課程名稱︰微積分乙下
課程性質︰系必修
課程教師︰王振男
開課學院：醫學院
開課系所︰醫學系
考試日期（年月日）︰2012/4/17
考試時限（分鐘）：110分鐘
是否需發放獎勵金：是，謝謝
（如未明確表示，則不予發放）
試題：
1.（10％）The derivative of f(x,y,z) at a point P is greatest in the direction
of v＝i＋j－k. In this direction, the value of derivative is 2√3.
（i） What is ▽f at P？
（ii） What is the derivative of f at P in the direction of (1/√6)(2i－j＋k).
2.（10％）Find the equation of the tangent plane to the surface given by
cos(πx)－(x^2)y＋e^(xz)＋yz＝4 at the point (0,1,2).
3.（i）（20％）Compute the curvature and the torsion of the helix
r(t)＝(acost)i＋(asint)j＋(bt)k, a,b≧0.
（ii）（5％）For r(t) in （i）, express dＮ/ds in terms of the unit tangent
vector Ｔ and the unit binormal Ｂ.
4.（20％）Determine whether any one of the following two functions is
differentable at (0,0).
（i）f(x,y)＝√｜xy｜;
（ii）f(x,y)＝｜y｜(1＋x^2).
5.（20％）Classify all critical points of f(x,y)＝x^4＋y^4－x^2－y^2＋10, i.e.,
determine whether a critical point is a local maximum, a local minimum, or a
saddle point.
6.（15％）Find the shortest distance from the point (1,0) to the parabola
y^2＝4x.
參考答案：
1.（i）2i＋2j－2k；（ii）0
2.2x＋2y＋z＝4
3.（i）curvature＝a/(a^2＋b^2)；torsion＝b/(a^2＋b^2)；
（ii）－a/(a^2＋b^2)Ｔ＋b/(a^2＋b^2)Ｂ
4.（i）not differentiable；（ii）not differentiable
5.(0,0)→local maximum
(0,1/√2)、(0,－1/√2)、(1/√2,0)、(－1/√2,0)→saddle point
(1/√2,1/√2)、(1/√2,－1/√2)、(－1/√2,1/√2)、(－1/√2,－1/√2)→local
6.1 minimum
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