《2024江苏省大学生数学竞赛本1-3试题与评分标准》
一．填空题（每小题4分，共20分）(1)设f?u??arctan
1?u2?lnxdy,??x??,y?f???x??,则1?uxdx4n
3n
x?1
?1/5(本一)(1)设a?0，则lim(1?2
n???
?3?4)=42nn
1
n9(本二)(本三)(本一)(1)lim(1?2
n???
3n2n
?3)=2
1nn(2)??sinx?cos2x?x???
?20
dx?
?/2?2/3ax
(2)设a?0，则limln(1?e)ln(1?)?
axa2
(本二三)(3)?0
??
1
?1?x?22dx??/4(本一)(3)设f(u)?arctan
dy12,?(x)?,y?f(?(x)),则dx1?uxx?1
?1/10(本二三)??0,0,0??3,函数(4)已知函数F?u,v,w?可微，Fu??0,0,0??1,Fv??0,0,0??2,Fwz?f?x,y?由F2x?y?3z,4x2?y2?z2,xyz?0确定，满足f?1,2??0,则(本一)??fx??1,2??
(4)-2.?
+?
0
x
dx?23(1?x)1/4(本二三)(5)设?是区域??x,y?｜x
2
?y2?4,0?y?x?的边界曲线，取逆时针方向,则???x?y??3??y?1?eydx??x?y??xyeydy???3?6?.(本一)(5)已知F(u,v)可微，Fu'(0,0)?1,Fv'
(0,0)?2，函数z?f(x,y)由方程F(2x?y?3z,4x2?y2?z2)?0确定，满足f(1,2)?0,则fx'(1,2)?
-6二.解下列两题（每小题5分，共10分）2(1)求极限lim?1?3????2n?3???2n?1??n
?????2?4????2n?2???2n????;(2)求极限x2?xy?y2xlim?y?sinx4???x4?y4??y4?.2
解(1)记a12?32????2n?1?1n?
???2k?122?42????2n?2,因为?2k??2k?2??1?k?N*?,所以0?a1?3?2n?3???2n?1?n?22?3?542?5?762????2n?2?2?2n?1?2n?2?2n?1?2n?2,因为lim2n?1n??
?2n?2?0,应用夹逼准则得limn??an?0.(2)应用不等式的性质得x2?xy?y2?x2?y2?2xy?2?x2?y2?,x4?y4?2x2y2,
0?x2?xy?y2x4?y4?sin?x4?y4
??2?x2?y2?112x2y2?y2?x2，
因为lim?11?xx2?xyy??????y2?x2???0,应用夹逼准则得?y2xlimy????x4?y4?sin?x4?y4??0.二.解下列两题（每小题5分，共10分）?(1)求定积分?
20
(cosx?cos2x)2dx;
(2)求极限xlim
x?y???x2sin(x2?xy?y2
).
y???
?xy?y2(本二三)(本一)(本二)二.解下列两题（每小题5分，共10分）(1)求定积分(本三)?
?20
(cosx?cos2x)2dx;
(2)求极限lim
x?y
sin(x2?xy?y2).22x???x?xy?y
y???
2024年江苏高等数学竞赛本1-3参考答案
