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In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function , the Taylor polynomial is the truncation at the order k {\textstyle k} of the Taylor series of the function.
Multi-binomial theorem ( x + y ) α = ∑ ν ≤ α ( α ν ) x ν y α − ν . {\displaystyle (x+y)^{\alpha }=\sum _{\nu \leq \alpha }{\binom {\alpha }{\nu }}\,x^{\nu }y^{\alpha -\nu }.} Note that, since x + y is a vector and α is a multi-index, the expression on the left is short for ( x 1 + y 1 ) α 1 ⋯( x n + y n ) α n .
Taylor's theorem can be used to obtain a bound on the size of the remainder. In general, Taylor series need not be convergent at all. In fact, the set of functions with a convergent Taylor series is a meager set in the Fréchet space of smooth functions.
Gaussian optics is a technique in geometrical optics that describes the behaviour of light rays in optical systems by using the paraxial approximation, in which only rays which make small angles with the optical axis of the system are considered. [2] In this approximation, trigonometric functions can be expressed as linear functions of the angles.
The notes that Marx took have been collected into four independent treatises: On the Concept of the Derived Function, On the Differential, On the History of Differential Calculus, and Taylor's Theorem, MacLaurin's Theorem, and Lagrange's Theory of Derived Functions, along with several notes, additional drafts, and supplements to these four ...
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The country's top doctor wants a new warning added to alcohol that would alert drinkers about links to cancer, but don't expect cigarette-style warning labels any time soon. U.S. Surgeon General ...
In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite. A simulation-based alternative to this approximation is the application of Monte Carlo simulations.
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