課程名稱︰微積分甲下 課程性質︰暑修 課程教師︰朱樺 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰106/09/08 考試時限（分鐘）：170 試題 : 2 -1 3 ∂ tan y ln(y+1)+cos x (1) (5%) Evaluate ------ (----------------------). ∂x∂y e^(sin y) -1 x (2) (5%) Find unit vectors u in which the rate change of f(x,y) = tan (---) y at the point P(3,4) in the direction u has value 4/25. (3) (10%) Let S1 be the surface z = x^2+y^2 and S2 the surface 2x^2+y^2+z^2=7. (a) Find the angle between S1 and S2 at the point P(-1,1,2). (b) let C be the curve of intersection of S1 and S2. Find the equation of tangent line to C at P. (4) (10%) Let u = f(x,y) where x = e^s cos t and y = e^s sin t. Assume that u has continuous second order partial derivatives. If 2 2 2 2 2 ∂u ∂u ∂u ∂u ∂u A ------ + B ------ = ------ + ------ + C ------ , 2 2 2 2 ∂s ∂t ∂x ∂y ∂x∂y find A, B and C. (5) (10%) Find the extreme values of f(x,y,z) = xy+z^2 on the ball x^2+y^2+(z-1/2)^2≦1. 1 y y^2 (6) (10%) Evaluate ∫∫ ----------- dx dy. 0 y^3 √(1-x^2) (7) (10%) Write five other iterated integrals that are equal to the integral 1 1 1-y ∫∫ ∫ f(x,y,z) dz dy dx. 0 1-x 0 □ □ □ (a) ∫ ∫ ∫ f(x,y,z) dz dx dy □ □ □ □ □ □ (b) ∫ ∫ ∫ f(x,y,z) dy dx dz □ □ □ □ □ □ (c) ∫ ∫ ∫ f(x,y,z) dy dz dx □ □ □ □ □ □ (d) ∫ ∫ ∫ f(x,y,z) dx dz dy □ □ □ □ □ □ (e) ∫ ∫ ∫ f(x,y,z) dx dy dz □ □ □ (8) (10%) Let B be the solid bounded by 4x^2+4xy+2y^2+z^2 = 4. Find ∫∫∫ (x^2+y^2+z^2) dV. B x sin z y^z (9) (10%) Let F =〈x+xy+xyz, ∫ ------- dz, x 〉. Find ▽‧F and ▽×F. y z (10) (10%) Let C be the curve (x^2+y^2)^3 = x^2. Find ∫ ∣y∣ds. C x^2 y+xy^2 -x^3-x^2 y (11) (10%) Find ∫ F‧dr, where F =〈-------------, ------------〉and C is the C (x^2+y^2)^2 (x^2+y^2)^2 closed (not simple) curve formed by the graphs of y = (x^2-1)(x-2) and y = -(x^2-1)(x-2) oriented counter-clockwise. (12) (10%) Find the area of the part of the sphere x^2+y^2+z^2 = b^2 that lies inside x^2+y^2 = a^2 where 0<a<b. 2 x^2 2y^2 2 3z^2 (13) (10%) Evaluate ∫ F‧dr, where F =〈-y +e ,x+e ,z +e 〉and C is C the curve of intersection of x^2+y^2 = 1 and y+z = 2 oriented counter-clockwise as viewed from above. (14) (10%) Evaluate ∫∫ (x cos z + y^2 + z sin x) dS where S is the sphere S x^2+y^2+z^2 = 4. --

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