课程名称︰微积分乙下 课程性质︰必修 课程教师︰王振男 开课学院：医学院 开课系所︰医学系 考试日期（年月日）︰2011／06／21 考试时限（分钟）：10:20~12:30 , 130min 是否需发放奖励金：是 （如未明确表示，则不予发放） 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 後附一张 Table of the standard normail distribution. --

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