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2022年11月19日 星期六

107年台綜大轉學考-微積分A詳解

臺灣綜合大學系統107學年度學士班轉學生聯合招生考試

科目名稱:微積分A


解答(a)limnnn+1=limn11+1/n=1(b)limne1/x2(1x53x3)=limne1/x213x3x5=10=0
解答{Py=x3/2P(t,t3/2),tRQ(5/2,0)¯PQ=(t5/2)2+t3=t3+t25t+25/4f(t)=t3+t25t+25/4f(t)=3t2+2t5f(t)=6t+2f(t)=0(3t+5)(t1)=0t=1,5/3{f(1)>0f(5/3)<0t=1,¯PQ,P(1,1)
解答{x=3t2y=2t3{x=6ty=6t220(x)2+(y)2dt=2036t2+36t4dt=106t1+t2dt=213udu(u=1+t2)=[2u3/2]|21=25/22
解答(a)limn|an+1an|=limn|(n+1)2xn+1n2xn|=|x|<11(b)f(x)=11x=n=0xnf(x)=1(1x)2=n=1nxn1xf(x)=x(1x)2=n=1nxn(xf(x))=1+x(1x)3=n=1n2xn1g(x)=x(1+x)(1x)3=n=1n2xng(1/2)=6=n=1n22nn=1n22n=6
解答x2y+exyz2cos(xz)=0n|(1,1,0)=(2xy+yzexyz+2zsin(xz),x2+xzexyz,xyexyz+2xsin(xz))|(1,1,0)=(2,1,1):(2,1,1)(x1,y1,z)=02(x1)+(y1)+z=02x+y+z=3
解答{x=u2v2y=2uv{xu=2uxv=2vyu=2vyv=2u{gu=fxxu+fyyu=2ufx+2vfygv=fxxv+fyyv=2vfx+2ufy{guu=2fx+2u(xfxxu+yfxyu)+2v(xfyxu+yfyyu)=2fx+4u2fxx+8uvfxy+4v2fyygvv=2fx2v(xfxxv+yfxyv)+2u(xfyxv+yfyyv)=2fx+4v2fxx8uvfxy+4u2fyyguu+gvv=4(u2+v2)fxx+4(u2+v2)fyy=4(u2+v2)(fxx+fyy)=0
解答{u=x+yv=xy{x=(u+v)/2y=(uv)/2J=(x,y)(u,v)=|xuxvyuyv|=|1/21/21/21/2|=12Dsin(x+y)dA=0ππ012sinududv=0π1dv=π
解答{x=rcosθy=rsinθz=zEx2+y2dV=2π010r21r2dzdrdθ=2π010r4r2drdθ=2π0712dθ=76π
解答{M(x,y)=yex+sinyN(x,y)=ex+xcosy{My=ex+cosyNx=ex+cosyMy=Nxpotential function f滿{fx=Mfy=Nf=Mdx=N,dy{yex+sinydx=yex+xsiny+ϕ(y)ex+xcosydy=yex+xsiny+ρ(x)f(x,y)=yex+xsiny+C{p0=r(t=0)=(1,1)p1=r(t=2)=(5,3)=f(p1)f(p0)=3e5+5sin3(e+sin(1))=3e5+5sin3+e+sin1
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解題僅供參考,其他轉學考歷屆試題及詳解

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