教师考试数学：“式”通八达-常考数学公式

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1mj01UJJᦪ6.8ᦪf4/᝞:a>0,awl,M>0,N>0,;<(1)=>10gaM+10ga=10ga(MN)(2)?>loga"-loga—=loga^(3)ᦪ@nlogM=logMn(nG/?)Aaa(4)a1°gaN=NA(5)logbMn=^logM(bW0,nG/?)aa(6)logN=^(b>0,b^l)alogba(7)ᱯEloga1=0,loga=1,loga}=-1.aJ2.ᢣᦪ)-------------------------------------------------<](1)a0=l(a>0)(2)ar•0*2*4s67=ar+s(r,sGR,a>0)(3)஽=ar~s(r,sGR,a>0)(4)(aZ?)r=arbr(jGR,a,b>0)(5)(ar)s=ars(r,sGR,a>0)(6)a~r=^(reR,a>0)V____(7)as=Va^(r,sEN*,a>0,s>1).

2mj02UJJᦪᦪḄ0(1)sin*2a+cos2a=l(sin2a=1—cos2a,cos2a=1—sin2a)sina#(.sina(2)-----=tanasina=tanacosa,COSa=-----cosa----------\tanaf2.OPI<](1)sin^a±£)=sinacosp±cosasinp(2)cos(a±£)=cosacosp+sinasinptana±tan£(3)tan(a±S)=1+tanatan0I3.MJ1J(1)sin2a=2sinacosa
(2)cos2a=cos2a—sin2a=2cos2a—1=1—2sN2a2tana(3)tan2a=2l-tanaf4.QRSᳮIᙠAABCVWaஹbஹcᑖXY4Bஹ஺Ḅ8[WRYA/BC᜜]ᙊḄabc_`,ᑣᨵ:=2R.sinAsinBsinC

3mj03mjf5.eRSᳮiᙠAABCVWWa2=b2+c2-2bc222COSTI,b=a+c-2accosB,u2.,,2f2C2J_g2h2oo,oi—u~rC—Ct-஺+C—Dc=a£+bf£—2abcosCcosAA=----------,cosB=------------2bc2aca2+b2-c2COSC=2abIf6.ᦪḄijk--<].ct(+2$2Qtany1—tann>sina=-----------cosa=------------1+tan221+tan2q7.r☢#<]#%(1)s=-ah=-bh=-ch(%ஹ᝞ஹvᑖX⊤xaஹbஹcvḄy)Ananbnc(2)S=-absinC=-besinA--acsinB222,ஹa2sinBsinCb2sinCsini4c2sin?lsinB(3)S=-----------------=------------------=------------------2sin(B+C)2sin(C+A)2sinQ4+8)(4)S=2R2sinAsinBsinC(4Y᜜]ᙊḄ_`)Aabc(5)S~4R___________________________1(6)S=dp_d)(p-b)3—c)(p=-(a+Z?+c)).

4mj04UJJT8.ᦪḄᕜ{Iᦪy=4sin(co%+0),xER|ᦪy=4cos(3%+@),xER,(YᦪW}4W0,3>0)Ḅᕜ{7=*Aᦪy=3Atan(wx+g),x/c7T+|kEZ,(4a,9Yᦪ,}"஺zo,3>0)Ḅᕜ{T=~.3ᔣ91数量积HT1I0■OY(1)a-b=\a\\b\cos6(a^0,b^0,0°<0<180°)(2)ஹ3ᔣ,d=(%!71),b=(xy)/8)E2)2Ḅᜳᑣcos8=-=,2+k2—.'\a\\b\Jx12+y2/x2+y2'11y22(3)_L3O23=0=XX+y1y2=012(4)=a஺=.ᑮy-i

5mj05UJJᦪᑡk1.PᦪᑡḄ⚗f=%+(l)d(?i>2,dYᦪ)T2.PᦪᑡḄn⚗OL+a)71(71-1)nSn=--------------=9Ḅ+------------d;;13.ᦪᑡḄ⚗a==aqn-1(n22,qW0,}Yᦪ)=aqn~m[n>m)n1mIf4.ᦪᑡn⚗OI%y71al(q=1)Sn=Ḅ(1—Qn)a—aqrn(qwi)1—q1—QI5.ᦪᑡOT(1)-=—(1p

6mj06UJJ1J1J---------------------------a(a/+a2+—b஺/2++—b>(Ḅ%+ab+—F222aby,ᐸV%,a,…WaWb,…WbER+,}nn2n/2n==...=.bibb2ni2.n(||a||<\a+b\<\a\+\b\᩽▲Kᦈ¤¥ᑣ¦--------------------------------------0,¯ᙠ᦮ᦪN>0,±²᜜m>NWᨵ³-a\0,¯ᙠQᦪ5,0<-<6WX-^XQ\f(x)-A\

7mj07UJJJ3.ᑨS᩽▲¯ᙠḄ¶·¥ᑣ(1)ᜳ¸¥ᑣ¹ᦪ/(ஹ)Wg(x),hQ)ᙠ
Ḅº·»¦¼½t/°(x,r)ᑁ¿Àg(%)

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9mj09UJJJ9.ᦪ᩽▲Ḅ1óI(1)lim@/(x)An=lim(nGN),ᐸVᔜᦪ᩽▲ᙳ¯ᙠ஺+X-^XQ
X->%0(2)õ¨>ᑣᦪ¿Àöᑡᩩ«জlim/O)=limgO)=a,ᐸVa=஺ᡈa=oox^ax->aঝᙠÄaḄº»¦⚞½ᑁ¶ὅᙳÝ"W}g'Q)WOᑣᨵ{úûBᑖ6.ᑖVæSᳮT(1)(üý)Sᳮᦪ/ᙠᐸ᩽æÄG(a,b)ᜐÝ"Wᑣ¨ᨵf(%o)=0.(2)Rolle(þÿ)
ᳮᦪ/ᙠ@a,ᵪᙠ(a,b)ᑁ/(a)=/(b),ᙠfG(a,b),/'(f)=0.(3)Lagrange(ᨽ)
ᳮᦪ/ᙠ@a,bAᙠ(a,b)ᑁᙠfG(a,b),/'(f)=V).(4)Cauchy(!")
ᳮᦪ/#gᙠ$%@a,bAᙠ$%(a,b)ᑁ&'e(a,ᑗ),g(x)W0,+),ᙠfE(a,b),-=/H%.஺যg(b)-g(a)

10mj10UJJT2.1234?fM=/&)+/(J0)(K-%o)+(%-&)2+…+MN(X-A%o[)[n+[)RnM-ᐸQLagrange(RSᨽU)VW⚗/?„(%)=G-&)"+i(fE(%/())).3.Maclaurin6789᪍;34<](1)fM=/(0)+f(0)x+\/+…+2±^22nx+f(;+i)(?)+i(

11mjUJJᦪi.<@ᦪḄᦪ34i<](1)C'=o(2)(xny=nxn-1(3)(1HGI/'(%)=0,&f'(%o)=O):(i)᝞NᙠO▬QḄRSr(x)>o,TS(U)o,/(W)X᩽\.4,ᑨ_`aឋḄᐙᑖᩩfJ/(%)ᙠ$%@a,ᑗ,ᙠ(a,b)ᑁᐹᨵ▤#j▤ᦪ(1)kᙠ(a,b)ᑁ/(%)>0,ᑣ/(%)ᙠ@a,lX`Ḅ;(2)kᙠ(a,b)ᑁ/஻(%)<0,ᑣ/(%)ᙠ@a,ᵨXaḄ.

12mj12UJJoᑖ?1.p
oᑖḄឋqI<](1)/[/(%)+g(x)]dx=ff(x)dx+Jg(x)dx
(2)fkf(x)dx=kff(x)dx.I2.p
oᑖḄrBI(1)stᵨoᑖ34rBxa+1Jxadx+Cfsinxdx=—cosx+Ca+1[cosxdx=sinx+C
f——ஹdx=arctanx+CJl+x2fdx=arcsin%+Cfexdx=ex+C.(2)uvᑖC(wxᣚᐗC)1dx=d(x+C)dx=-d(kx+C)Kxdx=-d[x2+C)x2dx=-d(x3+C)
-1-dx=-d(—F—dx=2d(y[x+C)\x/-dx=d(lnx+C)exdx=d(ex+C)sinxdx=—d(cosx+C)cosxdx=d(sinx+C)1———dx=d(tanx+C)dx=d(arctanx+C).COS2X

13mj13UJJ(3)wjᣚᐗoᑖCOwᣚᐗoᑖCp{wjᣚᐗCEoᑖ)|ᐭ~Ḅ஺oᑖ%ᡂᐵ~ᦪ9&),ᓽᣚᐗw=(p(t).'=0(஺vw(t)ᙠᦪ'Jf((p(t))(p'(t)dt=F(t)+ff(x)dx=ff](p]((p'(6dt=F(t)+C=/)+Co(4)ᑖoᑖC஻(ᐹᨵᦪᑣᨵ[u(x)vf(x)dx=u(x)v(x)—jur(x)v(x)dxᡈfu(x)dv(x)=u(x)v(x)—[v(x)du(x)(1)ᦪ/(%),g(%)ᙠᘊo,ᑣr/(x)dx=T
/(x)dx
/[/(%)±g(%)]dx=Sa/(%)dx±Sag(X)dx
Ckf(x)dx=k,f(x)dx(kGR).(2)᝞Noᑖ$%ᑖᑖᑣᙠ᦮~$%Ḅ
oᑖ+ᑖ$%
oᑖ#ᓽ//(%)dx=r/(x)dx+dx.

14mj14UJJTT⚗ᦪᦣឋᑨ--------------------------------I(1)¡ᑨ_Cᜧᦈ£W\ᦈ\¤ᦣ£Wᜧ¤ᦣZ(2)O᪷ᑨ_C(P<1,ᦪ2j15ᦈZklim¦§=஺஺>1,ᦪ2ji%i¤ᦣlim஻n=+8©8uL-8n[P=l,ªᑨ_Cᜫᦔ[P<1,ᦪ©ᦈklimZ=pjp>1,ᦪ2—1%¤ᦣ,lim%=+871T8vIn->00IP=l,ªᑨ_Cᜫᦔ(3)Opᦪ¡E®1%=2¯i*>o,&p>i)ᦈ&P41)¤ᦣ.If2.°┯ᦪḄᦣឋ(²³´₈ᑨ_C)I°┯ᦪ2241)©1%1¶·%i2஻n+1&n>1,limuoonr0,ᑣEK1(—1¸1%1ᦈ.T3.¹ᦪᦈº»I¹ᦪ2j0¼(½-@¸ᑣᨵklim¾=/,ᑣᐸᦈº»©78an1,0(+oo,I=0

15I4.¹ᦪᦈÂfkÃᦪᙠw%o=RᜐᦈᑣᐸᦈÂ(-R+&,R+%A0kÃᦪᙠ%W=-Rᜐᦈ,ᑣᐸᦈÂ@-R+&,/?+%)0kÃᦪᙠÅ-'o=±Rᜐᦈ,ᑣᐸᦈÂ@—R+%OR+%OLÆᐗᦪvoᑖ1.jᐗᦪÇᦪÈk᩽▲lim"r"s᝞ᙠ,ᑣʪ᩽▲ᦪ/®y)ᙠX-^XQX-XQPo(%o,yo)ᜐᐵ'ḄÇᦪËÌ'(%(),y0).Hjᐗᦪᐰvᑖn---------------------------------------------------
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×I6n1/zÎy{dzr=ZᔳW/zS;A°tDi=i

16mj16UJJ14.jÖoᑖḄឋq£(1)ÙឋឋqI[a/o,y)+Sg(%y)]db=aIf(x,y)do+஺g(x,y)dcyJJj)JJDJJD(2)$ÂÚឋkÛ$Â஺ᑖ~Û$Â4ஹ஺2,Ó1J[[f(x,y)do=ff/(%,y)c/d+ff/(“)᝞JJDJJDJJD2T(3)¡
ᳮkᙠ஺/(x,y)

17mj17UJJঝ஺yoᑖ$Â஺æᵨp4wi(y)q%W%2(y),cwywd⊤ëᑣrrfdrX(y)2f(x,y)d(y=dyf(x,y)dxJJDJCঞ஺Xஹy᝞N஺ìpXxí$ÂîpXçí$Âᑣæ஺ᑖᡂkïᑖ,ðᑖXxíᡈç$ÂñᑭᵨjÖoᑖ$ÂÚឋ$Â஺ḄjÖoᑖóᓄᡂ+õ\$ÂḄjÖoᑖ#.(2)ᑭᵨöÊឋ÷ᓄrBজᑖ஺ᐵ!◱#$%&ᑣ0,k/(Îy)=f[jay)dxdy=*ffjay)dxdy,kf(%_y)=f(x,y)ᐸDi=Dn{x\0}ᑖ.ঝᑖ஺ᐵ!y#$%&ᑣ஻jo,k/(Oy)=J^f(x/y)dxdy-j2f(x,y)dxdy,k/(r,y)=f(x,y)ᐸDi=Dn{y஻0}ᑖ.ঞᑖ஺ᐵ!y=%$%&ᑖᐵ!)*+yᐹᨵ.ᣚ$%ឋ&ᑣf{x,y)dxdy=f(y,x}dxdyDτJJD

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(2)(A+F)C=AC+BC,C(A+B)=CA+CB
(3)k(AB)=(kA)B=A(kB)
(4)kA=(kE)A=A(kE).f3.Y:▣ḄSTUVtLJ(1)(AT)T=A
(2)(A+B)T=AT+BT
(3)("A=kAT
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20mj20UJJI6.:▣A"⌮Ḅᐙᑖs⌕ᩩvf1ভH0,wxy4"⌮zA-1=I7.⌮:▣ḄឋnI(i):▣a"⌮ᑣ⌮:▣B{|Ḅ}a—1y:▣/"⌮z⌮:▣/T~"⌮xQ4T)T=4(2):▣4"⌮ᑣ:▣5~"⌮x047)1=(41(3)A,B{▤"⌮:▣ᑣAB~"⌮xQ4B)T=8-.1(4):▣A"⌮kḄᦪᑣ""⌮x(")T=*4-1K(5)ভ=dW0,ᑣᨵ14Tl=/1=.IA|ifs.⌮:▣Ḅ(E[/T)<ᵨ48=£1ᡈ84=E,U"⌮x4-1=8.19.:▣Ḅ<](1)ᙠ:▣P,P/pT=B,ᑣᨵAB(2)Ḅᑨ¢Ḅᱯ¤ஹ¥(¦§4ᑮ¨©Ḅª«¬Ḅ)ஹ
ᑡḄ.

21iio.:▣Ḅᔠ¯(1)ᔠᙠ:▣P,PAPrB,ᑣᨵA°Bᔠ(2)ᔠḄᑨ¢±ஹ²ᱯ¤Ḅ/ᦪ.¬IឋI³´µ¶1.ᐗ·¸¬ឋ´¶4*=0¹Ḅº»----------------<(1)y¼Q4)=?!u>4%=஺ᨵ|¹¹(2)yrQ4)/?2@n_Aan&6?i-i@n᜜A(1)Bz&{02ÌÍÎÏ…“ᓫᓄÒ{VḄ/᪗Ô±ÃÊ.

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23mj23UJJI4.ë
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