In-service graduate students of China Renmin University have always been the first choice of many people, and there are many in-service graduate students in the school, such as law, computer, finance and so on. China Renmin University is an important base for higher education and research in humanities and social sciences in China, and is known as a banner of humanities and social sciences in China. So, what are the problem-solving skills of 20 17 postgraduate entrance examination math proof questions? Let's take a look at the on-the-job postgraduate education network.
There are less than two months before the January entrance examination of 20 17 in-service graduate students. It is also important for candidates to make good use of this time to review mathematics. Let me give you a brief introduction.
1, remember the basic principle in combination with geometric meaning.
Remember the basic principles, including conditions and conclusions, such as zero existence theorem, mean value theorem, Taylor formula and two criteria of limit existence, combined with geometric significance.
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 the monotonicity and boundedness of the sequence in this problem 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.
Another example is that 18 (1) of Mathematics I in 2005 is a proof of the existence theorem of zero. As long as the graphs of functions y=f(x) and y = 1] on [0,1] are set in Cartesian coordinate system, we can see that these two graphs intersect immediately.
It should also be seen from the figure that the size relationship between the two functions at the two endpoints is just the opposite, that is, the difference function values at the two endpoints are different in sign, and the existence theorem of zero points ensures that there is zero point in the interval, which proves the required result. If the second step really cannot solve the problem satisfactorily, go to the third step.
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.
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