[試題] 99下 王振男 微積分乙下 期末考

[shokanshorin (上官薔凜)]

課程名稱︰微積分乙下
課程性質︰必修
課程教師︰王振男
開課學院：醫學院
開課系所︰醫學系
考試日期（年月日）︰2011／06／21
考試時限（分鐘）：10:20~12:30 , 130min
是否需發放獎勵金：是
（如未明確表示，則不予發放）

1. Let Ω be a bounded domain in R^3 with smooth boundary. Assume that u(x,t)

   satisfies the following things

       ∂u     3   ∂^2 u
   ╭ —— － Σ  ———— ＋ αu ＝ 0 , for all (t,x)εR+ ×Ω,
   │  ∂x    j=1  ∂x_j^2
   │
   │ u(t,x) ＝ 0 , for all (t,x)εR+ ×∂Ω
   │
   ╰ u(0,x) ＝ u_0(x) , for all xεΩ (u_0(x) ＝0 , for all xε∂Ω)

   (ε = belong ; ∂Ω means the boundary of Ω)

   where R+ ＝ {tεR:t＞0} and α is a constant. Show that

   ∫∫∫u^2(t,x)≦(∫∫∫u_0(x)^2 dx )˙e^(-2αt) , for all tεR+.
     Ω               Ω

   (Hint: multiplying the equation by u and using the divergence theorem.)

   (15%)

2. Let the surface S ＝ {(x,y,z)：x^2＋y^2＋(z－4)^2＝25 , z＞0} and a vector
         →     →     →           →
   field F ＝ y i＋z^3 j＋√(1+z^4) k.
                    →   →            →
   Compute ∫∫curl F ˙ n  dS , where n  is the unit outer normal of S with
            S
   n(0,0,9) ＝ k. (10%)

3. Find the moment of inerita about the z-axis of a thin shell of constant

   density 1 cur(?) from the cone 4x^2＋4y^2－z^2＝0 , z≧0 , by the circular

   cylinder x^2＋y^2＝2x. (in HW#6) (15%)

4. The hazard rate function of an organism is given by

   λ(x)＝0.1＋0.5×e^(0.02x) , x≧0 , where x is measured in days.

   (a) What is the probability that the organism will live less than ten days?

   (b) What is the probability that the organism will live for another five

       days given that it survived the first five days?

   (in HW#8) (10%)

5. Suppose X_1 , X_2 , ... , X_n are i.i.d. random variables with uniform

   distribution on (0,1). Define X ＝ min(X_1,X_2,...,X_n).

   (a) Compute P(X＞x).

   (b) Show that P(X＞x/n) → e^-x as n → ∞. (in HW#8) (15%)

6. How often do you have to toss a coin to determine p (head) within 0.1 of its

   true value with probability at least 0.95? Estimate the sample size by using

   (a) the Law of Large Numbers and (b) the Central Limit Theorem. (20%)

7. Suppose that a narrow beam flashlight is spun around its center, which is

   located a unit distance from the x-axis. When the flashlight has stop

   spining, consider the point X (a random variable) at which the beam

   intersects the x-axis. (If the beam is not pointing toward the x-axis,

   repeat the experiment.)

   (a) Find the probability distribution function of X. (10%)

   (b) Does the random variable ︱X｜^1/2 have finite expected value?


                     ﹣﹣﹣﹣﹣⊙ ﹣﹣﹣﹣﹣
                               ┬↘
                               │θ↘
                              1│    ↘
                               │      ↘
   ──────────────┴──────────────────────
                               0          X                             x-axis

後附一張 Table of the standard normail distribution.

--
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◆ From: 140.112.241.132
※ 編輯: shokanshorin    來自: 140.112.241.132      (06/21 15:09)
推 sp050612    :原PO微積分超強!!!
→ shokanshorin:死定了啦...這次的考題不是普通的____...
推 woieyufan   :                               好睡
推 liltwnboiz  :超難 還出最後一次習題 沒做根本死光 =.....=
→ liltwnboiz  :我看到那閃閃發光的Ｃ在跟我揮手......
→ ALegmontnick:done
推 s30127s     :淚推微積分QQ
推 wayne315315 :這...第一題看不懂= =
好問題...我也看不懂...有小小改動幾個字這樣~
※ 編輯: shokanshorin    來自: 140.112.241.132      (06/21 23:41)
推 ALegmontnick:done,too