问题 问答题

设函数y(x)在任意点处的增量

,其中α是当△x→0时比△x较高阶的无穷小,且y(0)=π,则y(1)=______.

答案

参考答案:首先尝试从△y的表达式直接求y(1).为此,设x0=0,△x=1,于是△y=y(x0+△x)-y(x0)=y(1)-y(0)=y(1)-π,代入△y的表达式即得
[*]
由于仅仅知道当△x→0时α是比△x较高阶的无穷小,而不知道α的具体形式,因而从上式无法求出y(1)来.
由此可见,为了求出y(1)必须去掉△y的表达式中包含的α.利用函数的增量△y与其微分dy的关系可知,函数y(x)在任意点x处的微分
[*]
这是一个可分离变量的方程,它的满足初始条件y|x=0=π的特解正是本题中所说的函数y(x),解出y(x)即可得到y(1).
将方程[*]分离变量,得[*]
求积分可得[*]
解出y得方程的通解[*]
扫初始条件 y(0)=π可确定[*],于是
[*]

填空题
阅读理解

阅读理解。

     It is natural for young people, at times, to blame their parents for most of the misunderstandings between

them. They have always complained that their parents are out of touch with modern ways; that they do not

trust their children to deal with crises; that they talk too much about certain problems, and that they have no

sense of humor, at least in parent-child relationships. I think it is true that parents often look down upon their

teenage children and also forget how they themselves felt, when young.

     For example, young people like doing things without careful consideration: it is one of their ways of

showing that they can accept a challenge. Older people worry more easily: most of them plan things ahead, at

least in the back of their minds, and do not like their plans to be upset by something unexpected.

     So my advice to you is this: when you want to borrow the family car or get your mother to mend

something for you, you will have better success if you can possibly ask in advance.

     Young people often annoy their parents with their choices in clothes and hairstyles, in entertainers and

music. This is not the main reason. They feel cut off from the adult world. So they create a culture and society

of their own. Then, if it turns out that their music and entertainers, or vocabulary, clothes and hairstyles annoy

their parents, this gives them additional enjoyment. They feel they are better, at least in a small way, and they

are leaders in style and taste.

      Sometimes you don't obey because you do not want your parents to accept what you do. If they did

accept, it looks as if you are tricking your own age group. But in that case, you are supposing that you are the

underdog: you can't win but at least you can keep your honor. This is a passive way of looking at things. It is

natural enough, after long years of childhood, when you were completely under your parents' control. But it

ignores the fact that you are now beginning to be responsible for yourself.

      If you plan to control your life, co-operation can be part of the plan. You can impress others, especially

parents, with your sense of responsibility, so that they will give you the right to do what you want to do.

1. According to the passage, all the following complaints are true EXCEPT that ______.

A. parents have no sense of humor at all

B. parents are always behind the times

C. parents talk too much about certain problems

D. parents don't believe their children can deal with crises

2. What does the author suggest to young people to win support from their parents?

A. Ignoring them.

B. Controlling them.

C. Entertaining them.

D. Cooperating with them.

3. We can learn from the passage that ______.

A. young people are not allowed to join adult society because they have their own culture

B. it is impossible for young people to get along well with older people

C. older people tend to forget what they felt when they were young

D. older people often lead the way in fashion

4. The underlined word "underdog" in the 5th paragraph refers to a person who _____.

 A. keeps dogs as pets

B. seems less likely to succeed

C. likes to cheat his own age group

D. looks down upon others around him