作者xavier13540 (柊 四千)
看板NTU-Exam
標題[試題] 102下 陳榮凱 微積分甲下 期中考
時間Fri Apr 18 03:46:22 2014
課程名稱︰微積分甲下
課程性質︰數學系大一必帶
課程教師︰陳榮凱
開課學院:理學院
開課系所︰數學系
考試日期(年月日)︰2014/04/17
考試時限(分鐘):180
是否需發放獎勵金:是
(如未明確表示,則不予發放)
試題 :
The total is 105 points.
2 2 2 2
(1) (10 pts) Evaluate the volume of the solid bounded by x + y + z ≦ 4 and x +
2
y ≧ 1.
(2) (20 pts) Evaluate the following integrals.
________
/ 2
1 √(1 - x ) 2
(a) (5 pts) ∫∫ y dydx.
0 0
3
4 2 x
(b) (5 pts) ∫∫ _ e dxdy.
0 √y
2
∞ n -x
(c) (10 pts) ∫ x e dx for nεZ .
-∞ ≧0
2 2
x - y -2xy 2
(3) (15 pts) Consider L = ───── dx + ───── dy in R - {o}. Evaluate
2 2 2 2 2 2
(x + y ) (x + y )
∫L for the following curves.
γ
(a) γ is the unit circle starting from (1, 0) counterclockwise.
(b) γ consists of line segments from (1, 0) to (0, 1) to (-1, 0) to
(0, -1) then to (1, 0).
2
(4) (10 pts) Let S ㄈ R be a closed set. Prove that for any convergent
sequence {P } ㄈ S, its limit point P is contained also in S.
n
(5) (20 pts) Let g(x) be a continuous function in the interval a ≦ x ≦ b and
h(y) be a continuous function in the interval c ≦ y ≦ d. We consider the
function f(x, y) = g(x)h(y) in the region R: { a ≦ x ≦ b, c ≦ y ≦ d }.
(a) Prove that f(x, y) is continuous in R.
(b) If g(x) is differentiable at x = x , and h(y) is differentiable at y =
0
y . Prove that f(x, y) is differentiable at (x , y ).
0 0 0
b d
(c) Show that ∫∫f(x, y) dR = (∫ g(x) dx ) (∫ h(y) dy ).
R a c
2 2
(6) (15 pts) Consider f(x, y) = 2x + 3y - 4x + 6. Find its extreme values on
2 2
the region R: { x + y ≦ 16 }.
2 2
( Hint: Find local maxima or minima in { x + y < 16 } and find maxima or
2 2
minima in { x + y = 16 }. )
2
(7) (5 pts) Suppose that f(x, y) is differentiable. We consider g(s, t) = f(s
2 2 2 ∂g ∂g
- t , t - s ). Show that g satisfies t ── + s ── = 0.
∂s ∂t
2
(8) (10 pts) Prove that R - {o} is path connected but not simply connected.
註:'ε'和'ㄈ'都是屬於符號。
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第03話 已經沒什麼好期望了 第04話 被當、21都是存在的
第05話 怎麼可能會all pass 第06話 這考卷絕對有問題啊
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第09話 這樣成績,教授絕不會讓我過的 第10話 再也不依靠考古題
第11話 最後留下的補考 第12話 我最愛的學分
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→ xavier13540 :這次的題目好難QQ 04/18 03:51
※ 編輯: xavier13540 (140.112.196.111), 04/18/2014 03:55:29
推 t0444564 :已收錄 04/18 08:45
推 cmrafsts :我耍笨了QQ 04/21 10:07