※ 引述《ss355227 (前佑)》之铭言： : ※ 引述《sky2857 (楷中)》之铭言： : : 作者: SWW (鼠呜呜) 看板: NTU-Exam : : 标题: [试题] 98下 周青松 微积分甲下 期末考 : : 时间: Wed Jul 7 12:34:24 2010 : : 课程名称︰微积分甲下 : : 课程性质︰必修 : : 课程教师︰周青松 : : 开课学院：理学院 : : 开课系所︰ : : 考试日期（年月日）︰2010/6/21 : : 考试时限（分钟）：100分钟 : : 是否需发放奖励金：是 : : （如未明确表示，则不予发放） : : 试题 : : : A.(a) Findｆwith the conditionｆ'(t)=2ｆ(t) and ｆ(0)=i-k. : 小考= = : 欸我还是不太知道怎麽算 : help this is an essential type of ODE and PDE it's very important in engineering mathematics seperate the variables first d ---- f(t) = 2 f(t) dt df(t) => -------- = 2dt = > integrate both sides we get f(t) => ln| f(t) | = 2t + C => f(t) = exp[±(2t+c)] replace exp(±c) by a constant k we have f(t) = k *exp(±2t) since the negative term is wrong so f(t) = k*exp(2t) operate the process on each componet you'll get the answer : : (b) Letｆbe a differentiable vector-valued function. Show that whenver : : ∥ｆ(t)∥≠0. : : d ｆ'(t) ｆ(t)‧ｆ'(t) : : —(————— － ————————ｆ(t) : : dt ∥ｆ(t)∥ ∥ｆ(t)∥^3 : 中间是等於吗 = =? : : B.(a) Find the directional derivative of f(x,y,z)=Ax^2 + Bxyz+ Cy^2 at the : : point P(1,2,1) in the direction of Ai+Bj+Ck. : : (b) Assume that ▽f(x) exists. Prove that, for each integer n, : : n n-1 : : ▽f (x)=nf (x)▽f(x). : : Dose the result hold of n is replaced by an arbitary real number? : : C.(a) Use the chain rule to find the rate of change of : : f(x,y,z)=x^2 y+zcosx : : with repect to t along the twisted cubic ｒ(t)=ti+t^2 j+t^3 k. : : (b) Set r=∥ｒ∥, whereｒ=xi+yj+zk. If f is a continuously differentiable : : function of r, then : : ｒ : : ▽〔f(r)〕=f'(r)—, where r≠0 : : 　　　　　 r : : D.(a) Evaluate the double intergral : : __ : : ∫∫ √xy dxdy, Ω:0≦y≦1, y^2≦x≦y. : : Ω : 新题目耶 : 2/27 right : : (b) 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. 2 ab pi : : E.(a) Evaluate : : π/2 π/2 1 : : ∫ ∫ ∫e^z cosxsiny dzdydx : : 0 0 0 : pi^2 (e -1 )/ 4 : : (b) Evaluate the triple integral : : ∫∫∫2ye^x dxdydz, : : T : : where T is the solid given by 0≦y≦1, 0≦x≦y, 0≦z≦x+y. : 恩 --

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