First, the analysis of specific test sites
First of all, we must find out what is the theoretical basis of this proof, which is equivalent to our tool. What tools do we need?
1. Properties of closed interval continuous functions.
Maximum theorem: a closed interval continuous function must have a maximum and a minimum.
Inference: Boundedness (closed interval continuous function must be bounded).
Intermediate value theorem: Any number between the maximum value and the minimum value of a closed interval continuous function can find a point on the interval, so that the function value of this point corresponds to it.
Zero theorem: closed interval continuous function, if the end function values of the interval are different in sign, then there must be a point in the interval with zero function value.
Second: differential mean value theorem (one lemma, three theorems)
Fermat's Lemma: The function f(x) is defined in a neighborhood U(ξ) of point ξ, where it is derivable. For any x∈U(ξ), there is f(x)≤f(ξ) (or f (x) ≥ f (\).
Rolle Theorem: If the function f(x) satisfies:
(1) is continuous on the closed interval [a, b];
(2) Derivable in the open interval (a, b);
The function values at the end points of the interval are equal, that is, f(a)=f(b
Then there is at least one point ξ (a
Geometrically speaking, the condition of Rolle's theorem indicates that the curve arc (equation is) is a continuous curve arc, and there are tangents that are not perpendicular to the X axis everywhere except the end points, and the longitudinal coordinates of both ends are equal. And the theorem conclusion shows that:
There is at least one point on the arc where the tangent of the curve is horizontal.
Lagrange mean value theorem: If the function f(x) satisfies:
(1) is continuous on the closed interval [a, b];
(2) Derivable in the open interval (a, b);
The function values at the end points of the interval are equal, that is, f(a)=f(b),
Then there is at least one point ξ (a
Enhanced version: If the function f(x) is continuous on the integral interval [a, b], there is at least one point ξ on (a, b), so the following formula holds.
Fourth, the derivative theorem of variable limit integral: If the function f(x) is continuous in the interval [a, b], the upper limit function of the integral variable has a derivative in [a, b], and the derivative is:
Fifth, Newton-Leibniz formula: If the function f(x) is continuous in the interval [a, b] and the original function F(x) exists, then
The above theorem requires understanding and mastering the content of the theorem and the corresponding proof process.
Second, matters needing attention
For the specific test sites in the above article, Mr. Tong gave some points for attention, which is also a "small signal" in the proof question. I hope everyone can clearly understand and master:
1. Among all theorems, only the interval of ξ in the intermediate value theorem and the integral mean value theorem is a closed interval.
2. Lagrange mean value theorem is a bridge between function f(x) and derivative function f ′ (x).
3. The integral mean value theorem is a bridge between definite integral and function.
4. Rolle theorem and Lagrange mean value theorem deal with one function, while Cauchy mean value theorem deals with two functions. If there are two functions in the conclusion, the form is similar to Cauchy mean value theorem, then we should think of our Cauchy mean value theorem.
5. If the enhanced version of the integral mean value theorem is applied to theorem proving, it must be proved first.
Secondly, the proof of the mean value theorem is generally divided into two categories: first, the application of Rolle theorem can also be divided into two categories: simple and complex. Simple types are generally proved to be f'(ξ)=0, f'(ξ)=k (k is an arbitrary constant), and f' (ξ 1) = g'
Generally, a conclusion like this only needs to find the conditions of Rolle's theorem. Generally, the first two conditions of Rolle theorem are all discussed, but the function values of two different points need to be equal. In order to find this condition, knowledge points such as the properties of closed interval continuous function, integral mean value theorem, Lagrange mean value theorem, the properties of limit and the definition of derivative are generally used. Complex type means that the conclusion is complex and it is necessary to establish an auxiliary function, and then make the auxiliary function meet the conditions of Rolle's theorem. The establishment of auxiliary function generally depends on the idea of solving differential equations. The second is to satisfy an expression with two points. Lagrange mean value theorem and Cauchy mean value theorem are generally used in this kind of problems, and the idea is to put aside the same letters in the conclusion first.
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