[考古] 台大96A 參考解答

看板trans_math作者yuyumagic424 (五月飛雪)時間17年前 (2007/07/24 04:52)推噓9(9推 0噓 0→)

留言9則, 8人參與討論串1/3 (看更多)

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 無實數解故不合 => 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 我不想拿筆算所以這個就留這樣了囧 -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 140.112.239.90

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rligaoi2

07/24 08:36, , 1^F

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pobm

07/24 10:56, , 2^F

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Frobenius

07/24 11:05, , 3^F

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