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→ shokanshorin:樓上拍拍別桑心XD 04/28 00:26
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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※ 編輯: shokanshorin 來自: 140.112.241.132 (04/27 16:04)