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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