推 geobrownbear:這張有給他難到 06/22 17:09
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※ [本文轉錄自 NTU-Exam 看板]
作者: sheepout (羊咩咩) 看板: NTU-Exam
標題: [試題] 94下 張秀瑜 微積分乙
時間: Thu Jun 22 14:25:25 2006
課程名稱︰微積分乙
課程性質︰共同必修
課程教師︰張秀瑜
開課系所︰管院
考試時間︰94學年度下學期6月22號
是否需發放獎勵金：
*[1;33m（如未明確表示，則不予發放）*[m
試題 :
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
第五題是要描述這個方法 而不是解釋它的原理
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