[試題] 104暑 蔡雅如 微積分甲下 第二次小考
課程名稱︰微積分甲下
課程性質︰必修
課程教師︰蔡雅如
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2016/08/19
考試時限（分鐘）：50
試題 :
1. (20%) Classify the quadric surface cx^2 + 2y^2 - 2cx + 47 - z = 0
2. (30%) Consider a smooth space curve →r(t) = <t-1, t^2, (3)^(1/2)t+3^(1/2)>
(a) Find the curvature of the curve, κ(t).
(b) Find the unit tangent, unit normal, and the binormal vectors →T(t),
→N(t), →B(t).
(c) Find the osculating plane of →r(t) at a general point. What can you
say about the curve?
3. (20%) Find the first partial derivatives of the following functions:
(a) f(x,y) = (2y)^(3x), for y > 0,
(b) f(x,y,z) = (x^2 + y^3 + z^4)^(1/3) at (0,0,0).
4. (30%) Let f(x,y) = { x^3/(x^2+y^2), (x,y) ≠ (0,0)
0, (x,y) = (0,0)
(a) Is f(x,y) continuous at (0,0)?
(b) Compute f_x(0,0) and f_y(0,0).
(c) Write down the linear approximation L(x,y) of f(x,y) at (0,0).
(d) Compute lim (x,y)→(0,0) |f(x,y) - L(x,y)|/(x^2+y^2)^(1/2). is f(x,y)
differentiable at (0,0)?
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