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The first-derivative test depends on the "increasing–decreasing test", which is itself ultimately a consequence of the mean value theorem. It is a direct consequence of the way the derivative is defined and its connection to decrease and increase of a function locally, combined with the previous section.
For this reason, a generalisation of the lemma can be used in the definition of differentiability in multivariable calculus. In particular, suppose f maps some subset of R n {\displaystyle \mathbb {R} ^{n}} to R {\displaystyle \mathbb {R} } .
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. [1] It is one of the two traditional divisions of calculus, the other being integral calculus —the study of the area beneath a curve.
The study of analogies of this type is known as umbral calculus. A general theory covering such relations, including the falling and rising factorial functions, is given by the theory of polynomial sequences of binomial type and Sheffer sequences .
for the infinite series. Note that if the function () is increasing, then the function () is decreasing and the above theorem applies.. Many textbooks require the function to be positive, [1] [2] [3] but this condition is not really necessary, since when is negative and decreasing both = and () diverge.
Calculus is also used to find approximate solutions to equations; in practice, it is the standard way to solve differential equations and do root finding in most applications. Examples are methods such as Newton's method, fixed point iteration, and linear approximation.
Steiner's problem, asked and answered by Steiner (1850), is the problem of finding the maximum of the function = /. [1] It is named after Jakob Steiner. The maximum is at =, where e denotes the base of the natural logarithm. One can determine that by solving the equivalent problem of maximizing
The theorem states that if you have an infinite matrix of non-negative real numbers , such that the rows are weakly increasing and each is bounded , where the bounds are summable < then, for each column, the non decreasing column sums , are bounded hence convergent, and the limit of the column sums is equal to the sum of the "limit column ...
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