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2024年2月1日 星期四

111年成大環工碩士班-微積分詳解

 國立成功大學111學年度碩士班招生考試

系所:環境工程學系
科目:微積分

解答:(1)f(x)=5x(43x)=5xe6xln2f(x)=5e6xln230ln2xe3xln4=5(43x)30ln2(43x)x(2)g(x)=log54x21x=log54log5x21x=2log522log5x12log5(1x)=2log522ln5lnx12ln5ln(1x)g(x)=2xln5+12(1x)ln5(3)h(t)=sin(arccost)=sin(arcsin1t2)=1t2h(t)=t1t2

解答:1+ln(xy)=exyy+xyxy=(1y)exyy(1)+1y(1)1y(1)=(1y(1))e1y(1)1+y(1)=1y(1)y(1)=0 tangent line:y=1



解答:V==13×A()×h()=Rθ=2πr(r)r=Rθ2πR2=r2+h2h=R2r2V=13r2π×R2r2V(θ)=π3(Rθ2π)2R2(Rθ2π)2=R2θ212πR2R2θ24π2V(θ)=R2θ6πR2R2θ24π2R4θ348π31R2R2θ24π2V(θ)=0R2R2θ24π2=R2θ28π23θ28π2=1θ=26π3

解答:


y¯OA=ar2=a2+(h2)2a2=r2(h/2)2V1==a2hπ=(r2h24)hπV2=PBy=h/20(r2y2)πdy=(r22h124h3)π=2V2V1=(r2h112h3)π(r2h24)hπ=16h3π



解答:x2a2+y2b2=1y2=b2b2a2x2y=±b2b2a2x2area of the ellipse: 4a0b2b2a2x201dydx=4a0b2b2a2x2dx=4ba011a2x2dx=4b0π/21cos2θ(asinθ)dθ=4ab0π/2sin2θdθ=2ab0π/21cos(2θ),dθ=2ab×(π2)=abπ


解答:(1)y=10x21+x3y=10x21+x3dx=10u13du=203u+c1y=2031+x3+c1(2)exy+4exy=1y+4y=ex integration factor I(x)=e4dx=e4xe4xy+4e4xy=e3x(e4xy)=e3xe4xy=e3xdx=13e3x+c1y=13ex+c1e4x(3){P(x,y)=x2+y2Q(x,y)=2xy{Py=2yQx=2yPxQyNot ExactPyQxQ=4y2xy=2xu=2xuintegration factor u(x)=1x2{uP=1+(y/x)2uQ=2y/x(uP)y=2yx2=(uQ)xpotential function Φ(x,y)=uPdx=uQdy1+y2x2dx=2yxdyxy2x+ϕ(y)=y2x+ρ(x)Φ=xy2x+c1=0y=±x2+c1x(4)v(t)=1yv=yy2y=y2v:y=y2v=320yy21600v=320y+11600=320v+11600v+320v=11600e3t20v+320e3t20v=11600e3t20(e3t/20v)=11600e3t20e3t/20v=1240e3t20+c1v=1y=c1e3t30+1240=1+c2e3t/20240y=2401+c2e3t/20y(0)=15=2401+c2c2=15y=2401+15e3t/20

解答:(1)π0tanθ3dθ=[3ln(cos(θ/3)]|π0=3ln12+3ln1=3ln2(2)02x2ex3/2dx=[23ex3/2]|02=23(11e4)(3)x2=5tan2u{x2+5=5(1+tan2u)=5sec2udx=5sec2udu1(x2+5)3/2dx=5sec2u(5sec2u)3/2du=5sec2u55sec3udu=15secudu=15cosudu=15sinu+c1=x5x2+5+c1(4){u=lnxdv=x5dx{du=dx/xv=x6/6x5lnxdx=16x6lnx16x5dx=16x6lnx136x6+c1(5)20dxx22x+2=20dx(x1)2+1=[tan1(x1)]|20=tan11tan1(1)=π4(π4)=π2
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解題僅供參考,其他歷年試題及詳解

2 則留言:

  1. 第三題從r那邊就錯了,r=R-Rθ/2π,麻煩勘誤了!

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