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(4−3x)=5xe−6xln2⇒f′(x)=5e−6xln2−30ln2xe−3xln4=5(4−3x)−30ln2(4−3x)x(2)g(x)=log54x2√1−x=log54−log5x2√1−x=2log52−2log5x−12log5(1−x)=2log52−2ln5lnx−12ln5ln(1−x)⇒g′(x)=−2xln5+12(1−x)ln5(3)h(t)=sin(arccost)=sin(arcsin√1−t2)=√1−t2⇒h′(t)=−t√1−t2
解答:V=圓錐體體積=13×A(底面圓面積)×h(高)底面圓周長=Rθ=2πr(r是底面圓半徑)⇒r=Rθ2π又R2=r2+h2⇒h=√R2−r2⇒V=13r2π×√R2−r2⇒V(θ)=π3(Rθ2π)2√R2−(Rθ2π)2=R2θ212π√R2−R2θ24π2⇒V′(θ)=R2θ6π√R2−R2θ24π2−R4θ348π3⋅1√R2−R2θ24π2因此V′(θ)=0⇒R2−R2θ24π2=R2θ28π2⇒3θ28π2=1⇒θ=2√6π3
解答:此題相當於求藍色面積繞y軸旋轉所得體積的兩倍假設¯OA=a⇒r2=a2+(h2)2⇒a2=r2−(h/2)2V1=圓柱體體積=a2hπ=(r2−h24)hπV2=⌢PB繞y軸旋轉體積=∫h/20(r2−y2)πdy=(r22h−124h3)π欲求之體積=2V2−V1=(r2h−112h3)π−(r2−h24)hπ=16h3π

解答:(1)y′=10x2√1+x3⇒y=∫10x2√1+x3dx=∫10√u⋅13du=203√u+c1⇒y=203√1+x3+c1(2)exy′+4exy=1⇒y′+4y=e−x⇒ integration factor I(x)=e∫4dx=e4x⇒e4xy′+4e4xy=e3x⇒(e4xy)′=e3x⇒e4xy=∫e3xdx=13e3x+c1⇒y=13e−x+c1e−4x(3){P(x,y)=x2+y2Q(x,y)=−2xy⇒{Py=2yQx=−2y⇒Px≠Qy⇒Not ExactPy−QxQ=4y−2xy=−2x⇒u′=−2xu⇒integration factor u(x)=1x2⇒{uP=1+(y/x)2uQ=−2y/x⇒(uP)y=2yx2=(uQ)x⇒potential function Φ(x,y)=∫uPdx=∫uQdy⇒∫1+y2x2dx=∫−2yxdy⇒x−y2x+ϕ(y)=−y2x+ρ(x)⇒Φ=x−y2x+c1=0⇒y=±√x2+c1x(4)v(t)=1y⇒v′=−y′y2⇒y′=−y2v′⇒原式:y′=−y2v′=320y−y21600⇒v′=−320y+11600=−320v+11600⇒v′+320v=11600⇒e3t20v′+320e3t20v=11600e3t20⇒(e3t/20v)′=11600e3t20⇒e3t/20v=1240e3t20+c1⇒v=1y=c1e−3t30+1240=1+c2e−3t/20240⇒y=2401+c2e−3t/20⇒y(0)=15=2401+c2⇒c2=15⇒y=2401+15e−3t/20
解答:(1)∫π0tanθ3dθ=[−3ln(cos(θ/3)]|π0=−3ln12+3ln1=3ln2(2)∫0−2x2ex3/2dx=[23ex3/2]|0−2=23(1−1e4)(3)x2=5tan2u⇒{x2+5=5(1+tan2u)=5sec2udx=√5sec2udu⇒∫1(x2+5)3/2dx=∫√5sec2u(5sec2u)3/2du=∫√5sec2u5√5sec3udu=∫15secudu=15∫cosudu=15sinu+c1=x5√x2+5+c1(4){u=lnxdv=x5dx⇒{du=dx/xv=x6/6⇒∫x5lnxdx=16x6lnx−∫16x5dx=16x6lnx−136x6+c1(5)∫20dxx2−2x+2=∫20dx(x−1)2+1=[tan−1(x−1)]|20=tan−11−tan−1(−1)=π4−(−π4)=π2
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解題僅供參考,其他歷年試題及詳解
第三題從r那邊就錯了,r=R-Rθ/2π,麻煩勘誤了!
回覆刪除會錯題意, 修改完畢, 謝謝!
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