以下是這次考完長庚轉學考微積分後整理的題型 大家考完閒閒沒事可以當做消遣算算吧 一、 填充題 70% (每題7分) 1, ∫ x / [(9-2x-x^2)^(1/2)] dx = _____ 2, n lim sigma (2/n)*[(5+2i/n)^10] = _____ n-> oo i=1 3, find the area which is inside r =3sinθ, outside r =1+sinθ = _____ 4, 求 x^3+2y^2=3xyz在(1,1,1)此點之切平面方程式 = _____ 5, 5x (d/dx) ∫ (sint)^5 dt = _____ x^5 6, find the interval of convergence ∞ sigma (-1)^n * {[(-3x)^n] /[ (n+1)^(1/2)]} = _____ n=0 7, 求 f(x)=x-x^2 與x軸 在第一象限內所圍面積 繞x=2 之體積 = _____ 8, 求f=(x^2)*y*z 在(1,1,1)此點之 Maximum directional derivative = _____ 9, f=x*(y^2)*(z^3) x=(3u^2)+2v y=4u - 2v^3 z=2u^2 – 3v^2 u=1, v=1 find (df/du) = _____ 10,求 f(x,y)= x^2 – (5/2)xy +y^2 在 x^2+y^2 <= 1之Maximum = _____ 二、 計算題 30% (每題10分) 1, if (x,y)不等於(0,0)時 f(x,y)= [(x^2)y]/[(x^4 +y^2)] (x,y)等於(0,0)時 f(x,y)= 0 find lim [(x^2)y]/[(x^4 +y^2)] (x,y)->(0,0) 2, if |g(x)| <= x^2 g(x) can derivative for all x prove g’(0) = 0 3, ∫∫ x^2 + y^2 dxdy =____ R: x^2+y^2 <=2y R --

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