AP微积分APCalculus公式大全完整版.docx

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1、11imit1.ImPOrta11tIimitSIimXfosinx1=1,xIimIimx-Q()APCa1cu1usBCIimG+t=e1oIimx-OsinaxabxXm+am4+ax+aabmm=nIimIim0,a1)f(x)XigWinnObn+bXnT+.+bx+bGiventwodifferentiab1efunctionsf(x)andg(x)(Xn)-nx-(4)(6)(e)e(a)=aIna(axfcinx)=cosx(tanx)-sec2(aQaD(cosx)=-sinx(CoQ-esc2(secx),=secxtanX(escx)=-escxcotx(arcsinx)

2、=-J=1(arccosx)=(arctanx)=211+x21-X2(arcotx)=一一1+x2(arcsecx)=IX17x2Tarc1(esc),=-xV2. RUIeS(1) Ify(X),g(x)arcdifferentia1,a. (f(x)g(x)=f(X)g(x)；b. (f(x)g(x)=f(x)g(x)f(x)gsw(Cisaconstam);w)=()0-()Ox.“1、g(x)c. 5fg2($Xg,，(g(x)0)espec,a1,y)gxgxgxQ/dydyduChainRU1ey=f1g(x)J=-dxdudx(4)Inversefunction互j(3)Imp

3、1icitDifferentiationf(x)I=-wheref(a)=b,thatisf(b)=a-I-Idy(5) Parametricfunctiondy_dtdxdx_)sincos(6) Po1arfunction二2一二-dx1()cossin(7) Vector()3.APD1iCatiO11SOfDeriVatiVe(1) MeanVa1ueTheorem,(c)二isdifferentiab1e,f1oca1naxmin:FirStDeriVatiVeTeStSeCondDeriVatiVeTeStf,changesfrom-to+,1oca1minf”(a)O11oca1

4、minf,changesfrom+to-,1oca1maxf(a)0,1oca1max(4)Abso1utemax/min:comparefunctionva1ueofcritica1pointandendpoints.Steps:1.findf;V2.so1ve1；pareva1ueatcritica1pointandendpoints.I11InIegra11. I11defi11iteInteRra1df(x)=f(x)+CDefinition:f,(x)dx=f(x)+CMethod2. MethOdSforIndefi1IitCInteera1不定积分方法Formu1as：kdx=k

5、x+C,dx=x+C(integrand为1),adx=-+CIna1dX=In1XI+C,xedx=e+C-dx=InIax+b+Cax+basnxdx=-cosx+Csi2xdx-csxdx=ix-isin2x+C2dx=x+-sin2x+C224224jecxdx=InSeCx+tanx+COCXdX=-1ncscx+cotx+COIXdX=InISinx+C.x,rdx二arcsin+Ca1111xj.Gx=arctan+C2aa11.1x,-dx=-arcsec+C22aaU-substitution(常见U的选择，分母，底数，指数，角度等)f(u(x)u,(x)dx=f(u)duS

6、teps:1.1etu=u(x),finddu2. expressintegrandintermsOfU,(即表示为U的函数f(u),3. Findtheantiderivativeoff(u).Partia1fraction(拆分)C21td(x-a)(x-b)dx=A.-B-xadx+Xfdx=A1nIX-8|+8111婕-1)|+(；(通分求人上)Integrationbypartsudv=uv-VdU,(oruv,dx=uv-vu,dxTabu1arIntegration,(storyabout1nx,sinx,andex).3. ImPrOMrIntsra1反常积分（两种形式，无穷积

7、分,瑕积分）Integra1oninfiniteinterva1fXdxba()=Iwnafxdxbf(x)dx=IimbK(b-a()=C()fXdxcfXdxbfxdxfxdxc()=m1CfXdXIntegrandwithinfinitediscontinuities()()a()dx-1m-af()dx,bf(x)dx=Iimbf(x)cbaCTCbf(x)dx=cf(x)dxbf(x)dx=Iimdf(x)dx+Iimbf(x)dx38cd*c-&d*,c*d4. RU1eSRiemannSum(1RAM,RRAM,MRAM,Trapezoida1Ru1e)(x)+1g(x)dx=k

8、bf(x)dx+1bg(x)dx,aaabcbaff(x)dx=2i1f(x)dx(f(x)iseven)o5. TheOrembfborf(b)=f(a)+bf,(x)dxaaa(),()-f(),(),where(,)cab.Rightfunction1effcticn-xf-d-x()a)f(t)dt=f(X)6. ADDIiCatiOnOfInteqra1MeanVa1uetheorem:(averageva1ueoffHorizonta1S1icedf(y)-g(y)dyArea：Vertica1S1icebf(x)g(x)dxaupperfunction0Actionba:2UA(

9、3) Vb1ume：withknowncrosssectionA,)istheareaofthecrosssection,aAxRevo1ution：1,*2-2dx,aOuternidiusInnernidiusShe11Method：2bydx(旋转轴为y-axis)or2bydy(旋转轴为x-axis)1engthIV.DiffCrentia1EaUatien1. SeParatiO11Variab1e二个N(y)dy-M(x)dx2. 1ogisticEquation=kP(1-)P(t)=,1imP(t)=dtK1e-kt-capacity.3. SI。DeFieIdX=X+Xn+1

10、n4. Eu1ersMethCd:dyy=y+xn+ndx3.Iin1=11nI2.anConvergesSnS-U_1.aAssumethatthefo11owing1imitexists:=Iimfnana)Ifaconvergesabso1ute1y,nn=11n=1=1K,whereistheenvironmenta1,wherefxispositive,decreasingandcontinuousfor1Integra1Test1etBoth()ifxdxand1.ITJHJ1A,AA(a)/2+.+(-a)nn2!,Z_f(n)Mac1aurinSeries：tW=Interva1ofconverge