課程名稱︰微積分甲下 課程性質︰必修 課程教師︰周青松 開課學院：管理學院、生農學院、理學院 開課系所︰工管系管系科管組、生工系、生機系、地質系 考試日期（年月日）︰2012/06/18 考試時限（分鐘）：110分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : I. A.(10%)Find f(t) given that f'(t)=2cost i - tsin(t^2) j + 2t k and f(0)=i+3k B.(10%)Find f(t) given that f'(t)=αf(t),where α is a real number, and f(0)=c. II. A.(10%)Find ▽(sin r) and ▽(e^r), where r = (x^2 + y^2 + z^2)^(1/2). B.(10%)Set f(x,y) = 1 + x^2 +y^2, find the points (x,y) at which ▽f(x,y)=0. III. A.(10%)Find that directional derivative of (x,y,z)=Ax^2 + Bxyz + Cy^2 at the point P(1,2,1) in the direction of Ai + Bj + Ck. B.(10%)Find that directional derivative of (x,y,z)=e^x cosπyz at the point P(0,1,1/2) in the directions parallel to the line in which the planes x + y - z = 5 and 4x - y - z = 2 intersect. IV. A.(10%)Evaluate ∫∫e^-(y^2/2) dx dy, where Ω is the triangular region bounded Ω by the y-axis, 2y = x, y = 1. B.(10%)Evaluate ∫∫∫y^2 dx dy dx, where T is the tetrahedron in the first-octant T solid bounded by the coordinate planes and the plane 2x + 3y + z = 6. V. A.(10%)Calculate the volume within the cylinder x^2 + y^2 = b^2 between the planes y + z = a and z = 0 given that a≧b＞0. B.(10%)Find the mass of a solid right circular cylinder of radius r and height h given that the mass density directly proportional to the distance from the lower base. --

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