第二届全国大学生数学竞赛预赛试卷参考答案(数学类).pdf

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1、1?3I?)m?m?Y(a)?!(10)?(0,1),x0=a,xn+1=a+sinxn(n=0,1,2,.).y:=limn+xn?3,?x sinx=a?.yyy:5?|(sinx)0|=|cosx|1,d?n,?k|sinx siny|x y|,x,y?.(2)|xn+2 xn+1|=|(sinxn+1 sinxn)|xn+1 xn|,n=0,1,2,.(4)l?|xn+1 xn|n|x1 x0|,n=0,1,2,.u?Xn=0(xn+1 xn)?,l?=limn+xn?3.(6)u4?“xn+1=a+sinxn?4?=?x sinx=a?.(8)?,?x sinx=a,=sin=a?,

2、K|=|sin sin|.d (0,1)?=.=x sinx=a?.y.(10)2?!(15)?B=01030002010000.y X2=B),p X?n?E?.11?(?8?)yyy:?y.?k),=?3E?A?A2=B.(2)?5?B?A?0,?3.(4)?A?A?,K 2?B?A?.=0.l?A?A?0.(6)u A?Jordan IO.?U?J1=000000000,J2=010000000J3=010001000.(10)l?A2?Jordan IO.?U?J1=J21=J22 J2=J23.(12)d A2?u 1,?B=A2?2 g.X2=B).y.(15)2nnn!(10)?

3、D?2?,f(x,y)?.y:f(x,y)3 D?Y.5:f(x,y)?(0,1)9(x1,y1),(x2,y2)D,f(x1+(1 )x2,y1+(1 )y2)f(x1,y1)+(1 )f(x2,y2).yyy:(?.?y(?.(i)u 0 9 x0,x0+?g(x),N?y x(x0,x0+):g(x0)g(x0)dg(x)g(x0)x x0g(x0+)g(x0).12?(?8?).(2)xyx0 x0 x0+xxl?flflflg(x)g(x0)x x0flflflflflflg(x0+)g(x0)flflfl+flflflg(x0)g(x0)flflfl,x (x0,x0+).dd=?

4、g(x)3 x0Y.?/,?mm?Y.(4)(ii)?(x0,y0)D.Kk 0?E x0,x0+y0,y0+D.(5)5?x y?,f(x,y)?,d(i)?(?,f(x,y0),f(x,y0+),f(x,y0)?x x0,x0+?Y,l?k.,=?3 M 0?|f(x,y0+)f(x,y0)|+|f(x,y0)f(x,y0)|+|f(x0+,y0)f(x0,y0)|+|f(x0,y0)f(x0,y0)|M,x x0,x0+.13?(?8?).(7)?,d(i)?(?,u(x,y)E,|f(x,y)f(x0,y0)|f(x,y)f(x,y0)|+|f(x,y0)f(x0,y0)|f(x,y0

5、+)f(x,y0)|+|f(x,y0)f(x,y0)|y y0|+|f(x0+,y0)f(x0,y0)|+|f(x0,y0)f(x0,y0)|x x0|M|y y0|+M|x x0|.u f(x,y)3(x0,y0)Y.y.(10)2ooo!(10)?f(x)30,1?Riemann?,3 x=1?,f(1)=0,f0(1)=a.y:limn+n2Z10 xnf(x)dx=a.yyy:P M=supx0,1|f(x)|0,(0,1),?0?5=?limn+n2Z10 xnf(x)dx=a.y.(10)2!(15)fi?g (?z)Le:?A(1,0,0),B(1,1,2),C(1,1,2),D

6、(3,0,0),E(3,1,2),F(3,2,4),G(0,1,4),H(3,1,2),I(5,22,8).fl=?a”)?:?,A!B!C?,D!E!F?.(6)?k?g?U?3?n:?V?V?.(10),?w?ABC?DEF 1?.(12)15?(?8?)?q?V?U(V?x?1?x?1?),?U?V.(15)5:?(?)?“)(x 2)2+y2z24=1.888!(20)?A?n n?(?7),?n?(1,.,n),A 0(p L?=?),?3 n?,?A=0,?n?x?y,?xAy6=0?k xAy+yAx6=0.y:?n?v,?k vA=0.yyy:?r,dK?(v+r)A(v+r)

7、0.(8)=vAv+rvA+rAv+r2A 0.(12)=vAv+rvA+Av+r2A 0.(14)e vA6=0,Kk vA+Av6=0.d?r?vAv+rvA+Av+r2A 0,?3?0,1?(k?)?g(x),?,0,1,flflflZf(x)g(x)dxflflfl2.Am=hmn,mn+Zm+1nmnf(t)dt,g(x)=1,x n1m=0Am,0,x 6n1m=0Am.(5)u 0 1,?K?k?vkn k+1n,n +1n,KflflflZf(x)g(x)dxflflflZk+1n|f(x)g(x)|dx+flflflZnk+1nf(x)g(x)dxflflfl+Zn|f(x)

8、g(x)|dxZk+1n1dx+0+Zn1dx2n.y.(10)2lll!(10)fi?:(0,+)(0,+)?Ne?Y,?vlimt0+(t)=+.eZ+0(t)dt=Z+01(t)dt=a +,?1L?.y:Z+0(t)2dt+Z+01(t)2dt 12a32.yyy:-P=R+p(t)dt,Q=R+q1(t)dt,I=a P Q,?pq=a.(2)17?(?8?)KZ+01(t)2dt Zq01(t)2dt1qZq01(t)dt2=1q(a Q)2=1q(I+P)2,Z+0(t)2dt Zp0(t)2dt1pZp0(t)dt2=1p(a P)2=1p(I+Q)2.(6)d,Z+0(t)2dt+Z+01(t)2dt1p(I+Q)2+1q(I+P)22pq(I+P)(I+Q)=2aQP+aI.(8)?p,q?v P=Q=a I2,l?Z+0(t)2dt+Z+01(t)2dt1a(a I)24I+aI=22(a+I)2412a32.y.(10)218?(?8?)