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:∫π/20sin2xcosxdx=[13sin3x]|π/20=13
解答:∫101x2+2dx=∫101x2+(√2)2dx=[1√2tan−1x√2]|10=1√2tan−11√2
解答:{x=r(θ−sinθ)y=r(1−cosθ)⇒{dx=r(1−cosθ)dθdy=rsinθdθ⇒擺線長=∫π/20√(dxdθ)2+(dydθ)2dθ=∫π/20√r2(1−cosθ)2+r2sin2θdθ=∫π/20r√2(1−cosθ)dθ=∫π/20r√4sin2(θ/2)dθ=∫π/202rsin(θ/2)dθ=[−4rcos(θ/2)]|π/20=(4−2√2)r
解答:{ln(1+x)=∑∞n=11n(−1)n+1xntan−1x=∑∞m=012m+1(−1)mx2m+1⇒考量ln(1+x)tan−1x的x5係數:{m=0,n=4:−14⋅1=−14m=1,n=2:−12⋅(−13)=16⇒f(x)=ln(1+x)tan−1x的x5係數=(−14+16)=−112⇒f[5](0)=−112×5!=−10
解答:x−y+z=0⇒y=x+z⇒{x+2y+3z=3x+5zx2+y2=x2+(x+z)2取{g(x,z)=3x+5zh(x,z)=x2+(x+z)2−29,利用Lagrange 算子求解{gx=λhxgz=λhzh=0⇒{3=λ(4x+2z)5=λ(2x+2z)⇒35=4x+2z2x+2z⇒x=−27z⇒y=x+z=57z⇒h(−2z/7,z)=0⇒2949z2=29⇒{z=7⇒x=−2,y=5z=−7⇒x=2,y=−5⇒{f(−2,5,7)=29f(2,−5,−7)=−29⇒最大值為29
解答:此題相當於求曲線y=√x繞x軸旋轉(x=1→9)所得的表面積即∫912π√xdx=[43πx3/2]|91=1043π
解答:令{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}
==================== END =====================
解題僅供參考,其他歷年試題及詳解
沒有留言:
張貼留言