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Cauchy's mean value theorem, also known as the extended mean value theorem, is a generalization of the mean value theorem. [ 6 ] [ 7 ] It states: if the functions f {\displaystyle f} and g {\displaystyle g} are both continuous on the closed interval [ a , b ] {\displaystyle [a,b]} and differentiable on the open interval ( a , b ) {\displaystyle ...
The term symbolic is used to distinguish this problem from that of numerical integration, where the value of F is sought at a particular input or set of inputs, rather than a general formula for F.
The slope field of () = +, showing three of the infinitely many solutions that can be produced by varying the arbitrary constant c.. In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral [Note 1] of a continuous function f is a differentiable function F whose derivative is equal to the original function f.
By the first part of the theorem, we know G is also an antiderivative of f. Since F′ − G′ = 0 the mean value theorem implies that F − G is a constant function, that is, there is a number c such that G(x) = F(x) + c for all x in [a, b].
In mathematics, Vinogradov's mean value theorem is an estimate for the number of equal sums of powers. It is an important inequality in analytic number theory , named for I. M. Vinogradov . More specifically, let J s , k ( X ) {\displaystyle J_{s,k}(X)} count the number of solutions to the system of k {\displaystyle k} simultaneous Diophantine ...
The value g(x)-g(y) is always nonzero for distinct x and y in the interval, for if it was not, the mean value theorem would imply the existence of a p between x and y such that g' (p)=0. The definition of m(x) and M(x) will result in an extended real number, and so it is possible for them to take on the values ±∞.
In general, by replacing constants with locally constant functions, one can extend this theorem to disconnected domains. For example, there are two constants of integration for ∫ d x / x {\textstyle \int dx/x} , and infinitely many for ∫ tan x d x {\textstyle \int \tan x\,dx} , so for example, the general form for the integral of 1/ x ...
This is known as the squeeze theorem. [1] [2] This applies even in the cases that f(x) and g(x) take on different values at c, or are discontinuous at c.