課程名稱︰ 暑修微積分甲下第二次期中考
課程性質︰
課程教師︰ 周青松
開課學院：
開課系所︰
考試日期（年月日）︰2009/9/3
考試時限（分鐘）：2小時
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Make sure to give sufficient reason in each problem or you will NOT get any
credit for your answer.
A.
(a)
Set a = a1i + a2j + a3k
and b = b1i + b2j + b3k
Show that a x b = (a2b3-a3b2)-i(a1b3-a3b1)j+(a1b2-a2b1)k
(b)
Set a = a1i + a2j + a3k
b = b1i + b2j + b3k
c = c1i + c2j + c3k
| a1 a2 a3 |
Show that (a x b) ‧ c = | b1 b2 b3 |
| c1 c2 c3 |
B.
(a)Find f(t) given that f'(t) = 2costi-tsint^2+2tk and f(0)=i+3k
(b)Find f(t) given that f'(t) = ti-t(1+t^2)^(-1/2)+te^tk and f(0)=i+2j+3k
C.
(a)
Show that if γ is a differentiable vector function of t ,then the function
r = ║γ║ is differentiable at where it is not zero and
dγ dr
γ‧ ── = r ──
dt dt
(b)
Show that for each integer n and all γ≠0 we have ▽r^n = nr^(n-2)γ
D.
(xy)^2
Set g(x,y) = ─── if (x,y) ≠ (0,0)
x^4+y^4
0 if (x,y) = (0,0)
dg dg
(a)Show that ─(0,0) and ─(0,0) both exist and evaluate their values
dx dy
(b)Show that lim g(x,y) doesn't exist
(x,y)→(0,0)
E.
(a)
Find all function with gradient yzi + (xz+2yz)j + (xy+y^2)k
(b)
2xy
Set f(x,y) = ─── if (x,y) ≠ (0,0)
x^2+y^2
0 if (x,y) = (0,0)
Show that f is not differentiable at (0,0)
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