# 武汉大学2016年基础数学复试笔试试题

Eufisky posted @ 2016年4月09日 17:53 in 考研 with tags 考研 武汉大学 , 914 阅读

1.设$f(x)$在$(a,b)$上可微,且$f(x)$在$a$点右连续,试证:

(1) 若导函数$f'(x)$的右函数极限存在且为$A$,证明导函数$f'(x)$在$a$点的右侧存在且${{f'}_ + }\left( a \right) = \mathop {\lim }\limits_{x \to {a^ + }} f'\left( x \right) = A.$

(2) $f'(x)$在$(a,b)$上不存在第一类间断点.

2.若级数$\sum_{n=1}^\infty a_n$收敛,证明级数$\sum\limits_{n = 1}^\infty {\frac{{{a_n}}}{{\sqrt {{r_{n - 1}}} + \sqrt {{r_n}} }}}$收敛,其中${r_n} = \sum_{k = n + 1}^\infty {{a_k}}$.

3.讨论$\lambda$取何值时, $y''+\lambda y=0$有非零的初值解,其中$y(0)=y(1)=0$.

4.$A$为正定矩阵, $A-B$为半正定矩阵,试证明:

(1) 方程$|\lambda B-A|=0$关于根$\lambda\geq1$;

(2) $|B|\leq |A|$.

5.讨论积分$\int_0^1 x^{p-1}\ln^2 xdx$在下列情况下的一致收敛性.

(1) $p\geq p_0>0$;

(2) $p>0$..

6. 设非负函数$f(x,y)$在区域$D$上可积,证明积分$\iint\limits_D f(x,y)dx=0$充分必要条件为$f(x,y)$在$D$上的连续点上等于$0$.

JKBOSE 12th Question 说:
2022年8月25日 01:31

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