中国科学技术大学数学分析2006年考研真题考研试题
¢¡¢£¢¤¢¥¢¦¢¤¨§¨©¨¨£¨¤¨¨¨¥ 2006 ¢¢ ¢¢¢¨¨¨¨¨ !#"#$&%('0)#1#2#35461#$#7#8 310 9 (3 @#A#B#C#D0E 2 150 2 ) F '0G#H#I#P#Q#R#S#T#I#U#V#W D S#T#X#Y#W#`#a#b#V#W#c#d#e#f#g 1 (15 h ) i a, b prqrsrtrurv x = 0 wrxryr y = ax + b x ≥ 0; x2 + 1 x < 0. 2 (15 h ) an > 0,
ru ∞ n=1 1 an rr irr
ru ∞ n=1 1 an+1 rr 3 (15 h ) m, n ≥ 0 rru irrh 1 0 xm(1 − x)ndx r 4 (15 h ) a > 0, f(x) [−a, a] rrdrerrfrtru hga −a f(x) 1 + ex dx = a 0 f(x)dx. 5 (15 h ) itiu f(x) vijik [a, b] iliminpoiqirtsvuixvyvw f (a) = f (b) = 0, gix v ξ ∈ (a, b) pry |f (ξ)| ≥ 4 (b − a)2 |f(b) − f(a)|. 6 (15 h ) rzrrtru f(x) {r|r}r~ryrurvorq [a, b] rrdre h w f(a) = 0, g max x∈[a,b] |f(x)| ≤ √ b − a( b a |f (t)|2 dt)1/2 , b a f2 (x)dx ≤ 1 2 (b − a)2 b a |f (x)|2 dx. 7 (15 h ) n rro D r
rr C rr hg C ∂u ∂n ds = D ( ∂2u ∂x2 + ∂2u ∂y2 )dxdy. 8 (15 h ) Γ : x2 a2 + y2 b2 = 1 rrrrriiihri L S, r J = Γ(b2x2 + 2xy + a2y2)ds, g J = S2L π2 9 (15 h ) rrrh 1 0 dx 1+x3 r h rr uri
iu ∞ n=1 (−1)n−1 3n−2 r 10 (15 h i x = a cos3 t, y = a sin3 t(a > 0) ¡r¢r y = x £r¤rrrrr¥ri 1
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