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。