微观经济学数学基础第7章经典微积分和线性代数

1、第七章基础微积分和线性代数第七章基础微积分和线性代数 717471174171274271374371474472757217517227527237537247673761731762732763733764第七章基础微积分和线性代数 7.1集合和函数711(set)(elenlent)AB%yzx&AA*AxgAxA07-1)()A=12A=xxA=xlx2+3x+6=0ABABZ)AABBBAAB(subset)Au3AB(fixedincomesecurity)AuBA(zAAuBBZ)AABA=BAcBABAB(propersubset)0711A=13=1.2(rxx2-3x+2=

2、0AA0B=CrB4B0123BAnA2nA2aA(powerset)(intersection)(complement)(unoin)1)AB()刀ABeq3)AABAkJB1GUGUcUcZgABABABgGc“Gr-lCCj=0Z&jAAAc=xxeSxiA(1)AuB=BB)rD(1)Au(BrD)=(AuB)r(AuD)(2)An(BoD)=(AnS)u(AnD)CJ12C”W2limsupCnlimCnnTOOlimsupC,=niJCA=HveCnnMi*liminfCnnliminfCn=JQca=wIweCnnIkJiweliminfCnnNweCnnwelimsupCnli

3、minfCnUlimsupC”(liminfCn)c=limsupC：(limsupCZI)c=liminfC爲帀“刃limsupCn=Q|JCXnn-ljliminfCjt=Up|CxJtIkiCM=12(indexset)(difference)7=L2nABSAB=xeSxeAAgBAB(difference)(DeMorgan)(1)(AoB)c=AcnBc(2)(AcB)e=u第七章基础微积分和线性代数 liminfCn=limsupC”C=liminfCfl=limsupCnCGnqnG(cj()liminfC=CC,cC2cC3clC”(tribe)()J來cb712)QeT2)

4、3)TQ7131)QACT_3)Q(Borel)g-HmCZI=lJC=C”1T8HmCZI=nc=CJt-1T8(space)(subset)ucf./-IAeTAceT)-0(algebra)UC”i-i(field)o_4R$(R)(a.h)a)Reb第七章基础微积分和线性代数 (topology)州宀宀忑xnnaAA(closed(openset)b(boundedset)71466TcA-bayxx3)(transitivity)ayyzxz(binary)7-1(preferenceordering)xyXxyxyxyx-y%yXxy()AlAcRa.h=x:axb(a.h)=xi

5、axh第七章基础微积分和线性代数 #第七章基础微积分和线性代数 #eb7-4a.h)=x:axbY.+oo)第七章基础微积分和线性代数 第七章基础微积分和线性代数 713XY1)X2)XYfXYYcRfX(function)714XY()f(image)YXY(12711Y/：xY(functional)XfXY()xD)lx2xTy=f(x):RnTR“(0a=/(A)yM=FlayA(yy)(inverseimage)=/(A)(inversemapping)712厂：YTX()s=/(/)cc=g(S)ct(compoundfunction)C=gf(t)x2+y2=1f(x.y)=a/

6、XypxP713!714)()91)(continuity)(continousfunction)NicholasBourbiiki2)(monotonity)y=f(x),xeD/(X2)h(x)()f(x)/U)3)(linearity)a0(affinetransformation)(affine)f(x)=a+bx(linearfunction)(cardinalutility)4)(concavity)y=f(x)9xeD714(convexity)/X),x2&D0A妙(和+(l-A)/(x2)f(x)10第七章基础微积分和线性代数 #第七章基础微积分和线性代数 #io7-37-4

7、a201015W15)cE(x)=0.5以(10)+05(20)ABB7-4b7.6.2第七章基础徽积分和线性代数 II01Bo1o-!1乙亠in!(r)/2.7uo【旳（旧)(l(X)/x吸gjr(p+XU)X)(.Y40-.Y)x任(X)/(Z(1（V）/(W=(o+v)/=(o-M/(r)/=(x)/r-r【Wl吒(X)/=Xr=(x)/iunii().YXjx()X吹gjJ8JJMXN|-Hx|8No000(order)f(x)g(x)x5r/(X)Inn=Zfg(x)1)0vzoo/U)g(Q()g(x)2)z=0/U)g(x)f(x)=Og(x)3)z=oo/(X)g(x)f(x)

8、=Og(x)(7-6)XTOX2X0722y=/U)721liin=人A.vAy=/U)(derivative)7-7Ay/ZLvABtgpzBAABfg(xAAC第七章基础微积分和线性代数 1870II017-7y=/(x)-vA/（X）722dnydxn722C=0.01少-03少+400+1000(marginal)12=0.00302-0.60+40dQdQ1=0.0060-0.6()o()7-6DE()0(0)()()0()Max一s.t()7-47-4I第七章基础微积分和线性代数 14II01()()/：RTRXHX-/(A)X/(A)第七章基础微积分和线性代数 #14II01第七

9、章基础微积分和线性代数 #14II010/F0广(F)=o/(a*)0R1)2)3)/(x)014723%(x)=x+ax1,a0W(x)=1+lax=0.v=-0.5a”(x)=2dS0(regularity)第七章基础微积分和线性代数 II01第七章基础微积分和线性代数 II01r=g(t)723(Itoformulas)A-=/_，(.y)y=f(x)7247-10Lx7-9(chainrule)BB=flg(0dBdg(t)dtdg(f)dtdydyIdxyA%xAv/A.vrdxfxdxxdy=fxdxLx/(a+Aa)/(x)+/aAvx.v+A.vBdy(tgaLx)AvBCLx9TII01(Tayodsformula)x)xHxefs芒f(X)HM3+fGo)(xAo)72570n0co)+(xo)(xX。)+5