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2023年9月4日 星期一

112年臺綜大轉學考-微積分C詳解

臺灣綜合大學系統112學年度學士班轉學生聯合招生

科目名稱:微積分C

解答limx1x1x1=limx1x1(x1)(x+1)=limx11x+1=12
解答π/20sin2xcosxdx=[13sin3x]|π/20=13
解答101x2+2dx=101x2+(2)2dx=[12tan1x2]|10=12tan112
解答{x=r(θsinθ)y=r(1cosθ){dx=r(1cosθ)dθdy=rsinθdθ=π/20(dxdθ)2+(dydθ)2dθ=π/20r2(1cosθ)2+r2sin2θdθ=π/20r2(1cosθ)dθ=π/20r4sin2(θ/2)dθ=π/202rsin(θ/2)dθ=[4rcos(θ/2)]|π/20=(422)r
解答{ln(1+x)=n=11n(1)n+1xntan1x=m=012m+1(1)mx2m+1ln(1+x)tan1xx5:{m=0,n=4:141=14m=1,n=2:12(13)=16f(x)=ln(1+x)tan1xx5=(14+16)=112f[5](0)=112×5!=10
解答xy+z=0y=x+z{x+2y+3z=3x+5zx2+y2=x2+(x+z)2{g(x,z)=3x+5zh(x,z)=x2+(x+z)229Lagrange {gx=λhxgz=λhzh=0{3=λ(4x+2z)5=λ(2x+2z)35=4x+2z2x+2zx=27zy=x+z=57zh(2z/7,z)=02949z2=29{z=7x=2,y=5z=7x=2,y=5{f(2,5,7)=29f(2,5,7)=2929
解答y=xx(x=19)912πxdx=[43πx3/2]|91=1043π
解答{x=rsinϕcosθy=rsinϕsinθz=rcosϕ
解答利用\text{ Green theorem }求解,令\cases{P(x,y))= y^2+\sin x\\ Q(x,y)=3xy-e^y} \Rightarrow \cases{{\partial \over \partial y}P=2y\\ {\partial \over \partial x} Q=3y} \\ \Rightarrow \oint_C (y^2+\sin x)dx+(3xy-e^y)dy = \iint_D \left({\partial Q\over \partial x} -{\partial P\over \partial y}\right)dxdy = \iint_D y\,dxdy \\=\int_0^\pi \int_2^3 r^2\sin \theta \,drd\theta   =\int_0^\pi {19\over 3}\sin \theta\,d\theta =\bbox[red, 2pt]{38\over 3}
解答\mathbf F(x,y,z)=xy\mathbf i+(y^2+e^{xz})\mathbf j+ \cos(xy)\mathbf k \Rightarrow \text{div }\mathbf F =F_x +F_y+F_z = y+2y+0=3y\\ \cases{z=1-x^2\\ z=0\\ y=0\\ y+z=2} \Rightarrow E\{(x,y,z)\mid -1\le x\le 1,0\le y\le 2-z, 0\le z\le 1-x^2\}\\利用發散定理: \\\iint_S \mathbf F\cdot d\mathbf S =\iiint_E \text{div }\mathbf F\,dV =\int_{-1}^1 \int_0^{1-x^2} \int_0^{2-z} 3y\,dydzdx=\int_{-1}^1 \int_0^{1-x^2} {3\over 2}(z-2)^2\,dzdx \\= \int_{-1}^1 4-{1\over 2}(x^2+1)^3\,dx =\bbox[red, 2pt]{184\over 35}

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