※ [本文转录自 NTU-Exam 看板] 作者: jacky10liao (天才侦探) 看板: NTU-Exam 标题: [试题] 周青松 微积分甲下95 时间: Sat Jun 30 01:06:19 2007 课程名称︰ 微积分甲下 课程性质︰ 系必修 课程教师︰ 周青松 开课学院： 理学院 开课系所︰ 数学系(但是现地理、地质、生工、生机、工管等系修习) 考试日期（年月日）︰96.06.22 考试时限（分钟）：120 是否需发放奖励金：是，感谢 （如未明确表示，则不予发放） 试题 : I. 2 A) Find F(t) given that F'(t)=2costi-tsint j+2tk and F(0)=i+3k B) Let γ be a differentiable vector function of t and r=||γ||. d γ 1 dγ Show that ---- --- = ---[(γ╳----)╳γ], when r≠0. dt r r^3 dt II. A) Let γ=xi+yj+xk and r=||γ||. Show that, for each integer n and n n-2 all γ≠0, ▽r =nr γ. B) Find a function f with the gradient F: 2 F(x,y,z) = yzi+(xz+2yz)j+(xy+y )k. III. 2 2 y -z A) Find the directional derivative of f(x,y,z)=xe at (1,2,-2) t-1 in the direction of the path γ(t)=ti+2cos(t-1)j-2e k. B) Use the chain rule to find the rate of change of 2 f(x,y,z)=x y+zcosx with respect to t along the tuisted cubic 2 3 γ(t)=ti+t j+t k. IV. A) Use double integration to calculate the area of the region Ω 2 enclosed by y=x and x+y=2. 2 -y /2 B) Evalute the double integral ∫∫e dxdy, Ω the triangle Ω fomed by y-axis, 2y=x, y=1. V. A) Find the mean of the solid ight cicular cylinden of radius r and height h given that the mass density is directly proportional to the distance from the lower base. x B) Evaluate the triple integral ∫∫∫2ye dxdydz, where T is the T solid given by 0≦y≦1, 0≦x≦y, 0≦z≦x+y 每大题均20分 (每一小题10分) -- 清汤 热汤 面汤 甜汤 酸梅汤 杨桃汤 花生汤 薏仁汤 红豆汤 绿豆汤 黄豆汤 扁豆汤 冬瓜汤 南瓜汤 黄瓜汤 丝瓜汤 猪血汤 猪脑汤 猪肚汤 猪肝汤 牛舌汤 牛尾汤 牛肉汤 牛杂汤 中将汤 四物汤 汤ㄊ啷汤 汤汤汤 黄泉路上孟婆汤 喝下去你全忘光 --

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