※ 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
1
∫ (x^(-2/3))(lnx) dx
0
1
= lim ∫ (x^(-2/3))(lnx) dx
R→0+ R
|1 1 1
= lim (3)(x^(1/3))(lnx) | - ∫ (3)(x^(1/3))(---) dx
R→0+ |R R x
(令 u = lnx , dv = x^(-2/3) dx , 則 du = 1/x dx , v = (3)(x^(1/3)) )
1
= lim (-3)(R^(1/3))(ln(R)) - ∫ (3)(x^(-2/3)) dx
R→0+ R
ln(R) |1
= lim (-3)(----------) - (9)(x^(1/3)) |
R→0+ R^(-1/3) |R
1/(R)
= lim (-3)(------------------) - (9)(1 - R^(1/3))
R→0+ (-1/3)(R^(-4/3))
= -9 + (9)(lim R^(1/3))
R→0+
= -9 + (9)*(0) = -9
: ∞ 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.
: 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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