Đề thi tuyển sinh sau đại học môn Giải tích 2012 (Đợt 1)

sg cilo DUcvA DAorAo<br /> D4.I HQC HUE<br /> <br /> Ho vd tEnthl sinh: 56 bdo danh.<br /> <br /> ri'rm<br /> <br /> ( rUYEN SINHsAUDAI HQCNAM 2012 Egt 1) Mdnthi: GIAITiCH<br /> @anh cho cao hqc) Thdi gian lam bdi: 180 Phut<br /> <br /> CAu1. a) Khaos6ttinh khavi cuahdm f(x,!)=l*,*y' [o<br /> <br /> ,<br /> <br /> \<br /> <br /> ( 9'n6u (x; +(o;o) \ ' r y)<br /> n6u(x;!) = (0;o).<br /> <br /> b) Tim mi0nhQitu cuachu6inamlfly thua<br /> @<br /> <br /> L c) Tinh tich phdn ducrng<br /> <br /> Y<br /> <br /> (-2)" _1,), . (x<br /> n*L<br /> <br /> * I G.siny + 2xy)d.x (x' + e' cosY)dY<br /> \ r /' |<br /> <br /> tu ol trong d,6L lii cungcuaparab x - y2 chAy di6m 0 (0;0) d6nA(L; 1). Ciu 2. Cho A ld tdp con kh6c rSng trong kh6ng gian metric (X, d). Chrmg minh ring, him sO;, X + IRx6c dinh bdi<br /> <br /> x) f (r) = d(x;A) : IEId(a;<br /> trong X. A) tuc X liOn trOn vd taphapM - {* . X: 0 s (d(*; A))' + d.(x; s 2}- dong )<br /> Ciu 3. Xdt tApX g6m c6c hdm thgc x = x(t) liOntpc tr0n [0; +*) saocho<br /> <br /> a) Chtmg minh (X; ll<br /> <br /> ll) h<br /> <br /> < erlx(t)l +*' "i'Tl*' khdng gian dinh chuAnvoi<br /> l l r f lr e[ o ; + m )<br /> <br /> l s u p e t l x ( t ), v x e X .<br /> <br /> b) Xet phi6m ham f , X + IR, saocho f (x) - Io** tx(t) dt. Hdy ki6m tra sy x6c clinh cnaf Q) vd chrmgminh f tuy}ntinh li€n tpc. Tinh ll/ll. Cffu 4. a) Chokh6nggianHilbertH vitM ldt4p contrum4ttrongcuaH. Gid su x e H vit (x,")n ld ddy bi ch4ntrong H saocho v6i m6i y e M thi limrl-+@(xn;!l = (x;yl. Chimg minh ring ddy (x) r, hQitq y6u d€nx. b) Tr€n mQtkhdng gian Hilbert H, vsimoi a * 0 chrmgminh ring, phi6mhamtuy6n rinh li€n t.ucfr(x) = (x; al ldmQttodn6nhvd suy ra fo cingld mOt6nhx4 mo.<br /> <br /> Ghi chti: Cdn bQcoi thi khong giai thich gi th€m.<br /> <br />