问题 问答题

设f(x),g(x)在[a,b]上连续,在(a,b)内可导,证明:存在ξ∈(a,b),使得

答案

参考答案:[证明] 令


则F(x)满足拉格朗日中值定理的各个条件.
因F(a)=f(a)g(a)-g(a)f(a)=0,
F(b)=f(a)g(b)-g(a)f(b),
则存在ξ∈(a,b),使
F(b)F(a)=(b-a)F’(ξ),
即f(a)g(b)-g(a)f(b)=(b-a)[f(a)g’(ξ)-g(a)f’(ξ)],
亦即

解析: 从待证等式出现b-a因子,使人猜想到可能使用拉格朗日中值定理证之.但使用该定理的函数是什么如何找可将待证等式右边中的ξ变为x,去掉导数符号得到

=f(a)g(x)-g(a)f(x),
此为所要找的辅助函数.

阅读理解

The Beets were a new couple (夫妇). The things in their rooms, such as the television, the computer, the fridge, furniture(家具) and so on, were all left by their parents. They only paid some money for their food and clothes. But they tried their best to save money. If they had something important to tell their friends, they always visited them instead of calling them.

One day, it was a quarter past nine. Mr. Beet didn’t come back. His wife was a little worried. The supper was cold and she had waited for two hours. She was sitting by the table in the dark (黑暗) room.

Suddenly the woman heard someone coming upstairs. It was her husband’s step. She hurried to open the door. In the dark she could hear his deep breathing(呼吸). As soon as he sat down, she turned on the light and found he looked very tired.

“Why do you come back so late, dear?” asked the woman. “Haven’t you caught a bus?”

“Yes, but I didn’t take it.”

“Why not?”

“I ran after it so that I saved two dollars this evening.”

“That’s great!” the woman said happily. “Run after a taxi next time and you’ll save both more time and money.”

小题1:The supper was ready at ________ .

A.6:15

B.6:30

C.6:45

D.7:15小题2:Mrs. Beet sat in the dark because ___________        .

A.she wanted to save money.

B.she didn’t like the bright light.

C.she was afraid to be seen.

D.she wanted to have a good rest.小题3:When Mr. Beet got home that evening, __________   .

A.his wife was angry with him

B.he was very tired

C.he was too hungry to say a word

D.his wife felt sorry小题4:Mrs. Beet hoped _______________  .

A.her husband went home by bus the next time.

B.her husband could keep on doing exercise.

C.her husband could be stronger and stronger.

D.her husband could save more money and time.

单项选择题