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2022年9月24日 星期六

108年台綜大轉學考-工程數學D04詳解

臺灣綜合大學系統108學年度學士班轉學生聯合招生考試

科目名稱:工程數學
類組代碼:D04

解答
(a)

(b)f(x)={x+π,π<x<0x+π,0<x<πf(x)bn=0an;a0=12πππf(x)dx=12π(0πx+πdx+π0x+πdx)=12π×π2=π2an=1πππf(x)cos(nx)dx=1π(0π(x+π)cos(nx),dx+π0(x+π)cos(nx)dx)=1π([xnsin(nx)+1n2cos(nx)+πnsin(nx)]|0π+[xnsin(nx)1n2cos(nx)+πnsin(nx)]|π0)=1π2n2(1(1)n)={4n2π,n0,nf(x)=a0+n=1ancos(nx)f(x)=π2+2πn=11n2(1(1)n)cos(nx)
解答y+ω2y=0:λ2+ω2=0λ=±ωiyh=Acos(ωt)+Bsin(ωt)r(t)=sintω1;ω=0.5,1.2,2,10,1yp=Csintyp=CsintCsint+ω2Csint=sintC(ω21)=1C=1/(ω21)y=yh+ypy=Acos(ωt)+Bsin(ωt)+1ω21sint,AB
解答y=2xyydy=2xdx12y2=x2+Cy(1)=5252=1+CC=232y2=2x2+23
解答f=2exyz{fx=2yzexyzfy=2xzexyzfz=2xyexyz{fxx=2y2z2exyzfyy=2x2z2exyzfzz=2x2y2exyz2f=fxx+fyy+fzz=2exyz(x2y2+y2z2+z2x2)
解答{u=[abc]v=[def]u×v=[bfcecdafaebd](u×v)=[(bfce)(cdaf)(aebd)]=[bf+bfcececd+cdafafae+aebdbd](1){u×v=[abc]×[def]=[bfcecdafaebd]u×v=[abc]×[def]=[bfcecdafaebd]u×v+u×v=[bf+bfcececd+cdafafae+aebdbd](2)(1)=(2)u×v=u×v+u×v
解答A=[110110001]det(AλI)=0λ(λ+2)(λ1)=0{λ1=0λ2=2λ3=1λ1=0(Aλ1I)x=[110110001][x1x2x3]=0{x1+x2=0x3=0,v1=[110]λ2=2(Aλ1I)x=[110110003][x1x2x3]=0{x1=x2x3=0,v2=[110]λ3=1(Aλ1I)x=[210120000][x1x2x3]=0x1=x2=0,v3=[001]P=[v1v2v3]=[110110001]P1=[1212012120001]A=P[λ1000λ2000λ3]P1=[110110001][000020001][1212012120001]
解答{y1=y2+2u(t1)y2=y1+1u(t1)[y1y2]=[0110][y1y2]+[2u(t1)1u(t1)]y=Ay+bA=[0110]det(AλI)=λ2+1=0λ=±iλ1=i(Aλ1I)x=[i11i][x1x2]=0x1=ix2,v1=[i1]λ2=i(Aλ2I)x=[i11i][x1x2]=0x1=ix2,v2=[i1]y=Ay[y1y2]=C1eit[i1]+C2eit[i1]{y1=C1ieit+C2ieity2=C1eit+C2eit{y1=C1eit+C2eit+2u(t1)y2=C1ieitC2ieit+1u(t1){L{y1}=L{C1eit+C2eit+2u(t1)}L{y2}=L{C1ieitC2ieit+1u(t1)}{sY1(s)y1(0)=C1si+C2s+i+2ses/ssY2(s)y2(0)=C1isiC2is+i+1ses/s{Y1(s)=C1s(si)+C2s(s+i)+2s2+2ses/s2Y2(s)=C1is(si)C2is(s+i)+1s2+1ses/s2{Y1(s)=C1i(1si1s)+C2i(1s1s+i)+2s2+2ses/s2Y2(s)=C1(1si1s)C2(1s1s+i)+1s2+1ses/s2{y1=C1i(eit1)+C2i(1eit)+2t+2(t1)u(t1)y2=C1(eit1)C2(1eit)+t+1(t1)u(t1)
解答A=[103111010]rref(A)=[100010001]Rank(A)=3
解答(2,0,a1)(1,0,8)=2+8a1=0a1=14
解答{3x+y+2z=1xy+3z=3y2z=1[312113012][xyz]=[131]Ax=b{Ax=[112313112]Ay=[312133012]Az=[311113011]{det(Ax)=8det(Ay)=13det(Az)=6det(A)=1{x=det(Ax)/det(A)=8y=det(Ay)/det(A)=13z=det(Az)/det(A)=6{x=8y=13z=6

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