※ 引述《ss355227 (前祐)》之銘言：
: ※ 引述《sky2857 (楷中)》之銘言：
: : 作者: SWW (鼠嗚嗚) 看板: NTU-Exam
: : 標題: [試題] 98下 周青松 微積分甲下 期末考
: : 時間: Wed Jul 7 12:34:24 2010
: : 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：理學院
: : 開課系所︰
: : 考試日期（年月日）︰2010/6/21
: : 考試時限（分鐘）：100分鐘
: : 是否需發放獎勵金：是
: : （如未明確表示，則不予發放）
: : 試題 :
: : A.(a) Findｆwith the conditionｆ'(t)=2ｆ(t) and ｆ(0)=i-k.
: 小考= =
: 欸我還是不太知道怎麼算
: 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
: 中間是等於嗎 = =?
: : 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
: 新題目耶
: 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
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 114.34.202.142