課程名稱︰微積分乙下 課程性質︰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）︰2011／4／49 考試時限（分鐘）：120min 是否需發放獎勵金：是 （如未明確表示，則不予發放） 1. Let r(t) be a curve in R^3 with r'(t)≠0. Show that the curvature of r(t) │r'(t) ×r''(t)│ κ(t) ＝ ───────── ("×" means "cross product") │r'(t)│^3 Compute the curvature of r(t)＝(t,4t^(3/2),-t^2) at t＝1. (10%) 2. Let f(x,y,z) and g(x,y,z) be infinitely differentiable functions. Assume that f(0,0,0)＝g(0,0,0) and ▽f(0,0,0)＝(-1,2,1),▽g(0,0,0)＝(3,-1,2). f(x,x^2,x^3) Find the limit: lim ───────. (15%) x→0 g(x^3,x^2,x) 3. Let a three-dimensional surface in R^4 be described by f(w,x,y,z)＝w˙e^(wx)＋yz＋xz＋xy＝0. Find the equation of the tangent plane at (0,1,-1,1). (10%) 4. Evaluate the following iterated integral. (10%) 1 1 ∫ ∫ e^[x^(3/2)] dxdy. 0 y^2 5. A space probe in the shape of the ellipsoid 4x^2＋y^2＋4z^2＝16 enters Earth's atmosphere and its surface begins to heat. After 1 hour, the temperature at the point (x,y,z) on the probe's surface is T(x,y,z)＝8x^2＋4yz－16z＋600. Find the hottest point on the probe's surface. (20%) 6. Classify all critical points of f(x,y)＝xy˙e^(-x^2-y^2). (20%) 7. In the second derivative test, when f_xx˙f_yy－(f_xy)^2＝0 at the critical point, then the test is inconclusive. Give three examples where the critical is a local maximum, a local minimum, or a saddle point and that f_xx˙f_yy－(f_xy)^2＝0 at the critical point. (15%) (PS: These three examples can be explained either by one function or by three different functions respectively.) --

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