿ/3)—/(஺)]=(஺2)/).7./(x)ᙠ(a,+8)+lim/(x)=lim/(x)=A஺!"ᙠ§G(a,+8),v_i.z»vߟkm26

29+fC)=o஺8./(x)!/(x)ᙠ/01234ᨵ/(x)+f(x)Ḅ01.9.8ᦪ/(x)ᙠ:▬<=>◀:1᜜limf(x)=A,!/'(%)"ᙠf(Xo)=A.10.F/(x)ᙠG%HIJ/'(஺)K/'(b)23ḄMNᦪᑣPQ"ᙠ1Jw(஺,:_f&)=k.11.8ᦪ/(x)ᙠ(aT)ᑁ/'(X)ᓫWX/'(X)ᙠᓃ)=>.12.F8ᦪ/(x),g(x)KT([ᙠ\a,b]=>,ᙠ(a,b)X"ᙠ^€`,T),+af(a)g(a)h(a)f(b)gS)k(b)=0.fG)g'C)/)13./(x)ᙠ(8,+8)=>lim/(x)=4-00,X!/(x)ᙠ(8,+0஺)eᑮX->±O0jḄᨬlm.14.f(x)ᙠ஺T)=>,lim/(x)=B.x-^b(1)F"ᙠX]e\a,b),+/(X1)ுB,ᑣ/(x)ᙠ\a,6)rᑮᨬᜧmt(2)᝞v"ᙠᑁe\a,b),+/(xJ=B,wᔲyz/(x)ᙠ\a,b)rᑮᨬᜧm?15./(x)ᙠ\a,+8)ᨵ}f(x)"ᙠlimf(x)=/?.ᓃ=0.16.:arcsinx+arccosx=l1).§2ᑖm4ᳮᐸᵨ1.^ᑡ4Ḅ᩽▲:ஹ..tanax(1)lim-----;i°sinbx1-COSX2(2)lims0x3sinx27

30ln(l+x)-x(3)hm-----------;♦0cosx-1/,ஹ1.tanx-x(4)hm---——;;3°x-sinxIncosax(6)lim--------ioIncos/7x..tanx-6lim---------secx+52X(9)limQr—x)tan—;XTn2(10)lim/%;,V->1X(11)lim——(a,b>0);x—>+coe"71---arctanx(12)lim2------——;sin—xlncX(13)lim——(b,c>0);(14)limx"In'x(b,c>0);x->0+1-2sinx(15)hm---------x—cos3x6..Inx(16)lim----;XT஺’cotx(1+x)x-e(17)limA—>0x(18)lim"t10+(19)lim|In—A->0+X28

31limv->0lim.v->0x2sin2xlimsinxlnx.10+2.8ᦪf(x)ᙠ\0,x]ᵨᨽ¢m4ᳮᨵ/W-/(O)=7'(^)x^e(O,l).¥^ᑡ8ᦪ¦ᑣ=2প/(x)=ln(l+X);(2)f(x)=e\3./(x)¨▤!lim"x+2஻)_2ஹ(x+/7)+2(x)/»->0T24.^ᑡ8ᦪ«wᵨ¬r®ᑣ᩽▲!2♦1xsin—(1)lim------;♦0sinx/0vx+sinx(2)hm--------;*°X-COSX2x4-sin2xlim------------(2x+sinx)*±x§38ᦪḄᓣ³ஹ´ឋK8ᦪ¶·1.ᵨ8ᦪḄᓫWឋX^ᑡ«¸¹:(1)—x

32x(3)x-----0;(5)2>3----,x>1.2.»4^ᑡ8ᦪḄᓫW¼3!(1)y=x3—6x;(2)y-yjlx-x1;(3)y=2x2-Inx;(4)y=X;x(5)y=2x2-sinx;(6)y=xne~x(«>0,x>0).3.^ᑡ8ᦪḄ᩽m!(1)y=x-ln(l+x);(2)y=x+—;l+3xy=."+5/(Inx)(5)y=2x3-x4;12(6)y=arctanx——ln(l+x).4.JO)=<(1)X!x=0½8ᦪḄ᩽lm1t(2)¾Xᙠ/Ḅ᩽lm1x=0ᜐ½ᔲÀÁ᩽mḄÂᐙᑖᩩÅᡈ¨ᐙᑖᩩÅ.30

335.X!F8ᦪ/,(X)ᙠ1/ᜐᨵf+(Xo)/(X)g(x),>0.f(x)g(x)!/(x)=0Ḅ/N᪷234ᨵg(x)=0Ḅ᪷.9.»4^ᑡ8ᦪḄ´ឋ¼3:Ø1!(1)y=3x2-x3;(2)y=x2+-;x(3)y-ln(l+x2);(4)y-Jl+f.10.XÚÛy=ᨵÜJÝÞÛḄßÔØ1.11.Îa,஻HàmÐ1(1,3)HÚÛynad+bfḄØ1â12.X!(1)F/(x)H^´8ᦪ/IHãäNᦪᑣ/l/(x)H^´8ᦪt(2)F/(x)ஹg(x)ᙳH^´8ᦪ,ᑣ/(x)+g(x)H^´8ᦪt(3)F/(x)H¼3/Ḅ^´8ᦪg(x)HJḄ^´⌴ç8ᦪᑣgo/(X)HIḄ^´8ᦪ.13./(X)H¼3/è´8ᦪX!FX஺€/Hé(X)Ḅ᩽lm1ᑣX஺H/(X)ᙠ/êḄ᩽lm1.14.ᵨ^´8ᦪᭆìX᝞^«¸¹:(1)MíNᦪ஺T,ᨵ31

34”<-(ea+*);2(2)Màãä8ᦪ஺T,ᨵ-a+b,2arctan---->arctana+arctanb.215.᝞à⌱ïðᦪT>0,Òw+ÚÛ_Lἕ2ycᙠX=±b(஺>0Hó4Ḅôᦪ)ᜐᨵØ1.2X16.y—Ḅ᩽mØ1ÍØ1ᜐḄᑗÛÒÓ.%2+117.¶Ì^ᑡ8ᦪḄ·ö!(1)y=x3-6x;(2)y=e-"T)2tব>=x—1.1+X(4)y=in----;1-x(5)y=y=x-2arctanx;(6)y=xe~x;-2x-3y=x2+1(x-1)3.(8)(X+1)3'X4(9)(1+X)3§48ᦪḄᨬᜧmᨬlmÎ⚪1.^ᑡ8ᦪᙠᢣ4¼3Ḅᨬᜧm:ᨬlm(1))=5É4+5/+1,[-1,2];32

350஻(2)y=2tanx-tan**x,[0,—);(3)y=Vxlnx,(0,+oo);(4)y=|x2-3x+2|,[-10,10];(5)y=eஹ3|,[-5,5].2.ó4ûH/ḄÛü¥ýjᑖᡂ/ü+ÿ
ᡠᡂḄ☢ᨬᜧ.3.ᵨᘤ஻!"ᦪ$%,',…,%*+,᪵Ḅᦪ.x⊤1ᡠ⌕Ḅ3.45678஻9ᦪ:;Ḅ<=>ᨬ?.224.BᑁDEFᙊH+J=1M<Eᙶ᪗PḄ☢ᨬᜧḄ.ab5.RM(p,p)ᑮXᱥZy2=2pxᨬ\]^.6.`a9ᙊb┡᮱efghV
j☢ᩞᧇḄmᓫo☢pqaᐗ.sᩞᧇḄmᓫo☢pqᓃᐗ*┡᮱Ḅuv8wḄxyEz{⌼pᨬḕ~7.᩟ᑜaᩩ☢☢4஻/Ḅ⍝s☢Ḅᚁ(ᓽ8w43ᜳ6tan6==),88w/zᕜᨬ?.(᪷$"ᕜᨬ?⍝45¡ᨬᜧ.)8.¢£Ḅ¤a,¢¥Ḅ¦§%஻?/s,¢£ª«¬R¢ª«f=0,°±²³¡¢¥Ḅ´µ=¶·x=tvcosa012y=tvsina--gfoº¦§%°»*᝞½¾᦮¢£Ḅ¤a,6¢¥À¶ᨬÁ.!"#ᑖ§1°ÃᑖḄᭆÆ1.BÇᑡ°Ãᑖ·(1)|(x5+x3-^-)dxÊ(2)j(5-x)3dxÊ33

36(3)J(V^+i—+q-)dxpdx(4)J7(177)r3x2(5)L+JdxÊ(6)J-dxÊ(7)j(2sinx-4cosx)rfxÊ(8)J(3-sec2x)dxÊ(9)J(tan2x+3)JxÊ2+sin2x(10)dxÊcos2xtanx(11)——dxÊ2X.cos—sin22,cos2x(12)■dxÊcosx-sinxdx(13)I1+cos2x(14)J(5V+1)2JXÊ(15)J(2+(16)21f(17)^Xyfxdx;(18)(19)\22x3xdx<34

37-3+sinx)Jx.(20)"a"2.Ba×Zy=/(x),7ᙠR(xj(x))ᜐḄᑗZḄ᳛2x,ÜÝR(2,5).3.ef/(x)ÞÃḄᐵàáâB/(x)·(1)xf\x)=1(x>0)Êফ=1(x>0)ÊXব/(x)/,U)=l(x>0)ʧ2ᣚᐗᑖç8ᑖèᑖç1.ᵨéêᑖçBÇᑡ°Ãᑖ:(1)ᓰX(2)(3)~-----dxÊ+1+7x^1-^=+-=!=MX(4)T?;yj3-x2Vl-3x2(5)——-~rdxÊ2+3-(6)e2dxÊ(7)xe~xdxÊ(8)
dx<1+e*rdx(9)'ex+e-x+2-X(11)jtanxdxÊ35

38(12)jtan5xsec2xdxÊrl-2sinx.(13)-----—dx

392.ᵨᣚᐗᑖçBÇᑡ°Ãᑖ:(1)^y/x2-a2dx<(2)(3)[X=dxÊzV5+x-x2(4)^2+x-x2dxÊ(11)[.A=dxÊy/l-x2fd+2(12)'(7+17dx.3.ᵨᑖèᑖçBÇᑡ°Ãᑖ:(1)jx2cosxdxÊ(2)jx3Inxdx·(3)jinxdxÊ(4)[arctanxdxÊ37

40arctanxdx(5)Jl-x(6)\xarctanxdxÊ(8)jcos(lnx)dxÊ(9)jsec5xdxÊ(10)Ê(11)jxsin2xdxÊ(12)jxcos2xdxÊ(13)j[ln(lnx)4--——]dxÊ(15)J(arcsinx)2dxÊ/÷rx,(16)——Ê—dxÊJsin-x(17)jln(x+V1+x2)dxÊ(18)JVxIn2xdx.4.BÇᑡ°ÃᑖḄ⌴ùúá:(1)/,,=(2)I=j(lnx)ndxÊn(3)I=JtanMxdxÊn(4)I=j(arcsinx)wdr.n5.BÇᑡᨵᳮþᦪḄ°Ãᑖ:fX54-X4-8(1)Jx3-xdxÊ38

41dxdxÊ(x+l)(x~+1)2(3)fxdxJl-x**4dx(4).dx:1+x3rx—2(5)'x2-7x+12dxx+4(6)dx—l)(x+2)dx(7)MT+x2x-3(8)dx+2x+1dx,—--------dxx+4x+2dx(10)dx.8—2x—6.!ᑡ#$ᨵᳮ'Ḅ)ᑖ:(1)J4+5cosxrdx(2),5+4sin2x(3)J2+sirrx,ஹrsecxdx(4)I---------J(1+secx)dx(5)1+tanxdx(6)(2+cosx)sinx/ஹrsinxcosx,(7)-----------dx

42dx(10)J•Jsinxcosxrcosxdx.(11)---------------dx<7sinx+2cosxrsin2x(12)dx.J1+cos2X7.!ᑡ8ᳮ9ᦪḄ;<)ᑖ:dx(1)■dxx(l+24+=)xdx1+xx2xdx(10)8.!ᑡ;<)ᑖ:dx(1)yJ(x-aXx-b)dx(2)x2y^ax2+P(3)xexsinxdx40

43(4)^xexcosxdxdx(5)sin(x+஻)sin(x+/?)(6)jtanxtan(x+a)dxrx3arccosx,—.dxJVT7rtanx,-------------dx<•U+tanx+tarrx/Bஹfx+sinx.(10)dx0).2.`/(x)ᙠV஺+c,b+c[R)Sbc/(x+c)ᙠVa,bLQR)Sd41

44(f(x+c)dx=f"f(x)dx.J(iJa+c3.`“USx=c,ce(a,b),f(x)=<0,xea,c)0,/(X);ឤw,bcVf(x)dx>0.2.`/(x)ᙠVa,SS/2(ᑜ=0,bc/(x)ᙠVa,ᑗQឤw.3.c/2(x)ᙠ3,ᑗR)S/(X)ᙠiSᑗ;R).4.!ᑡᔜ<)ᑖḄᜧ(1)Jxdx,fx2dxᐔফFxdx,fsinxdxবdx,^'dx.5.bc!ᑡ;'(`ᡠḄ)ᑖᙠ)(1)Iwje'dxWeফ14¡7171ব—<<2-7F3"*=6.(4)*)Jv42

456.bc(1)limf——dx=O<—£1+¤7C(2)limpsin"xdx=0.n—J)7.`f(x),g(x)ᙠ3,,bc¦£/©©(ᓽ«=f/(x)g(x)dx‘i=lᐸ஺=/<᳝<<%=bZ=¯y;_[,஺S4£[x“Sx/(i=1,2,S஻),2=max(Ar3.18.`/'(x)ᙠ[a,Sd/(a)=0,b|f/(x)dr|<-^-max|/'(x)|,9.`0

4613.`/(x)ᙠV0,1[S/(x)>a>0,b:14.`y=e(x)(xN0)YÏÐᓫÒÓ=Ḅ9ᦪS0(O)=O,x=0(y)YuḄÙ9ᦪSbc(᜛(x)dx+10(y)dy2ob(a>0,&>0).15.ᵨyÖ

47বlimgrfJ஻y(4)lim—yln(n+1)•••(2/7+1)"f8n3.h/(x)S(x)(1)/(x)=[7áâ(2)ã(x)=f(3)F(x)=£e'dt4.!ᑡ᩽▲fcost2dt(1)lim“TooX(2)lim,…£5.`/(x)ᙠV0,+8)dᓫÒ⌴ÓSb9ᦪXᙠ(0,+8)QdᓫÒ⌴Ó஺§4<)ᑖḄÛÜ1.ÛÜ!ᑡ<)ᑖ2(x+l)(x2-3).f(1)-----Þ----dx<13x245

48^4-x2dx:ম(6)^x2>Ja2-x2dxnÞsinmxcosnxdxdx
°(x2-x+l)3/2Sexdx4r)i+Vi+7(10)c-cosx(II)2Jl+sinx(12)fxarctanx6k(13)(14)jx2cos2xdx(15)[x2cos2xdxåæ-஻(16)(17)[sinmxcosnxdx(18)\--dx(6/>0)VQ+X0(19)L—(20)px3(l-5x2),0Jx2.ÛÜ!ᑡ<)ᑖপPsin9xdxফjsin,xdx(3)j"cos6Mx46

49mভÞcos7xdx(5)£(a2-x2)"dx(6)“I-”—3.bcḄ᜻9ᦪḄyᑗê9ᦪḆwᏔ9ᦪSḄᏔ9ᦪḄê9ᦪᨵdíᨵyow᜻9ᦪ.4.`/(x)ᙠᡠîOPQY9ᦪSbc:পf/(sinx)dx=F/(cosx)dxফÞx/(sinx)dx=~/(sinx)dx2/a9ஹdxব(4))//S)dx=^xf(x)dx(a>0)5.ÛÜ)ᑖp———dx.#cosx4-sinx6.ᑭᵨᑖñ)ᑖòbc:1f(4)(x—ll)du=f[f(tytdu7.`/"(x)ᙠ[a,b]Sd/(a)=/S)=0,b:(1)f/(x)dx=f/"(x)(x-a)(x-b)dxফ|ff(x)dx<("0max|/"(x)||]2a

50lim-[f(t)dt=IX->OCXJo§5<)ᑖᙠᱥᳮḄùᵨúû1.ᨵyↈ᱐•+4413ுöSþÿ
ᔣᐭḄ.a~b~2.ᜧ᫑᫑⌕ᐜ஺!ᙊ#$Ḅ%&20m,*27m,,-☢3m,⌕01234567ᡠ9Ḅ:஺3.;<Ḅ=>?@$AB6m,B2m,,10m,EF=>ᡠ⌕Ḅ஺!ḄG7&\000kg/m3.4.%&rḄᳫKᐭLM☢NOᳫḄG7&1,PQᳫRS-⌕9TU:V5.0XYZ[ᡠ◤ḄMXYḄ][ᡂ_G஺`aḄbcXY][1cm,e0XYZ[10cm⌕9TU:V6.ᨵ[&aḄhiLᙠᔜlᜐḄnopMNq;rlḄqstᡂ_GuhiḄtᙳop.§6wxᑖḄz{341.|a=~0xᑖ0,1ᑖᡂ10ᑖ'ᑖᵨ@$ᱥn34Ḅz{ᑮᦪl.2.0xᑖ10ᑖᵨᱥn34ᑡxᑖḄz{ᑮᦪl(1)f(2).ᦪ§11.ᑏ-ᑡᦪᙠx=0Ḅ£¤¥¦§⚗Ḅ©ª(1)e2x¬2(2)cosX;(3)In(l-x)¬(4)—¶¬(1+x)48

51+2x+1(5)-x-1(6)sin3x¬x¸2ᑍ24-x—1.1+x(8)In-----l-2x2.ᑏ-ᑡᦪᙠx=0Ḅ¿ᡠᢣḄ▤ᦪ:(1)ফIncosx,(x6)¬(3)sinx(4)3.ᑡᦪᙠ%=1Ḅ©ª:(1)Inx¬ফবP(x)=Ä-2x~+3x+5¬4.wÆᦪ஺b,cxf0(1)/(%)=(஺+஺È05])5111ᐗË&ËḄ5▤ÌÍ;/(%)=/ᓃÏ&xḄ3▤Ìͬফ1+bx5.ᑭᵨ᩽▲fl11r(1)lim--------¬sinxyJ-l-dফlimX->00sin62xবlim஻+—In1+—2y

52l-cos(sinx)ভlim---------z—Xfoo21n(l+x2)49

53(5)lim(Wx+3x--2x)¬!/(x)ᙠÚlḄÛܶÝÞßà/(0),/,(0),/"(0)¬limx->0!/(X)ᙠâãAäåÝÞßæ/(x)=/(x2),ç:è=0,é@2=5(2஻)ên\8.!P(x)&nÝT⚗(1)P(a),P'(a),…í(”)3)Ḇ&_ᦪçðP(x)ᙠ(a,+8)AÌ᪷¬ফò(2'(^…?(”’(_ö÷NçðP(x)ᙠ(—8,a)AÌ᪷;9.ç(2)e?Ìᳮᦪ¬10.!/(x)ᙠa,úAᨵ¶▤ßᦪ,à/'(a)=/0)=0,ᑣüᙠce(a"),c11.!/(x)ᙠal▬z¶ÝÞßà/"(a)#0,ᵫᑖᳮ:f(a+/i)—f(a)=f'(a+0h)h,0lhmO--20212.ᦪ/(x)ᙠa,#$ឤᨵf'\x)o0,ᑣᙠ(ᑗᑁ+,-./(/)+/(஽)ு,x,+xf22~J250

54§26ᑖᙠ789ᱥᳮḄ<ᵨ1>ᑡᔜABᡠDᡂḄFG☢6(1)V=4(x+l),V=4(l-x);(2)yTlnxl,y=O(0.1ᑡᵨ᩽ᙶ᪗⊤^ḄABᡠDFGḄ☢6(1)_`BaZ=a2cos2(p\(2)efgᴮBr=asin3(p;(3)iBr=acos0+b(b>a).3.>ᑡᵨjᦪkl⊤^ḄABᡠDFGḄ☢6(1)m=2,nay=2ana°;(2)᤮Bx=a(f-sin/),y=a(l-cosr)(00.4.yBy=x}~ᙊf+3y2=6)Ḅ☢6ᑖᡂ-ᑖA(Ḅnᙽ)8(Ḅnᙽ).B5,r=3cos஻r=1+cos஺ᡠDḄᐳᑖḄ☢6.6,>ᑡḄ6X2y2(1)~ᙊ+4=1sWab(2)y=sinx,y=0(0

55পxs(ii)ysW(3)Bx=a(t—sint),y-a(\-cosr)(0ᑡᔜA⊈ᡠDᡂḄ78Ḅ6(1)┵Ḅ6ᐸ$>Ḇ~ᙊ~ᙊḄsᑖ¡A,8a,b,¢£h<(2)¤ᙊ¥ᐸ$>ᑖ¦x§aஹbḄᙊ¢ᐸḄª«/?.8.®ᳫx§R,±£/?Ḅᳫ²6(/?WR).9->ᑡABḄµ(1)y=x2,00,00);(7)½¾Br=a(l+cos0),Q<00.10.>ᑡᔜABᙠᢣ.ḄA᳛A᳛x§(1)xy=4ᙠ.(2,2)W(2)ஹ=111Áᙠ.(1,0).11.>ᑡABḄA᳛9A᳛x§(1)ÂᱥBy2=2px(pு0);x2y2(2)_ABÄnÄ=1Wab222(3)·GBᡝ+ᡝ=½W12.>ᑡjᦪklÆÇḄABḄA᳛A᳛x§(1)Bx=a(f-sin/),y=஺(1ncost)(a>0);52

56(2)~ᙊx=acosf,y=bsinf>0);(3)ᙊḄuvBx=a(cosf+tsinf),y=a(sint-tcost).13.>ᑡÉ᩽ᙶ᪗⊤^ḄABḄA᳛x§(1)½¾Br=a(l+cos6)(a>0);(2)_`Br-2a2cos20(a>0);(3)ÊᦪËBr=a*(2>0).14.ÌAB¦ᵨ᩽ᙶ᪗klr=")ÆÇÍÄ▤ÏÐÑᙠ.6ᜐA᳛0lr2+2r'2-rr"lK=------------3—-(r2+r'2)215.ÂᱥBy=a/+Ô+ᙠ⚔.ᜐḄA᳛x§ᨬ.16.ABy=2(x-1-ḄᨬA᳛x§.17.ABy=e*$A᳛ᨬᜧḄ..18.>ᑡÙ☢ABsᡠÚA☢Ḅ☢6(1)y=sinx,0Wx4ÛxsW(2)x=a(r-sint),y=a(l-cost),a>0,06)XsWa~b~(4)x=acos,y=asir?fxsW(5)r=2a2cos2஺᩽s.•19.>ᑡABßḄà½(1)x§r,µá(a44)ḄᙳᒴᙊµW2(2)ÊᦪËB=(஺ு0,ä>0)$ᵫ.(0,஺)ᑮ.(஺/)ḄᙳᒴµßW(3)ÉA(0,0),3(0,1),C(2,1),DQ,0)⚔.ḄæGᕜèAB$+n.Ḅéê¡ë.ᑮì.ª«Ḅ2íW(4)x=a(f-sinf),y=a(l-cosf)04f42Û,a>0,éêîᦪ.53

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58y=0;(3)3x2+5x-5y'=0;(4)xydx+(x2+l)dy=0;মy—xy'=a(y2+y)(6)(y+3)dx+cotxdy=0;=10ᔩax(8)xsecydx+(x+\)dy=0;(9)(ex+y-ex)dx+(ex+y+ey)dy=0;(10)yInxdx+xInydy=0.2,!"#ᑖ%&'()*ᩩ,Ḅᱯ/:(1)?sinx=ylny,y|=e

59axx=2ফy5=e2",v|=08x=054

60(3)^--^-Jy=O,y|=1-1+y1+xx=஺3.=>?1gḄ=ABCDᵨDFGHI5JCKLMᡂOP5K=AHIḄQRᡂSP5ᙠf=10sL5QRVW50cm/s,C?4X10-5N.\]HI^*_`abᑖcdḄQRefgh4.iḄ⊦kᨵ᝞nḄopqiḄ⊦kQRriᡠtuḄ>RᡂOP5w_xᩞᧇ{|,i_`1600}d5~*>R஺Ḅb5iḄ>RrLMfḄᦪᐵ.ᦪ§1ᦪឋḄV1.ᦪᑡJnḄஹnq(1)x=1--8nn(2)X„=H[2+(-2)"];ব=%5஽*+1=1+1¡=L2,3,…);k"+1(4)x„=[1+(-1)-]—;n(5)=Vl+2n(-1)5,;n-\2nᐔযx“=--cos--.n+132.¨/(x)ᙠ஺|©5ª:(1)sup{-/(x)}=-inf/(x);xeDxw஺(2)inf{-/(x)}=-sup/(x).9xeD3.¨/=supE5±᜛eE5ª³E´µ⌱·ᦪᑡ{x஻}±¹º»¼½5¾limx஻=(3q஺GE,ᑣÀÁ᝞Âh4.ªᦈÄᦪᑡÅᨵKn5ÆW+8ḄᦪᑡÅᨵn5ÆW-00ḄᦪᑡÅᨵ.5.ᑖÇÈÉ'(nᑡᩩ,Ḅᦪᑡq(1)ᨵÊnḄᦪᑡ8(2)ËᨵÌ»ËᨵnḄᦪᑡ8(3)ÍËᨵÎËᨵnḄᦪᑡ8(4)Í»ËᨵλËᨵnḄᦪᑡ5ᐸ´ஹnÐᨵ▲.55

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4ᨵḄᐙ⌕ᩩ,e5%ḄþÂøᦪᑡ
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621i2.ᦪC=1+k.FI஻700MḄ᩽▲Nᙠ.3.ᑭᵨRSᦈU
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63(2)᩽▲x=/(x)Ḅ᪷#Ḅ.9.f(x)ᙠ஺ᙳᩩ>:(1)I/(x)-/(y)\8(2)£¤x=/(x)Ḅ¥ᙠ[஺,#Ḅ§0¥¨#᩽▲.§4ªX«¬(12ᦪḄឋ1./(x)ᙠ®12¯ᨬᜧx0#•Ḅ³´lim/(µ)=/(%),XT8limx=xn0X->82.f(x)ᙠ¶,12·³(1)min/(x)0,:ᙠ¾€(஺À),´/©)=(),/(x)ு0eপ/(஺)=%/(£)="?ᡈ/(஺)=m"(£)="{ফ஻?

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66limjfI/(x+/i)-/(x)Ic/x=0,ᐸæA00Jci12.'/(x)ᙠD,EF8!"(1):ᙠGH;ᑡI5஺/KLA&+iD+Ju(%,)u(a,b),MN(3",P)

67§2ᦪ⚗_ᦪḄᦈឋᐸឋ1.ᑡ_ᦪḄ"1পE--------------9£(5஻-4)(5஻+1)001_ফ4n2-fn=l§(-1বM=IN002n-l(4)z/l=l2"(5)£r"sin஻x,"l

684.'_ᦪ.ᔜ⚗£¤Ḅ¥_ᦪḄ⚗¦§¨ᔠªRᑮ¬_ᦪ“ᓽ஻=1n=lU“+I=¯°+%+2+…+“ᓃ஻=o,1,2,…ᐸ³´=0,µ<´<&<…<´<´+1<….e¶஺”ᦈ,!d·ᩭḄ_ᦪDᦈ.”=1§3¤⚗_ᦪ1.ᑨºᑡ_ᦪḄᦈឋ"81পZ»9஻=17n+஻ফ⁐(2஻—.I9ব£2஻-1ভ¾si¿;n=lম(a>1);"=]1+஻1য஻=inyjnঝ1ÄZ(E";”=[2/2+1E1(8)y————9Æ[ln(஻+l)r”=1(10)^2"sin49஻=௃3163

698(11)(n

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10916."Jᑡᦪᢣ▤ḄDEᦪ:u=x\iny+y3sinx,"^পdx3dy3஻=arctan"ᡠᨵ¸▤DEᦪ;ফ1—xy•/22ஹ4a%3%বw=sin(x+)ு)";Sp+^ru஻=xyze”z"__¿_(4)dxpdyqdzr“q…"j(5)X-ydxnidy"ஹdm+nu(6)u=ln(ax+by)"-------dxmdyn17.Â#JᑡᦪÃÄd2ud~u.—7+—T=°dxdy(1)u=ln(x2+y2)'22ফu=x-y'(3)u-ecosy'y(4)u-arctan—.x18.ᦪ஻=0(x+”(y)),#)dud2u_dud2udxdxdydydx219.,ᙠ]Åx஺,%ÆḄÇz=ᑁeᙠ4ᙠ]ÅX஺,%ÆnGᑣᨵ%Åᐭ00ᡭÆ=£"Å/%Ƨ2ÍᔠᦪF◚ᦪGᑖÐ1."JᑡᦪḄᡠᨵ▤DEᦪ$100

110(1)u=f(ax,by)'(2)u=f(x+y,x-y)'বu=f(xy2,x2y)-,ভ“=/(—,—)'yz(5)«=/(x2+y2+z2)'(6)uf(x+y,xy,-).2.Ó=—Jᔆᐸ6/1nGᦪÂ#/(x-y)13z+1Ôxdxydyy3.^=8஺Õ)c¡¨ᦪᦪg▤nEr=J/'+y'+z'஺rcd2vd2vd2v1d2v1--T--T--T=--T'.dx-dy-dzcdr4.*ᦪ/(x,y,z)-xÖ×ᦪÃÄᐵÙf(tx,ty,tz')=tnf(x,y,z),ᑣÚ(x,y,z)¡஻ÔÛÔᦪ./(x,y,z)nGÜ#)/(x,y,z)¡nÔÛÔᦪḄᐙ⌕ᩩà15.Â#Jᑡᔜ$(1)u=(p(x2+y2),5!iJy--x—=0$dxdy-/2ஹduxu(2)஻=)᜛(ᔆy-),ᑣ)ã+Õ='dxdyy(3)u=x(p(x+y)+y”(x+y)ä!j—r-2——+—r=0'101

111(4)u=x(p(-)+(2)ᑣX2^4+2xy+y2å=0.xxdxdxdydy6.஻=/(x,y)nGᙠ᩽ᙶ᪗/ᣚx=rcosd,y=rsin6F,#)ëஹ—íÆ*ïðÚÅ|£Æ2+Åò17d/0.dxdy8.ᦪ“=/Åó,ÆÆÃÄôõôö÷ød2ud~udx2+dy20,n/TLa,a2"n#)ᙠJᑡ/ᣚùúûᢝd/,ᓽᨵ■+=°.পX-—2~y—o2S+஻S+஻ফx=escost,y=essint'x=Ys,tx,y=w[s,tx~T-~~~=~.!"#$%&.বosdtdtos9.()*+Ḅ*ᣚ./,஻!1)*+:4=%=/+2,*ᣚ:$%ᒹ=

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113w2-v=3x+y,4dududvdvব0)s²$Ḅᑗ.³¥¦´ᑗµᙶ᪗¸ᡠ¹/Ḅº»¼.5.Y☢/+2>2+322=21Ḅᑗ☢.½¾¿À☢x+4y+6z=0.104

1146.¥¦☢Á(x—az,y—8z)=0Ḅᑗ☢ÂÃ$Ä¿.ᐸz஺Å!Æᦪ.X7.¥¦☢Z=x>ḄÇ$ᑗ☢ÈÉÊË.8.YÌ☢Í(x,y,z)=O,G(x,y,z)=0Ḅ¬ᙠ஺Ï☢sḄᢗÑḄᑗ.§4ᔣcᦪ1.P/(x,y,z)=x+/+z3,Y/ᙠᐘ(1,1,1)Õᑮ/=(2,—2,1)Ḅᔣcᦪ.2.Y^ᦪ஻=|zᙠA(5,l,2)ᜐÕᑮ6(9,4,14)ḄᔣABsḄᔣcᦪ.c#du3.Y—dl(Jto?o>(1)M=ln(x2+y2),(x,y)=(1,1),/Âx¸ÚᔣḄᜳ¯!60°<00ফM=xe'v,(x,y)=(l,l),/Âᔣ+(1,1)®ᔣ.004.P^ᦪ/(x,y)ᙠ(x°,%)Ük,ᓫÞᔣ+/]=,4=($(/.%)=1,à(ᐭ00â)=0,]/½ãdl2—0,ᡭ)_7dl-5å.5.P/ᙠ%(2,0)Ük./(ᓃ).)ᙠçᢣᔣ[=(2,-2)Ḅᔣcᦪê1,ᢣᔣËḄᔣcᦪê$3,ëìí(1)ᢣᔣî=(2,1)Ḅᔣcᦪêïðñ(2)ᢣᔣî=(3,2)Ḅᔣcᦪêïð?§5òóôõ105

1151.ᑏ÷Zᑡ^ᦪᙠᢣḄòóôõ(1)f(x,y)=2x2-xy-y2-6x-3y+5,ᙠ(1,-2).(2)f(x,y)-x2+xy+y2+3x-2y+4,ᙠ(-1,1).Y2.Y^ᦪ/(x,y)=-ᙠ(1,1)øùḄ஻▤úûüᨽþÿ⚗Ḅ.yy23.ᦪ/(x,y)=ᙠ(1,-1)Ḅ▤ᑏ!ᨽ#$⚗.x4.&ᑡᦪᙠ(0,0)Ḅ)▤*(1)/(x,y)=sin(x2+y2)0ফf(x,y)=exln(l+y)0(3)/(x,y)=yj\+x2+y2;(4)f(x,y)=excosy,■5.56Ḅ78ឋ*:£&x'y'+o(“')=0(2—0),i+j=0ᐸA᜛='.5&=o(i,jDEF᦮ᦪi+/=0,l,…஻)ᑭᵨ78ឋ/(x,y)=ln(l+x+y)M!ᨽ#$⚗Ḅn▤NO.6.QRS/஺,y)=5UV(*05)?ᵨAXYᳮ56[ᙠ஺6(0,1),\37V7107l0TC.7tO.7t6---cos——cos------sin——sin——.43366367.^/(x,y)ᙠ_஺ᑁᨵbcᦪ[ᙠd(x,y)=/,(x,y)e0.56/(x,y)ᙠ஺Dfᦪ.8.:|i|,|jklmḄnc&ᑡᦪopᑮr⚗Ḅst:cosx1+x+y(1)-----(2)arctan-------cosy1-xy9.^ᦪ/(x,y)ᨵzᑮ”▤|}bcᦪ,~5஻(f)=/(a+஺+)Ḅ஻▤cᦪm)প=(+&1f(a+kt,b+kt).dx8y106

11610.^/(x,y)D஻rrᦪ56aa(x—+k—ynf=n(n-\)•••(n-m+1)/.dxdy11.^/(x,y)=஻(ax+by),ᐸA஺DfᦪᙠᒹḄᑁ“ᨵg▤|}cᦪ.5*ᙠ(0,0)Ḅk/ay)=£"G3)Sy)"+(ay)k=0k•j=oᓝ◚ᦪ[ᙠYᳮ§1ᓫḄI.^ᦪ(x,y)(1)ᙠ_஺*ᐭ0-aWx4Xo+a,y஺-64y4%+b¤|}0ফ/?(x^o)=o0o(3)§x¨Y©ᦪF(x,y)kyḄ«!ᓫ¬ᦪ*ᑣ®¯ᑮ°±²³´~56µ.2.V+y+sin(·)=0ᙠ▬s¹ᔲᵨ᝞¼=/(x)Ḅ⊤¾´¿¹ᔲᵨ᝞x=g(y)Ḅ⊤¾´3.(x,y)=y2—/(]—V2)=஺ᙠÁÂḄ▬s®78ᙢpYᓫXஹ|}ஹdᨵ|}cᦪḄᦪy=/(x).4.56ᨵ78®cḄᦪy=y(x)sin+sinhy=x,cᦪy(x),ᐸey-e~yAsinhy=——-——.5.Æ+zlny+e"=1ᙠÉ(0,1,1)Ḅᑁ¹ᔲpY8ÊnkË᜜ÍÊnḄᦪ.6.^/k8ᐗᦪ~Ï/а±ᩩÒ2/(·)=/(x)+/(),)ᙠ(1,1)Ḅᑁ¹pY78ḄyDxḄᦪ.107

1177.^ᨵ*x=y+0(y),ᐸA᜛(0)=0,d§-a0,§-b

118u=f(x,y,z,t),'g(y,z/)=O,஻(z,f)=O.ᙠ°±ᩩÒ&஻kx,yḄᦪ´ᒹᒹ.dxdy7.^ᦪ஻=஻(x)÷×”=/(x,y,z),஻,u@ḄABC2D@ḄEBC2Bl+WV1+WVVᣚGH109

119IᓝKL᩽NOᩩ"᩽N§1᩽NOᨬQRST1.1ᑡᦪḄ᩽ᜧNW᩽QNX(1)f(x,y)=(x-y+\)2-,(2)/(x,y)=3axy-x3-yy(a>0)#22J1_-/“^ு0)#(4)/(x,y)=e2\x+y2+2y);(5)/(x,y)=sinxcos.co(x->-)(0-<^);++S(6)/(x,y)=(7^2+y2-i)2.2.op:=62+"+஺,stuvᦪw(xy),1=1,2z᝞ᑭᵨᨬQRST2~ᦪa,b,cᡠḄᐗGHv.3.op☢3ᨵnḄᙶ᪗ᑖ4(,%)(%2)2…24,)2~2OnḄGWᨬQ.4.~1ᑡᦪᙠᢣDᑁḄᨬᜧNWᨬQNX(1)f(x,y)=x2-y2,D={(x,y)\x2+y2<4};(2)/(x,y)=x2-xy+y2,D={(x,y)llxl+lyl0,D=R'.5.~¡X(1)/3>)=4:+25¢+஺2+26+2Q+£ᙠ¤ᨵᨬQN2¥ᨬᜧN2ᐸA>0,B2

120F(XoyoZo)=Oª(XoyoZoWO.«¬ᵫF(x,y,z)=0Ḅ◚ᦪz=/(x,y)ᙠ(x,%)°Ḅ᩽NḄ±⌕Wᐙᑖᩩ".~0ᵫx2+y2+z2-2x+2y-4z-10=0ᡠḄz=f(x,y)Ḅ᩽N.7.~1ᑡ◚ᦪḄ᩽ᜧNW᩽QNX(1)(x+y)2+(y+z)2+(z+x)2=3#(2)z2+xyz-x2-xy2-9=0.8.ᙠopᕜ¶2PḄᑗ¹02~☢ºᨬᜧḄ¹0.9.ᨵᙽ¼ᱏ2¾6=24cm,⌕ÀḄÁÂᢚÄÅᡂÇ2uȺᨬᜧ2~ÉÂḄʹaWᢚÄḄ¾Ëx(ÌFÍ).§2ᩩ"᩽NOÎÏᨽÑSᦪT1.~1ᑡᦪᙠᡠÒᩩ"1Ḅ᩽NX(1)/=X+y,Ôf+y2=Ö#(2)f-+y~tÔx+y-1=0#(3)f=x-2y+2z,Ôf+J+z'l#(4)f——I—,Ôx+y=2#xy(5)f=xyz>Ô/+)/+F=12x+y+z=0#(6)f=ax2+by2+2hxy,Ôf+)2=1#(7)f=x2+y2+z2>Ô(/+y+z?)2+b2y2+,212,iy=Q.x+m+nzill

1212.~/=/~Ü2ᙠᩩ"x+y+z=஻2a>0m>0«>0,p>0,x>0,ffy>0,z>0Þ1ḄᨬᜧN.3.~ᦪz=g(x"+y")ᙠᩩ"x+y=/(/>0,஻21)Þ1Ḅ᩽N2¨¡àX6஺20,á0,஻21â(a+8j<ä+4.~⊤☢ºæçºᨬᜧḄ¶Gç.5.~çºæ⊤☢ºᨬQḄ¶Gç.6.~ᙊḄ᜜ᑗ¹0☢ºᨬQὅ.7.¶஺Ḅ¼ëᑗᡂÁì2ìᡂíG02îìᡂᙊ஺ÁìḄ¶ᔜðñâ,òᡠíG0☢ºWᙊ☢ºÞWᨬQ.8.~óᑮR☢4x+4y+CR+4=02஺2*+%)'+஺21+"2=0ḄøùḄᨬú஺9.~ûᱥùy=FWýùx—y=lþḄᨬú.10.~x>0,y>0,z>0âᦪ/(x,y,z)=lnx+21ny+31nzᙠᳫ☢f+V+*=6ᡝḄ᩽ᜧ.a,b,cᦪ,23,a+b+c^abc<108-------.I6J11.ᦪ/(x,y,“,v),F(x,y,u,v),G(x,y,w,v)(▤*+,,-./▣1GF.FIG,GG.Gvv>42.L(x,y,u,v)=f(x,y,u,v)+2,F(x,y,u,v)+4G(x,y,u,v),89(:,%%,%)<ᦪLḄ>?@ABC(D)ு(<)0F<ᦪ/ᙠHIᩩKF(x,y,u,v)=0,G(x,y,u,v)=0LḄᩩK᩽1(ᜧ)@.MᓝOPQRSTḄUᑖ112

122§1QRSTḄWUᑖ1.XLᑡ᩽▲B(1)limf\x2+a2dx^“foJ-iফlim[x2cosaxdxinxব/(X)=fஹᒹM/y^(4)£jBf(t,s\lsllt.3.f(x)rsᦪ,Fপ=Af[IAx+4+t'Xd(x).4.ẆvᦪḄrsឋᐸh/(x)<[0,1]rsyḄᦪ.5.{ᵨUᑖ}LX~XLᑡUᑖB(1)Pln(tz2-sin2x)dx(tz>1);(2)£ln(l—2acosx+/Mx(1஻l<1);(3)Pln(6Z2sin2x+/?2cos2x)dx3,bw0)^arctan(atanx)ஹf(4)P------------dx(lal<1).tonY113

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125/(x)=f(x,y)dyᙠÑa,6)ᦈ¾ᙠx=bÒᦣA/(x)ᙠÑa)F¼ᦈ¾.6.QRSTḄ¶·Uᑖ/(x)=1'/(x,y)dyᙠÑa,ȼᦈ¾Ḅᐙ⌕ᩩK°)^"ax_-bxeফ--------sinmxdx(a>0,Z?>0)^x(3)¨xe~axsinbxdx(a>0).9.ᵨÖRᦪḄUᑖæçLᑡUᑖ:oo-ax~--bxree(I)[---------dxCa>O.h>0)^09ߟ0-᱐.ফ-------sinmxdx(a>0,/?>0).xᑭᵨ(:æçíîíïUᑖ10.l+x200cosax.L=-----axl+x2äéᐗsinax.rLj=I-----—dx.11l+x2ILᑭᵨe-^dy(x>0)æçᏝòóôUᑖ7ro.2/1r0°sinxF1=sinxax=———dxJ)21)õ116

126ä%os8cosx,^412.ᑭᵨö÷Uᑖ8sin%,71é_2yjᐔr-----dx=—,eax=----O21)2æçLᑡUᑖ:,4sinx,(1)———ax^xফ21sinycosyx᜜ᐔ,)y(3).x2e~ax2dx(a>0)^(4)¨d31*x(a>0)^(H-OC-U+=)(5)erdx(<7>0).J-0013.XLᑡUᑖ:(I)£-~—costdt^¨sln(l+x2)ফL1+x2dx.14.AB(1)fin(ù)dyᙠ[/,/?]S>1)¼ᦈ¾;(2)fûᙠ(ߟ8,ᑗþ<1)¼ᦈ¾.J)>Y§3ᑖ1.ᑭᵨᑖ.ᑡᑖ117

127(4)[x2y]a2-x2)dx(஺>0)(5)jrsin6xcos4xdx"#-e-'x'஻)*᦮ᦪ-;9dx/°A/3-COSXᐔ2sin2"xdx'஻)*᦮ᦪ-;(10)dx'஻)*᦮ᦪ-.2.34ᑡᑖᵨᑖ⊤6789:ᑖḄ<ᙠ>:m-1(1)-----dx஺2+x”-ஹf*dx(2)(3)PtanwxdxভJ£)dx<(5)#xpe~axInxdx(cr>0).3.MN(i)r%-r"jx=-r(-)(஻>o)6nn(2)limPe~xdx=\.n—>+x>J-oo4.MN118

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129MN/(x,y)ᙠ஺U|i.§2[ᑖᓄ¡ᑖ1.4ᑡX[ᑖ(1),D-[3,5]x[l,2]DD=0,1x[0,ᨴফDxyex+y2dxdy,D=[«,/?]বD(4)-dxdy)D—[0,1]x[0,1].£1+£2.3X[ᑖJJ7(x,y)dxdyᓄ)|¥¦§Ḅ¡ᑖ:D(1)஺ᵫx©ªf+y2=/(>>0)ᡠ¬ᡂ(2)஺ᵫy=x,x=2®y='(x>0)ᡠ¬ᡂx(3)஺ᵫ>=2x3,y=l¯y=2¬ᡂ(4)O={(x,y)M+|y|ீl}.3.ᦋ·4ᑡ¡ᑖḄ¡§প[dy^f{x,y)dx<বIdx£(x,>)¹+fdx"/(x,y)dy.4.j/(x,y)ᙠᡠᑖḄd>஺Uef7MNfdx£f(x,y)dy=fdyf/(x,y)dx.5.4ᑡX[ᑖ:120

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