Re: 幾題
※ 引述《dreamenjoy (今天也很爽)》之銘言：
: 1.Given an ε-δ proof for the limit x^2+3x-2=8
: x→2
Let ε > 0.
|(x^2+3x-2)-8| = |x^2+3x-10| = |x+5||x-2|
(Now we have to estimate |x+5|)
Assume |x-2| < 1, then 1 < x < 3.
=> |x+5| < 8
=> |(x^2+3x-2)-8| < 8|x-2|
Take δ = min{1,ε/8}, then for all x in 0 < |x-2| < δ,
|(x^2+3x-2)-8| < ε.
2
Therefore lim (x +3x-2) = 8.
x→2
: 2.Find a and b such that lim(sinax/x^3+b/x^2)=1/6
: x→0
: 3
: 3.∫ x^2-4x+3/x^3+2x^2+x dx
: 1
2 2
x -4x+3 x -4x+3 A B Cx+D
-------- = ----------- = --- + ----- + -------
3 2 2 2
x +2x +x x(x+1) x x+1 (x+1)
2 2
=> x -4x+3 = A(x+1) +Bx(x+1)+(Cx+D)x
2 2
= A(x +2x+1)+B(x^2+x)+(Cx +Dx)
2
= (A+B+C)x +(2A+B+D)x+A
=> A+B+C=1,2A+B+D=-4, A=3
We may get one more equation by replacing x with -1
=> 8 = C-D
Solve the system for A,B,C and D.
The rest is left to you.
: k
: 4.Find the interval of convergence of the series (-1) k
: Σ ———(x+2)
: 2 k
: k 3
: 5.f(x.y)= xy(y^2-x^2)/x^2+y^2 , (x,y)≠(0,0) Show that Fxy(0,0)≠Fyx(0,0)
: 0 , (x,y)＝(0,0)
: 這題是把XY分別微出來比較就好了是嗎??
f=F?
Yes, if you mean to check the both sides of the ineqality,
,but note that you should do it with the definition.
: 2 √4-y^2 √4-x^2-y^2
: 6. ∫∫ ∫ z√x^2+y^2+z^2 dzdxdy
: 0 0 0
Change to spherical coordinates.
Drawing the graph of the domain of integration may help you.
Let S = {(ρ,θ,φ) | 0≦ρ≦2, 0≦θ≦π/2, 0≦φ≦π/2},
then T = {P(ρ,θ,φ) | (ρ,θ,φ) is in S}.
2 √4-y^2 √4-x^2-y^2
=> ∫∫ ∫ z√x^2+y^2+z^2 dzdxdy
0 0 0
= ∫∫∫ z√x^2+y^2+z^2 dxdydz
T
2 2
= ∫∫∫ ρ cosφ *ρ sinφ dρdθdφ
S
The rest is left to you.
: 因為考古題有些沒解答 只好來請教高手了
: 感恩
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※ 編輯: mk426375 來自: 220.139.131.44 (06/04 17:10)
※ 編輯: mk426375 來自: 220.139.131.44 (06/04 17:19)
※ 編輯: mk426375 來自: 220.139.131.44 (06/04 17:24)
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