※ 引述《dreamenjoy (今天也很爽)》之铭言： : 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) : 这题是把XY分别微出来比较就好了是吗?? 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. : 因为考古题有些没解答 只好来请教高手了 : 感恩 --

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