推 axisaxes:真是有求必應呀XD 07/06 19:01
課程名稱︰微積分甲
課程性質︰數學 - 微積分
課程教師︰周青松
開課學院：(如下)
開課系所︰生機、生工、地質、地理、工管等
考試日期（年月日）︰ 2008/06/13黑色星期五
考試時限（分鐘）：8:10---10:00 遲到20分鐘不得進場
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Ｉ. For vector ａ＝ａ1 ｉ + ａ2 ｊ + ａ3 ｋ
and ｂ＝ｂ1 ｉ + ｂ2 ｊ + ｂ3 ｋ
A) Show that
｜ ｉ ｊ ｋ｜
ａ x ｂ ＝ ｜ａ1 ａ2 ａ3｜
｜ｂ1 ｂ2 ｂ3｜
2 2 2 2
B) Verify : ∥ａ x ｂ∥ + (ａ · ｂ) ＝ ∥ａ∥∥ｂ∥
II.
2
A) Find f(t) given that f'(t) = 2costｉ - tsint ｊ + 2tｋ , and
f(0) = ｊ + 3ｋ
B) Let γ be a differentiable vector function of t and set r = ║γ║.
Show that, where r ≠ 0
d γ 1 dγ
── (──) = ── [(γ x ──) x γ]
dt r r^3 dt
III.
2
A) Find the function with gradient F(x,y,z) = yzｉ+ (xz + 2yz)ｊ+ (xy + y )ｋ
B) Find the directional derivative of f(x,y,x) = z㏑(x/y) at (1,1,2)
toward the point (2,2,1).
IV.
A) Let U be an open connected set and let f be a differentiable function on U.
If ▽f(x) = 0 for all x in U, then f is constant on U.
1 3 3
B) Use the chain rule to find the derivative of f(x,y) =—(x + y )
3
with sespect to t along the ellipse γ(t) = acostｉ+bsintｊ
V.
A) Evaluate the double integral
3
∫∫ (3xy - y) dxdy ,Ω is the region between y = |x| and y = -|x|,
Ω
x in [-1,1].
B) Evaluate the trible integral
x
∫∫∫ 2ye dxdydz, where T is the solial
T
∕ given by 0 ≦ y ≦ 1 , 0 ≦ x ≦ y
﹨ and 0 ≦ z ≦ x + y.
---每大題均20分---
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