※ [本文转录自 NTU-Exam 看板] 作者: sheepout (羊咩咩) 看板: NTU-Exam 标题: [试题] 94下 张秀瑜 微积分乙 时间: Thu Jun 22 14:25:25 2006 课程名称︰微积分乙 课程性质︰共同必修 课程教师︰张秀瑜 开课系所︰管院 考试时间︰94学年度下学期6月22号 是否需发放奖励金： *[1;33m（如未明确表示，则不予发放）*[m 试题 : 1. use the chain rule to find δz/δs and δ^2z/δtδs (1) z=x^2y+3xy^4, x=s^2t, y=s lnt (2) z=f(s+at)+g(s-at), where a is a constant and f,g are differentiable. (3) z=f(x,y) is differentiable, where x=cost e^t ,y=sint e^t 2. if f is homogeneous of degree n (that is f(tx,ty)=t^n(x,y) for all t), show that f (tx,ty)= t^(n-1)f (x,y) x x 3. find the points on the ellipsoid x^2+2y^2+3z^2=1 where the tangent plane is parallel to the plane 3x-y+3z=1 4. find the local maximum and minimum values of f=x^3-3x-y^3+12y 5. explain the method of lagrange multipliers works in finding the extreme values of f(x,y,z) subject to the constraint g(x,y,z)=k 6. use a double integral to find the volume of the tetrahedron bounded by the planes x+y+z=2, x=3y, x=0, and z=o 7. sketch the regions and evaluate the integrals: (1) change the order to : 1 1 ________ ∫ ∫ √x^3 + 1 dxdy 0 √y ________ (2) convert to polar coordinates: 0 √a^2-y^2 ∫ ∫ (a^2-x^2-y^2)^2/3 dxdy -a 0 ∞ ___ ∞ 8. use the result: ∫ e^(-x^2) dx = √π to evaluate ∫ x^2 e^(-x^2) dx -∞ 0 第五题是要描述这个方法 而不是解释它的原理 --

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