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2024年3月23日 星期六

112年清華生命科學碩士班-微積分詳解

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)f(x)=g(x)f(x)=g(x)+C(B)g(x)=x3g1(x)=x1/3ddxg1(x)=13x2/3ddxg1(0)g1(x)x0

解答:limx0tan1xx=limx0(tan1x)(x)=limx011+x2=1
解答:(A)xy+y3=1y3(0)=1y(0)=1(xy+y3)=(1)y+xy+3y2y=0(x+3y2)y=yy=yx+3y2y(0)=13y(0)=13(B)y1=y(0)(x0)x+3y=3
解答:


ABC,{¯AB=¯ACABC=ACB=θBAC=π2θ¯OP¯AC¯AP=¯OAcosOAP=rcos(π2θ)=rsinθ¯AC=2rsinθABC=f(θ)=12¯AB¯ACsinBAC=2r2sin2θsin(π2θ)=2r2sin2θsin(2θ)f(θ)=4r2sinθcosθsin(2θ)+4r2sin2θcos(2θ)=4r2sinθ(cosθsin(2θ)+sinθcos(2θ)=4r2sinθsin(3θ)=03θ=πθ=π/3f(π/3)=343r2
解答:I=1x4+4dx=(1/4x/8x22x+2+1/4+x/8x2+2x+2)dx=I1+I2,where I1=182xx22x+2dx,I2=182+xx2+2x+2dxI1=18(1xx22x+2+1x22x+2)dx=18(12ln(x22x+2)+tan1(x1))+c1By the same way, I2=18(12ln(x2+2x+2)+tan1(x+1))+c2I=I1+I2=116(ln(x2+2x+2)ln(x22x+2))+18(tan1(x1)+tan1(x+1))+c3=116lnx2+2x+2x22x+2+14tan1x2x2+C
解答:I=1+(y)2dx=1+(23x1/3)2dx=1+49x2/3dxLet u=x1/3, then du=13x4/3dx=13u4dxdx=3duu4I=31+49u2u4duLet u=32sinhvdu=32coshvdvI=3coshv8116sinh4v32coshvdv=89cosh2vsinh4vdv=891tanh4vcosh2vdvLet w=tanhvdw=sec2vdvI=891w4dw=827w3+C=x27(9+4x2/3)3/2+C811+(y)2dx=827103/2127133/2=802710132713
解答:dydx=x2ycosx1ydy=x2cosxdxlny=(x22)sinx+2xcosx+c1y=c2e(x22)sinx+2xcosxy(0)=c2=2c2=2y=2e(x22)sinx+2xcosx
解答:(A)cosθ=ab|a||b|=321θ=π6(B)a×(c×d)=(1,0,3)×((2,3,4)×(3,1,1))=(1,0,3)×(7,14,7)=(143,7+73,14)
解答:{x=rcosθy=rsinθΩxy2x2+y2dA=π/2π/430r3cosθsin2θrrdrdθ=π/2π/430r5cosθsin2θdrdθ=π/2π/492cosθsin2θdθ=[32sin3θ]|π/2π/4=32(124)


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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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