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2023年9月4日 星期一

112年臺綜大轉學考-微積分C詳解

臺灣綜合大學系統112學年度學士班轉學生聯合招生

科目名稱:微積分C

解答limx1x1x1=limx1x1(x1)(x+1)=limx11x+1=12
解答π/20sin2xcosxdx=[13sin3x]|π/20=13
解答101x2+2dx=101x2+(2)2dx=[12tan1x2]|10=12tan112
解答{x=r(θsinθ)y=r(1cosθ){dx=r(1cosθ)dθdy=rsinθdθ=π/20(dxdθ)2+(dydθ)2dθ=π/20r2(1cosθ)2+r2sin2θdθ=π/20r2(1cosθ)dθ=π/20r4sin2(θ/2)dθ=π/202rsin(θ/2)dθ=[4rcos(θ/2)]|π/20=(422)r
解答{ln(1+x)=n=11n(1)n+1xntan1x=m=012m+1(1)mx2m+1ln(1+x)tan1xx5:{m=0,n=4:141=14m=1,n=2:12(13)=16f(x)=ln(1+x)tan1xx5=(14+16)=112f[5](0)=112×5!=10
解答xy+z=0y=x+z{x+2y+3z=3x+5zx2+y2=x2+(x+z)2{g(x,z)=3x+5zh(x,z)=x2+(x+z)229Lagrange {gx=λhxgz=λhzh=0{3=λ(4x+2z)5=λ(2x+2z)35=4x+2z2x+2zx=27zy=x+z=57zh(2z/7,z)=02949z2=29{z=7x=2,y=5z=7x=2,y=5{f(2,5,7)=29f(2,5,7)=2929
解答y=xx(x=19)912πxdx=[43πx3/2]|91=1043π
解答{x=rsinϕcosθy=rsinϕsinθz=rcosϕBe(x2+y2+z2)3/2dV=2π0π010er3r2sinϕdrdϕdθ=13(e1)2π0π0sinϕdϕdθ=23(e1)2π01dθ=43(e1)π
解答 Green theorem {P(x,y))=y2+sinxQ(x,y)=3xyey{yP=2yxQ=3yC(y2+sinx)dx+(3xyey)dy=D(QxPy)dxdy=Dydxdy=π032r2sinθdrdθ=π0193sinθdθ=383
解答F(x,y,z)=xyi+(y2+exz)j+cos(xy)kdiv F=Fx+Fy+Fz=y+2y+0=3y{z=1x2z=0y=0y+z=2E{(x,y,z)1x1,0y2z,0z1x2}:SFdS=Ediv FdV=111x202z03ydydzdx=111x2032(z2)2dzdx=11412(x2+1)3dx=18435

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解題僅供參考,其他歷年試題及詳解

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