Re: [考古] 95成大
※ 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
令 x = (r)(cosθ) , y = (r)(sinθ) , 則 |J| = r
0 ≦ r ≦ 1 , 0 ≦ θ ≦ π/2
1 √(1 - x^2)
∫ ∫ sin(x^2 + y^2) dydx
0 0
π/2 1
= ∫ ∫ (sin(r^2))(r) drdθ
0 0
π/2 -1 |r = 1
= ∫ (---)(cos(r^2)) | dθ
0 2 |r = 0
π/2 1
= ∫ (---)(1 - cos1) dθ
0 2
1 |π/2 π
= (---)(1 - cos1)(θ) | = (---)(1 - cos1)
2 |0 4
: 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.
: 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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