2024年3月15日 星期五

112年政大風險管理精算組碩士班-微積分詳解

 國立政治大學112學年度碩士班暨碩士在職專班

考試科目:微積分
系所別:風險管理與保險系精算科學組



解答:(a)xdx82x2x4=xdx9(x2+1)2=xdx31(x2+13)2Let u=x2+13du=23xdxxdx=32duxdx31(x2+13)2=32du31u2=1211u2du=12sin1u+c=12sin1x2+13+c(b)21x2+4x4+3x3+2x2dx=213x+5x+1+2x+2+2x2dx=[3lnx+5ln(x+1)2ln(x+2)2x]|21=7ln312ln2+1(c){u=x2dv=exdx{du=2xdxv=exx2exdx=x2ex+2xexdx=x2ex2xex2ex+c0x2exdx=[x2ex2xex2ex]|0=2

解答:limn|xn+1ln(n+2)ln(n+1)xn|=limn|ln(n+1)ln(n+2)x|<1|x|<1
解答:(a)limx0x20t1+t2dtx4=limx0(x20t1+t2dt)(x4)=limx0x21+x42x4x3=limx0241+x4=12(b)L=(ex1)1lnxlnL=ln(ex1)lnxlimx0+lnL=limx0+(ln(ex1))(lnx)=limx0+xexex1=limx0+(xex)(ex1)=limx0+xex+exex=limx0+(x+1)=1limx0+L=e1=e(c)L=(1+1x)xlnL=xln(1+1x)limxlnL=limxln(1+1x)1x=limx(ln(1+1x))(1x)=limx1x(x+1)1x2=limxxx+1=1limxL=e1=e(d)limn(1n+1+1n+2++1n+2n)=limnnk=11n+k=limnnk=11/n1+k/n=1011+xdx=ln2
解答:(a)By the triangle inequality, we have ||x||y|||xy|. Now, limxaf(x)=L. That is, ϵ>0,δ>0 such that |xa|<δ|f(x)L|<ϵ||f(x)||L|||f(x)L|<ϵlimxa|f(x)|=|L|,Q.E.D.(b)An reverse example: Let f(x)={2xa2x<a, then we have limxa|f(x)|=|2|, but limxaf(x) does not exist

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解題僅供參考,其他歷年試題及詳解

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