What's the use of differential mean value theorem? Seeing that there are many things mentioned in the book, such as concavity and convexity, monotony, extremum and so on, which are often used in economics, and they are coherent before and after. Only in this way can I feel that learning is beneficial. However, from the beginning to the end, I didn't find the function of the mean value theorem. It seems that I have said it in isolation, and the meaning is very clear, But after that, there is no following. I didn't find any coherence in it. I know it must be that I didn't find it. I hope you know

Maybe you write too simply, or you skip the strict deduction of concavity, monotonicity, extremum and so on
First of all, from a geometric point of view, the mean value theorem can be used to describe geometric intuition. For example, Rolle's theorem, Lagrange's theorem and Cauchy's theorem all have the geometric meaning of "there is a tangent parallel to the secant", while Taylor's theorem has a complex geometric meaning, which can be understood as replacing the secant with a curve of higher degree rather than a straight line
If you just look at the specific discussion of monotonicity, concavity, convexity and other properties that you think are particularly useful, you will find that it is not easy to strictly prove these geometrically intuitive properties, or generally speaking, many things that are very obvious are not logically clear, and the mean value theorem can just overcome those difficulties, and explain geometrically intuitive clearly, You may as well prove that f '(x) is always greater than 0 on the interval (a, b), then f (x) is strictly monotonic increasing on (a, b). If you don't use the mean value theorem, the proof is very difficult (Mr. Hua Luogeng tried to avoid the mean value theorem, but he didn't do it completely)
From the point of view of mathematics itself, the existence theorem is basically the most important, and the mean value theorem is the existence theorem without exception, and its technical value is far more than the expression of geometric intuition. Basically, it can be said that at least half of the buildings of the first generation of calculus were built by various mean value theorems (including the integral mean value theorem), The second generation of calculus mainly makes up for the deficiency on the basis of logic, but there is not much improvement on its practicability. At present, some scholars are studying the third generation of calculus which avoids the limit and the mean value theorem, but I think it is only for the beginners of non mathematics majors to get started faster, and it is not omnipotent to replace the equation with inequality

Is differential and differential mean value theorem related

Differential mean value theorem is based on the operational properties of differential and some theorems derived
The common ones are Rolle mean value theorem, Lagrange mean value theorem, Cauchy mean value theorem, etc

The history and development of differential mean value theorem

People's understanding of the differential mean value theorem can be traced back to the ancient Greek era
In geometry research, we get the following conclusion: "the tangent passing through the vertex of parabolic arch must be parallel to the vertex of parabolic arch
This is a special case of Lagrange's theorem. Archimedes, a famous Greek mathematician
It is the ingenious use of this conclusion that the area of parabolic arch can be obtained
Cavalieri, Italy, gave an interesting lemma to deal with the tangent of plane and solid figures in Volume I of "non-component geometry" (1635). Lemma 3, based on the geometric point of view, also describes the same fact: the tangent of a point on a curve must be parallel to the chord of the curve. This is a geometric differential mean value theorem, which is called Cavalieri's theorem
In 1637, Fermat, a famous French mathematician, gave Fermat's theorem in the method of finding maximum and minimum. In textbooks, people usually call it Fermat's theorem, Rolle, a French mathematician, gave Rolle's theorem in the form of polynomial in his paper solution of equations. In 1797, Lagrange, a French mathematician, gave Lagrange's theorem in his book analytic function theory, and gave the initial proof. It was Cauchy, a French mathematician, who systematically studied the differential mean value theorem. He was the promoter of mathematical analysis's rigorous movement, His three great works, analysis course, introduction to infinitesimal calculation course (1823) and differential calculation course (1829), reconstructed calculus theory with the main goal of strictness. He first gave the mean value theorem an important role and made it the core theorem of calculus, Cauchy first proved the Lagrange theorem strictly, and then extended it to Cauchy's theorem in the course of differential computation. Thus, the last differential mean value theorem was found

245 divided by 35

245÷35
=（49×5）÷（7×5）
=49÷7
=7
245 is easily divisible by 5, so extracting the common factor of 5 is the same as 35

Given that the straight line L passes through the point (0,3) and the inclination angle is twice of the straight line y = 2x + 1, the equation of the straight line L is obtained

The slope of the solution line y = 2x + 1 is k = 2, that is, Tan α = 2
That is to say, the inclination angle of the straight line is 2 α
That is, the slope of the straight line tan2 α = 2tan α / (1-tan & # 178; α) = - 4 / 3
That is, the equation Y-3 = - 4 / 3 (x-0)
That is 4x + 3y-9 = 0

9.3x = 0.3 (x + 6)

9.3x=0.3（x+6)
9.3x=0.3x+1.8
9.3x-0.3x=1.8
9x=1.8
x=0.2

It is known that all x of proposition p belong to [1,2], x ^ 2-A "0, and proposition q exists that x belongs to R, x ^ 2 + 2aX + 2-A = 0?

Both propositions are true. Proposition p is true. Proposition x ^ 2-A ≥ 0 holds on [1,2], so a ≤ {x ^ 2} min = 1 (i.e. the minimum value of a ≤ x ^ 2), i.e. proposition q is true. Proposition x belongs to R, x ^ 2 + 2aX + 2-A = 0, then Δ = (2a) ^ 2-4 (2-A) = 4A ^ 2 + 4a-8 ≥ 0, so a ≤ - 2 or a ≥ 1 take the intersection to get a ≤ - 2 or a = 1, i.e. the range of a is {a | a ≤ -

1、 (1) 3 / 5 / 4 / 1 / 6 / 15 / 14
(2) 9 of 4.25 × 20-1 of 2 × 0.45 + 9 of 4 ／ 20 of 9

(1) 3 / 5 / 4 / 1 / 6 / 15 / 14
=3/5x1/4x1/6x15/14
=3/5x1/6x1/4x15/14
=1/10x1/4x15/14
=1/40x15/14
=15/560
=3/112
(2) 9 of 4.25 × 20-1 of 2 × 0.45 + 9 of 4 ／ 20 of 9
=4.25x0.45-0.5x0.45+2.25x0.45
=(4.25-0.5+2.25)x0.45
=6x0.45
=2.7
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If the chord length of the line L with slope 1 cut by circle x2 + y2 = 4 is 2, then the equation of the line L is______ ．

Let the equation of a straight line be y = x + B, and the distance from the center of a circle to the straight line be d = | B | 2. Then, from the square of the radius equal to the sum of the square of the distance from the center of a circle to the straight line and the square of half the chord length, we get (| B | 2) 2 + 1 = 4, and the solution is b = ± 6, so the answer is y = x ± 6

30+31+32+33+34+35+36+37+38+39+.+60

1395