[考古] 95成大
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.
以上也煩請版上的高手大大們幫忙解惑一下
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