推 dodo1654:謝謝L大的指教 124.8.138.193 07/12 01:25
※ 引述《dodo1654 (secret)》之銘言：
: 1.Compute the following limit:
: １ ｎ
: (a) lim － Σ √[(3i/n)^7+(3i/n)^4]
: n→∞ ｎ i=1
: X 1
: (b) lim [ (-----)-(-----)].
: n→∞ lnX X-1
: x
: e ln(x+e) 1
: 2.Let f(x)=∫ -------------- dx defined on (-1,∞).
: 1 √(x^4+x^2+5)
: (a) Show that f is a strictly increasing function.
: -1
: (b) Find (f )'(0).
: 1 -2/3
: 3.Compute the improper integral ∫ x lnx dx.
: 0
: ∞ n n n
: 4.Find the radius of convergence of the infinite series Σ (------) ｘ .
: n=0 2n+1
: 5.Let f be a real valued function defined on R with f"(x)>0 for all x. Show
: that f(x) ≧ f'(0)x + f(0) for all x.
: 1 √(1-x^2 ) 2 2
: 6.Compute the iterated integral ∫∫ sin(x +y )dydx.
: 0 0
: １ ８
: 7.Find the local extreme values of the function f(x,y)= xy +－+－ for xy ≠ 0.
: ｘ ｙ
: 8.Use the method of Lagrange Multiplier to find the maximum value of the
: 3
: function f(x,y,z) = x+3y-2z defined on R subject to the constraint
: 2 2 2
: x +y +z =14.
令 g(x,y,z) = x^2 + y^2 + z^2 - 14
F(x,y,z) = f(x,y,z) + (λ)(g(x,y,z))
= x + 3y - 2z + (λ)(x^2 + y^2 + z^2 - 14)
Fx = 1 + (2λ)(x) = 0 => λ = -1/(2x)
Fy = 3 + (2λ)(y) = 0 => λ = (-3/2)(1/y)
Fz = -2 + (2λ)(z) = 0 => λ = 1/z
-1 -3 1 1
---- = (---)(---) = ---
2x 2 y z
1 3 -2
--- = --- = --- = t , t 屬於 R
x y z
x = t , y = 3t , z = -2t 代入 x^2 + y^2 + z^2 = 14
(14)(t^2) = 14 , t^2 = 1 , t^2 - 1 = 0 , (t + 1)(t - 1) = 0
t = 1 或 -1
(x,y,z) = (1 , 3 , -2) , (-1 , -3 , 2)
f(1 , 3 , -2) = 1 + 9 + 4 = 14
f(-1 , -3 , 2) = -1 - 9 - 4 = -14
當 (x,y,z) = (1 , 3 , -2)時
f(x,y,z) = x + 3y - 2z 有最大值 14
: 以上也煩請版上的高手大大們幫忙解惑一下
: 感謝!
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