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. --

※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 140.112.241.132 ※ 編輯: shokanshorin 來自: 140.112.241.132 (06/21 15:09) 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)