Proof of 1, the limit of sequence
The proof of the limit of sequence is the focus of number one and number two, especially the number two has been tested very frequently in recent years, and several big proof questions have been tested. The big problem generally involves the proof of the limit of sequence, and the method used is monotone bounded discrimination.
2. Relevant proofs of differential mean value theorem.
3, the problem of equation root
Include that uniqueness of the equation root and the number of the equation root.
4. Proof of inequality
5. Proof of definite integral equality and inequality
The main methods involved are differential calculus: constant variation method; Integral method: method of substitution and distributed integral method.
6. Five Equivalent Conditions of Path-independent Integral
This part is the focus of the top scholar exam, which has not been designed in recent years, so we should focus on it.
Method article
These are the places where it is easy to give proof questions, and students focus on summarizing the solutions of these questions when reviewing. So, what methods should we use to solve this kind of proof problem?
1, remember the basic principle in combination with geometric meaning.
Understanding the basic principle is the basis of proof, and different understanding levels (that is, the depth of understanding the theorem) will lead to different reasoning abilities. For example, in 2006, the real math question 16 (1) was to prove the existence of limit and find the limit.
As long as the existence of the limit is proved, the evaluation is easy, but if the first step is not proved, even if the limit value is found, you can't score. Because mathematical reasoning is closely linked, if the first step is inconclusive, then the second step is castles in the air.
This topic is very simple, using only one of the two criteria for the existence of limit: monotone bounded sequence must have limit. As long as we know this criterion, the problem can be easily solved, because for the series in this problem, "monotonicity" and "boundedness" have been well verified. There are not many proofs that the basic principles can be directly applied like this, but more need to use the second step.
2. Seek the proof method with the help of geometric meaning.
Many times, a proof problem can be correctly explained by its geometric meaning. Of course, the most basic thing is to correctly understand the meaning of the title text.
For example, the question 19 of Math I in 2007 is a proof of the mean value theorem, and we can draw a sketch of the function satisfying the problem conditions in the rectangular coordinate system. Then we can find that, in addition to the two endpoints, the two functions also have a point with the same function value, that is, a point between the points where the two functions take the maximum values respectively (correct inspection: the point where the two functions take the maximum values is not necessarily the same point). In this way, it is easy to think that the auxiliary function F(x)=f(x)-g(x) has three zeros, and the proved conclusion can be obtained by applying Rolle mean value theorem twice.
3. Inverse deduction method
Seek the proof method from the conclusion. For example, the problem 15 in 2004 is an inequality proof problem, which can be solved by applying the general steps of inequality proof: that is, constructing a function from the conclusion and deducing the conclusion by using the monotonicity of the function.
When judging the monotonicity of a function, we need to rely on the relationship between the sign of the derivative and monotonicity. In general, the monotonicity of a function can be judged only by the sign of the first derivative, but there are many abnormal situations (the example given here is abnormal). At this time, it is necessary to use the sign of the second derivative to judge the monotonicity of the first derivative, and then use the sign of the first derivative to judge the monotonicity of the original function, so as to get the result to be proved. Let F(x)=ln*x-ln*a-4(x-a)/e* in this problem, where eF(a) is the inequality to be proved.