The mean value theorem can be divided by definite integral (b-a), and according to the estimation theorem, the value is between m and m. According to the mean value theorem of continuous function, there is always ξ in f(x) to make its function value between the minimum value and the maximum value, and then b-a is multiplied.
The definite integral is the area of the shaded part, which is naturally the area between the lower part of the green line and the lower part of the red line; Mean value theorem: this area is equal to the area below the blue line between the minimum and maximum.
Extended data:
If it is the definite integral of the univariate function f(x) in the interval [a, b], just change the S in the formula of the above estimation theorem to the interval length b -a, such as the integral of the function f(x) in the monotonically decreasing interval [n+ 1, n], (n+1-n) * f (. =f(n) *(n+ 1-n), that is, any function continues its definite integral from the closed interval [a, b], where m is the minimum value of f(x) in the closed interval [a, b] and m is the maximum value.
The derivative only reflects the local characteristics of the function at one point; If we want to understand the global behavior of a function in its domain, we need to establish the relationship between the derivative and the function, and the differential mean value theorem is such a role. Differential mean value theorem, including Rolle theorem, Lagrange theorem, Cauchy theorem and Taylor theorem.
When comparing infinitesimal (large) quantities, we can see that the limit of the ratio of two infinitesimal (large) quantities may or may not exist. If it exists, its limit value is different. The limit of two infinitesimals or the ratio of two infinitesimals is called type limit or infinitive limit.
Baidu encyclopedia-mean value theorem
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