作者yuyumagic424 (五月飞雪)
看板trans_math
标题[考古] 台大96A 参考解答
时间Tue Jul 24 04:52:32 2007
http://www.lib.ntu.edu.tw/exam/undergra/96/96019.pdf
一.
(1) 令 t = 1-x
=> ∫x^n(1-x)^2 dx = ∫(1-t)^n t^2 (-dt) = -∫(1-x)^n x^2 dx
=> F(x) = -G(x) + c'
(2)
1 0 1
∫x^2(1-x)^21 dx = ∫(1-t)^2 (t)^21 (-dt) = ∫(1-t)^2 (t)^21 dt
0 1 0
t^22 2t^23 t^24 |1
= ----- - ----- + ----- | = 1/22 - 2/23 + 1/24
22 23 24 |0
= 1/262 - 2/23 = -501/6026
二.
f(x,y,z) = √( x^2+y^2+(z-2)^2 )
= √( 5 + z^2 - 4z + 4 )
= √( 9 + x^2 - 4xy + 4y^2 /25 - 4x + 8y/5 )
= √( 9 + (x^2 - 4xy + 4y^2 - 20x + 40y) /25 )
令 g(x,y) = x^2 - 4xy + 4y^2 - 20x + 40y
▽g(x,y) = λ▽(x^2 + y^2)
2x - 4y - 20 = 2λx
-4x + 8y + 40 = 2λy
x^2 + y^2 = 5
=>
λ= 0 or y = -2x
if λ= 0
then x = 2(y+5)
4(y+5)^2 + y^2 = 5
5y^2 + 40y + 95 = 0
y^2 + 8y + 19 = 0 无实数解故不合
=>
y = -2x
x^2 + 4x^2 = 5
x = ± 1
g(1,-2) = -75
g(-1,2) = 125
so the minimum of g(x,y) is -75
__________ __
=> f(x,y,z) = √9 - 75/25 = √6
三.
(1)
e e
In = t^2/2 (㏑t)^n | - n/2 ∫ t(㏑t)^(n-1) dt
1 1
= e^2 / 2 - n/2 In-1
(2)
e
I1 = e^2 / 2 - 1/2 ∫ t dt = e^2 / 4 + 1/2
1
I2 = e^2 / 2 - 2/2 I1 = e^2 / 4 - 1/2
I3 = e^2 / 2 - 3/2 I2 = e^2 / 8 + 3/4
I4 = e^2 / 2 - 4/2 I3 = e^2 / 4 - 3/2
四.
(1)
| (n+1) x^(3n+3) |
lim | ----------------| = |x^3| < 1 => |x| <1
n→∞| n x^(3n) |
=> r = 1
(2)
+∞
g(x) = Σ [ (n + 1/3) x^3n - 1/3 x^3n ]
n=1
= h(x) - 1/3 (x^3 + x^6 + ... )
1 ∞
= h(x) - 1/3 x^3 ---------- , where h(x) = Σ (n + 1/3) x^3n
1 - x^3 n=1
∞
∫h(x) dx = Σ 1/3 x^(3n+1) = 1/3 (x^4 + x^7 + x^10 + ... )
n=1
= 1/3 x^4 (1 + x^3 + x^6 + ... )
x^4
= ---------
3(1-x^3)
=>
d x^4 -21x^6 + 12x^3
h(x) = -- (----------) = ---------------
dx 3(1-x^3) 9(1-x^3)^2
-7x^6 + 4x^3 x^3
g(x)= ----------------- - -----------
3(1-x^3)^2 3(1-x^3)
-6x^6 + 3x^3 -2x^6 + x^3
= -------------------- = ---------------
3(1-x^3)^2 (1-x^3)^2
1/8 - 1/32
g(0.5) = --------------------- = 6/49
49/64
五.
2 ㏑x 3
∫ ∫ -------- dy dx
1 0 ㏑x
2
= ∫ 3 dy = 3
1
六.
→ → →
∫∫F ‧n ds = ∫∫∫▽‧F dV
S R
= ∫∫∫ ( 2 + 3y^2 + 2xy + 3x^2 + 3z^2 -2xy ) dx dy dz
R:x^2+y^2+z^2≦49
= ∫∫∫ 3(x^2 + y^2 + z^2) + 2 dx dy dz
R:x^2+y^2+z^2≦49
2π π 7
= ∫ ∫ ∫ ( 3r^2 + 2 ) r^2 sinφ dr dφ dθ
0 0 0
π 7
= 2π ∫ sinφ dφ ∫ 3r^4 + 2r^2 dr
0 0
3 r^5 2r^3 7
= 4π [ ---------- + ------- ]
5 3 0
9 x 7^5 + 10 x 7^3
= 4π -----------------------
15
= 4 x 243 x 451 π / 15
我不想拿笔算 所以这个就留这样了 囧
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