國立清華大學112學年度碩士班考試入學
系所班組別: 生命科學暨醫學院(丙組)
考試科目: 微積分
解答:(A){f(x)=lnxg(x)=ln|x|⇒f′(x)=1x=g′(x)也就是說f′(x)=g′(x)不代表f(x)=g(x)+C⇒不正確(B)g(x)=x3⇒g−1(x)=x1/3⇒ddxg−1(x)=13x2/3⇒ddxg−1(0)不存在⇒g−1(x)在x=0不可微⇒不正確
解答:limx→0tan−1xx=limx→0(tan−1x)′(x)′=limx→011+x2=1
解答:(A)xy+y3=1⇒y3(0)=1⇒y(0)=1(xy+y3)′=(1)′⇒y+xy′+3y2y′=0⇒(x+3y2)y′=−y⇒y′=−yx+3y2⇒y′(0)=−13y(0)=−13(B)y−1=y′(0)(x−0)⇒x+3y=3
解答:
等腰△ABC,其中{¯AB=¯AC∠ABC=∠ACB=θ⇒∠BAC=π−2θ假設¯OP⊥¯AC⇒¯AP=¯OAcos∠OAP=rcos(π2−θ)=rsinθ⇒¯AC=2rsinθ⇒△ABC面積=f(θ)=12¯AB¯ACsin∠BAC=2r2sin2θsin(π−2θ)=2r2sin2θsin(2θ)⇒f′(θ)=4r2sinθcosθsin(2θ)+4r2sin2θcos(2θ)=4r2sinθ(cosθsin(2θ)+sinθcos(2θ)=4r2sinθsin(3θ)=0⇒3θ=π⇒θ=π/3⇒f(π/3)=34√3r2
解答:I=∫1x4+4dx=∫(1/4−x/8x2−2x+2+1/4+x/8x2+2x+2)dx=I1+I2,where I1=18∫2−xx2−2x+2dx,I2=18∫2+xx2+2x+2dxI1=18∫(1−xx2−2x+2+1x2−2x+2)dx=18(−12ln(x2−2x+2)+tan−1(x−1))+c1By the same way, I2=18(12ln(x2+2x+2)+tan−1(x+1))+c2⇒I=I1+I2=116(ln(x2+2x+2)−ln(x2−2x+2))+18(tan−1(x−1)+tan−1(x+1))+c3=116lnx2+2x+2x2−2x+2+14tan−1x2−x2+C
解答:I=∫√1+(y′)2dx=∫√1+(23x−1/3)2dx=∫√1+49x−2/3dxLet u=x−1/3, then du=−13x−4/3dx=−13u4dx⇒dx=−3duu4⇒I=−3∫√1+49u2u4duLet u=32sinhv⇒du=32coshvdv⇒I=−3∫coshv8116sinh4v⋅32coshvdv=−89∫cosh2vsinh4vdv=−89∫1tanh4vcosh2vdvLet w=tanhv⇒dw=sec2vdv⇒I=−89∫1w4dw=827w−3+C=x27(9+4x−2/3)3/2+C⇒∫81√1+(y′)2dx=827103/2−127133/2=8027√10−1327√13
解答:dydx=x2ycosx⇒1ydy=x2cosxdx⇒lny=(x2−2)sinx+2xcosx+c1⇒y=c2e(x2−2)sinx+2xcosx⇒y(0)=c2=2⇒c2=2⇒y=2e(x2−2)sinx+2xcosx
解答:(A)cosθ=→a⋅→b|→a||→b|=√32⋅1⇒θ=π6(B)→a×(→c×→d)=(1,0,√3)×((−2,3,4)×(3,−1,1))=(1,0,√3)×(7,14,−7)=(−14√3,7+7√3,14)
解答:{x=rcosθy=rsinθ⇒∬Ωxy2√x2+y2dA=∫π/2π/4∫√30r3cosθsin2θ⋅r⋅rdrdθ=∫π/2π/4∫√30r5cosθsin2θdrdθ=∫π/2π/492cosθsin2θdθ=[32sin3θ]|π/2π/4=32(1−√24)
解答:I=∫1x4+4dx=∫(1/4−x/8x2−2x+2+1/4+x/8x2+2x+2)dx=I1+I2,where I1=18∫2−xx2−2x+2dx,I2=18∫2+xx2+2x+2dxI1=18∫(1−xx2−2x+2+1x2−2x+2)dx=18(−12ln(x2−2x+2)+tan−1(x−1))+c1By the same way, I2=18(12ln(x2+2x+2)+tan−1(x+1))+c2⇒I=I1+I2=116(ln(x2+2x+2)−ln(x2−2x+2))+18(tan−1(x−1)+tan−1(x+1))+c3=116lnx2+2x+2x2−2x+2+14tan−1x2−x2+C
解答:I=∫√1+(y′)2dx=∫√1+(23x−1/3)2dx=∫√1+49x−2/3dxLet u=x−1/3, then du=−13x−4/3dx=−13u4dx⇒dx=−3duu4⇒I=−3∫√1+49u2u4duLet u=32sinhv⇒du=32coshvdv⇒I=−3∫coshv8116sinh4v⋅32coshvdv=−89∫cosh2vsinh4vdv=−89∫1tanh4vcosh2vdvLet w=tanhv⇒dw=sec2vdv⇒I=−89∫1w4dw=827w−3+C=x27(9+4x−2/3)3/2+C⇒∫81√1+(y′)2dx=827103/2−127133/2=8027√10−1327√13
解答:dydx=x2ycosx⇒1ydy=x2cosxdx⇒lny=(x2−2)sinx+2xcosx+c1⇒y=c2e(x2−2)sinx+2xcosx⇒y(0)=c2=2⇒c2=2⇒y=2e(x2−2)sinx+2xcosx
解答:(A)cosθ=→a⋅→b|→a||→b|=√32⋅1⇒θ=π6(B)→a×(→c×→d)=(1,0,√3)×((−2,3,4)×(3,−1,1))=(1,0,√3)×(7,14,−7)=(−14√3,7+7√3,14)
解答:{x=rcosθy=rsinθ⇒∬Ωxy2√x2+y2dA=∫π/2π/4∫√30r3cosθsin2θ⋅r⋅rdrdθ=∫π/2π/4∫√30r5cosθsin2θdrdθ=∫π/2π/492cosθsin2θdθ=[32sin3θ]|π/2π/4=32(1−√24)
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解題僅供參考,其他歷年試題及詳解
第五題,題目給的是1/(x^4+4)不是1/(x^4+1),則答案不對.
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