课程名称︰微积分乙(下) 课程性质︰系必修 课程教师︰统一教学 开课学院：生农院 开课系所︰统一教学 考试日期（年月日）︰2008/04/17 考试时限（分钟）：15:30 ~ 17:20 (110min) 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 【考试须知】 1. 答案卷采正反双面印制，可用铅笔作答。 2. 踰越该题页面作答，不予计分。 3. 不能使用计算机及电子辞典。 4. 无论计算或证明题，皆应详述过程、理由；如未写出详细过程，一律不给分。 如引用未学过的定理或结果，应先予以证明。 1. (15%) Determine whether the improper integrals converge or not, and find the value of each that converges. (a) ∞ xdx (10%) (b) ∞ xdx (5%) ∫ ──── ∫ ──── 0 (1+x^2)^2 -∞ (1+x^2)^2 Ans: (a) 1/2 (convergent) (b) 1 (convergent) 2. (15%) Find dxdy ， where R is the region bounded by ∫∫ ─── R y y = x, xy = 1 and x = 2 Ans: 4ln2 - 2 3. (15%) Find 16 4 ∫ ∫ √(y^3 + 4)dydx 0 √2 16‧√(17)^3 - 16 Ans: ────────── 9 4. (20%) Find all critical points of f(x,y) = x^3 + y^2 + 2xy - 4x - 3y - 2 and determine which gives relative maximum, relative minimum or a saddle point. Ans: There is a minimum point (1, 1/2, -21/4); there is a saddle point (-1/3, 11/6, -439/108) 5. (20% total, 5% each) Let f(x,y) = 1/2 ln(x^2 + y^2). Find df df d2f d2f d2f ──, ───, ───, ── + ── dx dxdy dx^2 dx^2 dy^2 x -2xy y^2-x^2 Ans: ────, ──────, ──────, 0 x^2+y^2 (x^2+y^2)^2 (x^2+y^2)^2 ┌ ┐ 6. (15%) Find the eigenvalues and their corresponding eigenvector of│2 3│ │4 1│ └ ┘ ┌ ┐ ┌ ┐ Ans: Case 1, λ= 5 ; R =│1│ Case 2, λ=-2 │ 3│ │1│ │-4│ └ ┘ └ ┘ -- Work, as if you don't need money, Love, as if you've never been hurt, Dance, as if no one can see you, Sing, as if no one can hear. Live, as if the Earth was the Heaven...... --

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