2006年中国科学院数学分析与高等代数考研试题

? ? ? ? 2006 ? ? ? !#$%989?ABC 1 (15 D ) E a,bFHGHIHJHKHLx = 0MHNHOHP y = ax + b Q x 0; x2+ 1 Q x 0,UHK P n=1 1 an VHWHX EHYHUHK P n=1 1 an+1 Z VHWH 3 (15 D ) m,n 0HHK X EH_HD R1 0 xm(1 x)ndxHa 4 (15 D ) a 0,f(x) b a,acHHdHeHHfHJHK Xhg Z a a f(x) 1 + ex dx = Z a 0 f(x)dx. 5 (15 D )iJiKf(x)Lijika,biliminpoiqirtsvuiNvOvwf0(a) = f0(b) = 0, gix L (a,b)FHy |f00()| 4 (b a)2 |f(b) f(a)|. 6 (15 D )HzHaHJHKf(x)H|HHHOHKHLoHqa,bcHHdHe Xh w f(a) = 0, g max xa,b |f(x)| b a(Z b a |f0(t)|2dt)1/2, Z b a f2(x)dx 1 2 (b a)2 Z b a |f0(x)|2dx. 7 (15 D ) nbHHoTDH THHCHH Xhg I C u n ds = ZZ D ( 2u x2 + 2u y2 )dxdy. 8 (15 D )T : x2 a2 + y2 b2 = 1HHHHTHii_iDHibL S,HJ = H (b 2x2 + 2xy + a2y2)ds, g J = S2L 2 9 (15 D )HH_HD R1 0 dx 1+x3 Ha Xh YT ZHH KHiUiK P n=1 (1)n1 3n2 H 10 (15D ETx = acos3t,y = asin3t(a 0)HHy = xHHHHTHHHi_ 1 ? ? ? ? 2006? ? ? ? !#?$?%= - 6/?33A1B18093C3D 2E=F3G3H3I3J 03K3L H3M/N1O3; K3L 6373O3P3Q3R/N1O/S1T3U3V3D 1. (16 W )X=Y, , Z3_A = . . Rnn bacedbfg 2. (16 W )hjilklm#n a11x1+a12x2+a1nxn=0 a21x1+a22x2+a2nxn=0 an1,1x1+an1,2x2+an1,nxn=0 lo lp#qjZ A = a11a12a1n a21a22a2n an1,1an1,2an1,n . r Mj(j = 1,2, ,n)sltlp#qAuwv#xjyj clzll|# n 1j djgll (i) (M1,M2, ,(1)n1Mn)slklm#n ljl l (ii)A Z n 1,kmn s (M1,M2, ,(1)n1Mn) g 3. (16 W ) Z ll ll lim n+ 1 n n 1 !n . 4. (18 W ) r aZA = a1 a . . 1 a R100100, _ A50 y baeblg 1 1b 1 2 5. (18 W )=en1,2, ,s(s 2)hbieb*1+2,2+3, ,s1+ s,s+ 1hjil#ji g 6. (18 W )XwY#j?jjkmx2+ ay2+ z2+ 2bxy + 2xz + 2yz = 4#wll# x y z = P x 0 y 0 z 0 Zl#j#jklmy 02 + 4z 02 = 4. _ a,b jfl llp#qP. 7. (16 W ) rl #jlf(x) = xAx,?ux s x jl A s 3 3lllp qjll?jjklm A3 6A2+ 11A 6I = 0. ll max A max kxk=1 f(x). ?ukxk2= x2 1+ x 2 2+ x 2 3, y jlllf slllwkm z lpqA _ g 8. (16 W )A R20062006s q ( tp Ap= 0 Ap16= 0), Wl#hjilklm#nAx = 0 (x R2006) lll l ljf#lfg 9. (16 W ) r f s eb =#lV hbieblwfn s