2015年第26届北京市大学生数学竞赛试题(经济管理类)答案.pdf

?S?K ?X6O?y? 1?83?fi ?)m?(L+na) ?m?2015c10?24F?9:00?11:30 ?/“?4? ?Y ?!W?K (?K? 10 ?Kz?K 3 ? 30 ) 1. 4? lim x+ Rx 1 ?t2?e1/t 1? t?dt x2ln ?1 + 1 x ?= 1 2 “ 2. ?4?lim (x,y)(3,1 3) (2 + xy) 1 y+xy2 =e3“ 3. ?f(x)?N?Yk?f1(x)? Z f(x)dx = F(x) + C o Z f1(x)dx =xf1(x) F ?f1(x)? + C“ 4. Z 3/2 /2 sin10xdx =2 9! 10! “ 5. ?z = z(x,y)d?xyz + px2 + y2+ z2= 2(?z(1,0) = 1 Kz(x,y)3:(1, 0)?dz =dx 2dy “ 6. ?z = f(x,y)?kY?v 2f y2 = 2, f(x,0) = 1, f y(x,0) = xK f(x,y) =1 + xy + y2“ 7. Z 1 0 dy Z y y/2 sin(x2)dx + Z 2 1 dy Z 1 y/2 sin(x2)dx = 1 2(1 cos1) “ 8. ? + X n=1 Z 1/n 0 x 1 + x2 dx?5(u?)“ 9. ? : x + y = 1?S : x +y = 1K?S ?/7?= ?=N?NV = 2 15 “ 10. ?f(u)?k?Y?f(1) = 0q?z = f ?ex + ey?v? z x + z y = 2Kf(u) =lnu2“ 11?(?8?) 1?83?fi ?)m?(L+na) ?!(?K 8 ) ?f(x)3?D ?u“e?x0 D ?y = f(x)3:(x0, f(x0)?x = x09x?4 ?f(0) = 2f(x)“ )?f(x)3:(x0, f(x0)? y = f(x0) + f(x0)(x x0) -y = 0?x?:?Ix1= x0 f(x0) f(x0) ?Kk 1 2f(x0)(x0 x1) = 4 1 2f(x0) f(x0) f(x0) = 4 dd?y = f(x)?vXe? y= 1 8y 2 1 y = 1 8 x + C dy(0) = 2?C = 1 2 ? y = f(x) = 8 4 x 12?(?8?) 1?83?fi ?)m?(L+na) n!(?K 10 ) ?f(x)kn? lim x f(x) = A, lim x f(x) = 0y? lim x f(x) = lim x f(x) = 0 y?| Taylor m“? f(x 1) = f(x) f(x) + 1 2 f(x) 1 6 f(1), x 1 1?f(x)?NO?f(1) = 1 1 4 = 3 4 ?0 F/?!XJ?b?y = f(x)3(0, 1)S? ?f(x) 2f(x) x y?x0 (0, 1)?“ )?!k5?3?ey?8I?x0 (0, 1)? x0f(x0) = Z 1 x0 f(t)dt ?d?- g(x) = x Z 1 x f(t)dt, x 0, 1 Kg(x)3m0, 1?Y3(0, 1)S?kg(0) = g(1) = 0d Rolle n? 3x0 (0, 1)?g(x0) = 0“du g(x) = xf(x) Z 1 x f(t)dt k g(x0) = 0x0f(x0) = Z 1 x0 f(t)dt ?!g(x) = xf(x) Z 1 x f(t)dt?5?f(x)