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經濟數學方法壹、矩陣與行列式◎定義:−n×m階矩陣為一包括n列和m行的數字的方形排列，若以A代表此矩陣，則A=[a11a12…a1ma21a22⋯a2m⋮⋮⋯⋮an1an2⋯anm]=(aij)n×m例:A=[201312−1−2−3135],B=[−1113−3111]分別為4¿3和2¿4矩陣◎定義:若A=(aij)n×m,B=(bij)n×m則A+B=(aij+bij)n×m=(Cij)n×m=CαA=(αaij)n×m例:A=[213211],B=[−21251−3]則A+B=[2−21+13+22+51+11−3]=[02572−2]5A−B=5A+(−1)B=[105151055]+[2−1−2−5−13]=[12413548]A+A=[213211]+[213211]=[426422]=2[213211]=2A◎定義:若A=(aij)為n×m矩陣，B=(bij)為m×k矩陣，則A和B的乘積AB為n×k矩陣C例:A=[012201],B=[10021−1031]求AB及BAAB=[012201][10021−1031]=[0.1+1.2+2.00.0+1.1+2.30.0+1⋅(−1)+2⋅12⋅1+0.2+1.02.0+0.1+1.32.0+0⋅(−1)+1.1]=[271231]BA無法計算 3×32×3◎行列式:Cramer'sRule已知a11X1+a12X2=b1a21X1+a22X2=b2⇒X1¿=|b1a12b2a22||a11a12a21a22|=b1a22−b2a12a11a22−a12a21X2¿=|a11b1a21b2||a11a12a21a22|=a11b2−a21b1a11a22−a12a21例:解下列聯立方程式:[1−1112−1213][X1X2X3]=[520]⇒{X1−X2+X3=5X1+2X2−X3=22X1+X2+3X3=0X1¿=|5−1122−1013||1−1112−1213|=9=439X2¿=|15112−1203|9=−239X3¿=|1−15122210|9=−219±1貳、微分◎微分公式:Y=f(X)ΔYΔX=f'(X)=limΔX→0f(X+ΔX)−f(x)ΔX=dYdXΔ2YΔX2=d2YdX2=f''(X)◎若f(X)=Xn,∀X∈R⇒f'(X)=nXn−1,∀X∈R◎設f'(X)與g'(X)皆存在:xyexlnx11ddX{f(X)±g(X)}=df(X)dX±dg(X)dXddX{f(X)⋅g(X)}=f(X)dg(X)dX+g(X)df(X)dX[乘法公式]ddX{f(X)g(X)}=f'(X)g(X)−f(X)g'(X)g(X)2,g(X)≠0[除法公式]◎鏈鎖律(chainrule):設函數f與g皆可微分⇒ddXf(g(X))=f'(g(X))×g'(X)◎反函數(inversefunction):設函數f與g滿足f(g(Y))=Y⇔函數g為f之反函數g(f(X)=X且g=f−1⇒{f(f−1(Y))=Yf(f−1(X))=X◎偏微分:y=f(X1,X2)⇒∂y∂X1=f1(X1,X2)y=f(X1,X2)⇒∂y∂X2=f2(X1,X2)例:ddX3X2+2Y=6X◎全微分:y=f(X1,X2)dy=∂y∂X1dX1+∂y∂X2dX2例:TE=P¿Q⇒dTE=dP×2+dQ×P◎自然對數(e)與自然指數(ln):(x0,y0)y=f(x)αyx性質:(1)f(X)=ex⇒f(0)=1⇒limX→∞eX=0、limX→∞eX=∞f(X)=lnX⇒f(1)=0⇒limX→−∞lnX=−∞、limX→∞lnX=∞(2)ddXeX=eX(3)設f'存在⇒ddX(ef(X))=ef(X)⋅f'(X)(4)eX+Y=eX⋅eY,∀X,Y∈R(5)e−X=1eX(6)ddXlnX=1X,∀X>0(7)ddx|n||X|=1X,∀X≠0(8)ddX|n||f(X)|=f'(Xf(X)(9)lnX⋅Y=lnX+lnY(10)lnXY=lnX−lnY(11)lnXY=YlnX(12)elnX=X且lneX=X(13)YX=eXlnY◎切線與射線:給定切線上任一點(X,f/(x)>0X1X2f(x2)f(x1)XYabf/(x)<0X1X2f(x2)f(x1)XYabY)⇒y−f(X0)X−X0=f'(X)射線角度值tanα=y0X0◎函數的高階導數:d2YdX2=ddX{dYdX}、d3ydX3=ddX{d2YdX2}f''(X0)=limX→X0f'(X)−f'(X0)X−X0=limΔX0>0f'(X0+ΔX)−f'(X0)ΔX◎函數的臨界點及反曲點:(一)若X0∈Df(函數定義域)，f'(X0)=0(或f'(X0)不有在)，則X=X0為函數f之臨界點(二)函數f在[a,b]為嚴格遞增⇒X1X2f則(X1)f(X2)concaveupwardf/(x)>0f//(x)<0XY上凹f/(x)<0f//(x)>0XY上凹concavedownwardf/(x)<0f//(x)<0XY下凹f/(x)>0f//(x)<0XY上凹反曲點(inflectionpoint)上凹下凹xyC0(三)f''(X)>0∀X∈[a,b]⇔f函數在[a,b]為上凹f''(X)<0∀X∈[a,b]⇔f函數在[a,b]為下凹⇔故f'⇒函數遞增遞減性，f''⇒函數凹性(四)第一導數檢驗定理:f'(C)=0或f'(C)不存在xyf(C1)f(C2)C2局部最小值C1局部最大值XC切記f'-+f(C)為局部極小值f'+-f(C)為局部極大值f'--f'++f(C)為非局部極值第二導數檢驗定理:f'(C)=0f''(C)>0⇒f(C)為局部極小值f''(C)<0⇒f(C)為局部極大值f''(C)=0本定理失敗參、積分(一)不定積分(Indefiniteintegral)∫:f積分符號、(X):dX積分函數、:積分值⇒∫:f(X)dX而F'(X)=f(X)⇒f為F之導函數、F為f之xyf(x)ab∫dxxfba)(反導數故F為f之反導數⇒∫f(X)dX=F(X)+K(常數)◎性質:∫{f(X)±g(X)}dX=∫f(X)dX±∫g(X)dX∫Cf(X)dX=C∫f(X)dXddX{∫f(X)dX}=f(X)∫ddXf(X)dX=f(X)+C◎∫1XdX=|n|X|+C(二)定積分(definiteint...