課程名稱︰微積分乙(下)
課程性質︰系必修
課程教師︰統一教學
開課學院：生農院
開課系所︰統一教學
考試日期（年月日）︰2008/04/17
考試時限（分鐘）：15:30 ~ 17:20 (110min)
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
【考試須知】
1. 答案卷採正反雙面印製，可用鉛筆作答。
2. 踰越該題頁面作答，不予計分。
3. 不能使用計算機及電子辭典。
4. 無論計算或證明題，皆應詳述過程、理由；如未寫出詳細過程，一律不給分。
如引用未學過的定理或結果，應先予以證明。
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│
└ ┘ └ ┘
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Dance, as if no one can see you,
Sing, as if no one can hear.
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