推 petroheat :原po型男 少女殺手 06/22 15:40
課程名稱︰微積分甲下
課程性質︰大一系定必修
課程教師︰薛克民
開課學院：生農學院,理學院
開課系所︰生工系,生機系,地質系,工管系科管組
考試日期（年月日）︰98/6/15
考試時限（分鐘）：110
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.Consider the function f(x,y) = x^2 y + √y
(a)Find the directional derivative of f at (2,1) in the direction
toward the point (3,0) (7%)
(b)Find the maximum rate of change of f at (2,1), and the direction
in which it occurs. (8%)
2.Consider the function f(x,y) = xe^(-x^2-y^2) with variables x and y
(a)Find the minimum and maximum values of f when x and y belong to a
set of real numbers. (10%)
(b)Find the absolute minimum and maximum values of f when x and y are
in the region: x^2 + y^2 ≦ 1 (10%)
3.Find the volume and centroid of the solid D that lies above the cone z =
√(x^2 + y^2) and below the sphere x^2 + y^2 + z^2 = 1. (15%)
4.Consider the vector field F(x, y, z) = (e^y, xe^y + e^z, ye^z) in a
three-dimensional Cartesian coordinate system.
(a)Find curl F , and show that F is conservative. (5%)
(b)Find a potential function f(x, y, z) so that F = ▽f. (10%)
(c)Evaluate the line integral ∫ F‧dr, where C is the line segment from
c
(0, 2, 0) to (4, 0, 3). (5%)
5.Consider the line integral ∫ xy^2 dx - x^2 y dy , where C is a
c
closed loop that goes counterclockwisely from the line segment
(1, 1) to to (, 1) and then along the parabola y = x^2
(a)Integrate this line integral directly. (8%)
(b)Use Green’s theorem instead. (7%)
6.Find the area of the part of the surface z = 1 + 3x + 2y^2 that
lies above the triangle with vertices (0, 0), (0, 1), and (2, 1). (15%)
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※ 編輯: weison1221 來自: 122.124.103.32 (06/22 11:37)