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⇒∭Be(x2+y2+z2)3/2dV=∫2π0∫π0∫10er3⋅r2sinϕdrdϕdθ=13(e−1)∫2π0∫π0sinϕdϕdθ=23(e−1)∫2π01dθ=43(e−1)π
解答:利用 Green theorem 求解,令{P(x,y))=y2+sinxQ(x,y)=3xy−ey⇒{∂∂yP=2y∂∂xQ=3y⇒∮C(y2+sinx)dx+(3xy−ey)dy=∬D(∂Q∂x−∂P∂y)dxdy=∬Dydxdy=∫π0∫32r2sinθdrdθ=∫π0193sinθdθ=383
解答:F(x,y,z)=xyi+(y2+exz)j+cos(xy)k⇒div F=Fx+Fy+Fz=y+2y+0=3y{z=1−x2z=0y=0y+z=2⇒E{(x,y,z)∣−1≤x≤1,0≤y≤2−z,0≤z≤1−x2}利用發散定理:∬SF⋅dS=∭Ediv FdV=∫1−1∫1−x20∫2−z03ydydzdx=∫1−1∫1−x2032(z−2)2dzdx=∫1−14−12(x2+1)3dx=18435
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解題僅供參考,其他歷年試題及詳解
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