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Encyclopedia > Taylor's theorem

In calculus, Taylor's theorem gives a sequence of approximations of a differentiable function near a given point by polynomials (the Taylor polynomials of that function) whose coefficients depend only on the derivatives of the function at that point. The theorem also gives precise estimates on the size of the error in the approximation. The theorem is named after the mathematician Brook Taylor, who stated it in 1712, even though the result was first discovered 41 years earlier in 1671 by James Gregory. For other uses, see Calculus (disambiguation). ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... This article is about functions in mathematics. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... Brook Taylor (August 18, 1685 – November 30, 1731) was an English mathematician. ... James Gregory For other people with the same name, see James Gregory. ...

The exponential function y = ex (continuous red line) and the corresponding Taylor polynomial of degree four around the origin (dashed green line).
The exponential function y = ex (continuous red line) and the corresponding Taylor polynomial of degree four around the origin (dashed green line).

Contents

Image File history File links Taylorspolynomialexbig. ... Image File history File links Taylorspolynomialexbig. ...

Taylor's theorem in one variable

A simple example of Taylor's theorem is the approximation of the exponential function ex near x = 0: The exponential function is one of the most important functions in mathematics. ...

 textrm{e}^x approx 1 + x + frac{x^2}{2!} + frac{x^3}{3!} + cdots + frac{x^n}{n!}.

The precise statement of the theorem is as follows: If n ≥ 0 is an integer and f is a function which is n times continuously differentiable on the closed interval [a, x] and n + 1 times differentiable on the open interval (a, x), then we have Not to be confused with Natural number. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

 f(x) = f(a) + frac{f'(a)}{1!}(x - a) + frac{f^{(2)}(a)}{2!}(x - a)^2 + cdots + frac{f^{(n)}(a)}{n!}(x - a)^n + R_n(x).

Here, n! denotes the factorial of n, and Rn(x) is a remainder term, denoting the difference between the Taylor polynomial of degree n and the original function. The remainder term Rn(x) depends on x and is small if x is close enough to a. Several expressions are available for it. For factorial rings in mathematics, see unique factorisation domain. ...


The Lagrange form[1] of the remainder term states that there exists a number ξ between a and x such that Joseph-Louis, comte de Lagrange (January 25, 1736 Turin, Kingdom of Sardinia - April 10, 1813 Paris) was an Italian-French mathematician and astronomer who made important contributions to all fields of analysis and number theory and to classical and celestial mechanics as arguably the greatest mathematician of the 18th century. ...

 R_n(x) = frac{f^{(n+1)}(&# 0;}{(n+1)!} (x-a)^{n+1}.

This exposes Taylor's theorem as a generalization of the mean value theorem. In fact, the mean value theorem is used to prove Taylor's theorem with the Lagrange remainder term. In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal (parallel) to the average derivative of the section. ...


The Cauchy form[2] of the remainder term states that there exists a number ξ between a and x such that Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ...

 R_n(x) = frac{f^{(n+1)}(&# 0;}{n!}(x-&# 0;^n(x-a).

More generally, if G(t) is a continuous function on [a,x] which is differentiable with non-vanishing derivative on (a,x), then there exists a number ξ between a and x such that

 R_n(x) = frac{f^{(n+1)}(&# 0;}{n!}(x-&# 0;^ncdotfrac{G(x)-G(a)}{G'(&# 0;}.

This exposes Taylor's theorem as a generalization of the Cauchy mean value theorem. In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal (parallel) to the average derivative of the section. ...


The integral form[3] of the remainder term is This article is about the concept of integrals in calculus. ...

 R_n(x) = int_a^x frac{f^{(n+1)} (t)}{n!} (x - t)^n , dt,

provided, as is often the case, f(n) is absolutely continuous on [a,x]. This shows the theorem to be a generalization of the fundamental theorem of calculus. Absolute continuity of real functions In mathematics, a real_valued function f of a real variable is absolutely continuous if for every positive number ε, no matter how small, there is a positive number δ small enough so that whenever a sequence of pairwise disjoint intervals [xk, yk], k = 1, ..., n... The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ...


For some functions f(x), one can show that the remainder term Rn approaches zero as n approaches ∞; those functions can be expressed as a Taylor series in a neighbourhood of the point a and are called analytic. Series expansion redirects here. ... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ...


Taylor's theorem (with the integral formulation of the remainder term) is also valid if the function f has complex values or vector values. Furthermore, there is a version of Taylor's theorem for functions in several variables. In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...


For complex functions analytic in a region containing a circle C surrounding a and its interior, we have a contour integral expression for the remainder

 R_n(x) = frac{1}{2 pi i}int_C frac{f(z)}{(z-a)^{n+1}(z-x)}dz

valid inside of C.


Estimates of the remainder

Another common version of Taylor's theorem holds on an interval (a-r,a+r) where the variable x is assumed to take its values. This formulation of the theorem has the advantage that it is often possible to explicitly control the size of the remainder terms, and thus arrive at an approximation of a function valid in a whole interval with precise bounds on the quality of the approximation.


A precise version of Taylor's theorem in this form is as follows. Suppose f is a function which is n times continuously differentiable on the closed interval [a-r, a+r] and n + 1 times differentiable on the open interval (a-r, a+r). If there exists a positive real constant Mn such that |f(n+1)(x)| ≤ Mn for all x ∈ (a-r,a+r), then

 f(x) = f(a) + frac{f'(a)}{1!}(x - a) + frac{f^{(2)}(a)}{2!}(x - a)^2 + cdots + frac{f^{(n)}(a)}{n!}(x - a)^n + R_n(x),

where the remainder function Rn satisfies the inequality (known as Cauchy's estimate):

 |R_n(x)| le M_n frac{r^{n+1}}{(n+1)!}

for all x ∈ (a-r,a+r). This is called a uniform estimate of the error in the Taylor polynomial centered at a, because it holds uniformly for all x in the interval. In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. ...


If, in addition, f is infinitely differentiable on [a-r,a+r] and In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ...

 M_nfrac{r^{n+1}}{(n+1)!} rightarrow 0 as n rightarrow infin ,!

then f is analytic on (a-r,a+r). In other words, an analytic function is the uniform limit of its Taylor polynomials on an interval. This makes precise the idea that analytic functions are those which are equal to their Taylor series. In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. ...


Taylor's theorem for several variables

Topics in calculus

Fundamental theorem
Limits of functions
Continuity
Vector calculus
Matrix calculus
Mean value theorem For other uses, see Calculus (disambiguation). ... The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ... In mathematics, the limit of a function is a fundamental concept in analysis. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices, where it defines the matrix derivative. ... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal (parallel) to the average derivative of the section. ...

Differentiation

Product rule
Quotient rule
Chain rule
Implicit differentiation
Taylor's theorem
Related rates
List of differentiation identities For other uses, see Derivative (disambiguation). ... In calculus, the product rule also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ... In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist. ... In calculus, the chain rule is a formula for the derivative of the composite of two functions. ... In mathematics, an implicit function is a generalization for the concept of a function in which the dependent variable may not be given explicitly in terms of the independent variable. ... In differential calculus, related rates problems involve finding a rate that a quantity changes by relating that quantity to other quantities whose rates of change are known. ...

Integration

Lists of integrals
Improper integrals
Integration by:
parts, disks, cylindrical
shells
, substitution,
trigonometric substitution,
partial fractions This article is about the concept of integrals in calculus. ... See the following pages for lists of integrals: List of integrals of rational functions List of integrals of irrational functions List of integrals of trigonometric functions List of integrals of inverse trigonometric functions List of integrals of hyperbolic functions List of integrals of arc hyperbolic functions List of integrals of... It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ... In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals. ... In mathematics, in particular integral calculus, disk integration (the disk method) is a means of calculating the volume of a solid of revolution. ... Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution. ... In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ... In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. ... In integral calculus, the use of partial fractions is required to integrate the general rational function. ...

Taylor's theorem can be generalized to several variables as follows. Let B be a ball in RN centered at a point a, and f be a real-valued function defined on the closure bar{B} having n+1 continuous partial derivatives at every point. Taylor's theorem asserts that for any xin B, In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general. ... In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ...

f(x)=sum_{|alpha|=0}^nfrac{D^alpha f(a)}{alpha!}(x-a)^alpha+sum_{|alpha|=n+1}R_{alpha}(x)(x-a)^alpha

where the summation extends over multi-indices α (this formula uses the multi-index notation). The notion of multi-indices simplifies formulae used in the multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an array of indices. ...


The remainder terms satisfy the inequality

|R_{alpha}(x)|lesup_{yinbar{B} }left|frac{D^alpha f(y)}{alpha!}right|

for all α with |α|=n+1. As was the case with one variable, the remainder terms can be described explicitly. See the proof for details.


Proof: Taylor's theorem in one variable

Integral version

We first prove Taylor's theorem with the integral remainder term.[4]


The fundamental theorem of calculus states that The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ...

int_a^x , f'(t) , dt=f(x)-f(a),

which can be rearranged to:

f(x)=f(a)+ int_a^x , f'(t) , dt.

Now we can see that an application of Integration by parts yields: In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals. ...

 begin{align} f(x) &= f(a)+xf'(x)-af'(a)-int_a^x , tf''(t) , dt  &= f(a)+int_a^x , xf''(t) ,dt+xf'(a)-af'(a)-int_a^x , tf''(t) , dt  &= f(a)+(x-a)f'(a)+int_a^x , (x-t)f''(t) , dt. end{align}

(The first equation is arrived at by letting u = f'(t) and dv = dt; the second equation by noting that int_a^x , xf''(t) ,dt = xf'(x)-xf'(a); the third just factors out some common terms.)


Another application yields:

f(x)=f(a)+(x-a)f'(a)+ frac 1 2 (x-a)^2f''(a) + frac 1 2 int_a^x , (x-t)^2f'''(t) , dt.

By repeating this process, we may derive Taylor's theorem for higher values of n.


This can be formalized by applying the technique of induction. So, suppose that Taylor's theorem holds for a particular n, that is, suppose that Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...

 f(x) = f(a) + frac{f'(a)}{1!}(x - a) + cdots + frac{f^{(n)}(a)}{n!}(x - a)^n + int_a^x frac{f^{(n+1)} (t)}{n!} (x - t)^n , dt. qquad(*)

We can rewrite the integral using integration by parts. An antiderivative of (x − t)n as a function of t is given by −(xt)n+1 / (n + 1), so In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals. ...

 int_a^x frac{f^{(n+1)} (t)}{n!} (x - t)^n , dt
 {} = - left[ frac{f^{(n+1)} (t)}{(n+1)n!} (x - t)^{n+1} right]_a^x + int_a^x frac{f^{(n+2)} (t)}{(n+1)n!} (x - t)^{n+1} , dt
 {} = frac{f^{(n+1)} (a)}{(n+1)!} (x - a)^{n+1} + int_a^x frac{f^{(n+2)} (t)}{(n+1)!} (x - t)^{n+1} , dt.

Substituting this in (*) proves Taylor's theorem for n + 1, and hence for all nonnegative integers n.


The remainder term in the Lagrange form can be derived by the mean value theorem in the following way: In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal (parallel) to the average derivative of the section. ...

 R_n = int_a^x frac{f^{(n+1)} (t)}{n!} (x - t)^n , dt =f^{(n+1)}(&# 0; int_a^x frac{(x - t)^n }{n!} , dt.

The last integral can be solved immediately, which leads to

 R_n = frac{f^{(n+1)}(&# 0;}{(n+1)!} (x-a)^{n+1}.

Mean value theorem

An alternative proof, which holds under milder technical assumptions on the function f, can be supplied using the Cauchy mean value theorem. Let G be a real-valued function continuous on [a,x] and differentiable with non-vanishing derivative on (a,x). Let In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal (parallel) to the average derivative of the section. ...

 F(t) = f(t) + frac{f'(t)}{1!}(x-t) + dots + frac{f^{(n)}(t)}{n!}(x-t)^n.

By Cauchy's mean value theorem,

 frac{F'(&# 0;}{G'(&# 0;} = frac{F(x) - F(a)}{G(x) - G(a)}       (1)

for some ξ ∈ (a,x). Note that the numerator F(x) - F(a) = Rn is the remainder of the Taylor polynomial for f(x). On the other hand, computing F′(t),

 F'(t) = f'(t) - f'(t) + frac{f''(t)}{1!}(x-t) - frac{f''(t)}{1!}(x-t) + dots + frac{f^{(n+1)}(t)}{n!}(x-t)^n = frac{f^{(n+1)}(t)}{n!}(x-t)^n.

Putting these two facts together and rearranging the terms of (1) yields

 R_n = frac{f^{(n+1)}(&# 0;}{n!}(x-&# 0;^ncdotfrac{G(x)-G(a)}{G'(&# 0;}.

which was to be shown.


Note that the Lagrange form of the remainder comes from taking G(t) = (t-a)n+1, and the given Cauchy form of the remainder comes from taking G(t) = (t-a).


Proof: several variables

Let x=(x1,...,xN) lie in the ball B with center a. Parametrize the line segment between a and x by u(t)=a+t(x-a). We apply the one-variable version of Taylor's theorem to the function f(u(t)):

f(x)=f(u(1))=f(a)+sum_{k=1}^nleft.frac{1}{k!}frac{d^k}{dt^k}right|_{t=0}f(u(t)) + int_0^1 frac{(1-t)^n }{n!} frac{d^{n+1}}{dt^{n+1}} f(u(t)) dt.

By the chain rule for several variables, In calculus, the chain rule is a formula for the derivative of the composite of two functions. ...

frac{d^k}{dt^k}f(u(t)) = frac{d^k}{dt^k}f(a+t(x-a))=sum_{|alpha|=k}left(begin{matrix}k alphaend{matrix}right)(D^alpha f)(a+t(x-a))cdot (x-a)^alpha

where left(begin{matrix}k alphaend{matrix}right) is the multinomial coefficient for the multi-index α. Since frac{1}{k!}left(begin{matrix}k alphaend{matrix}right)=frac{1}{alpha!}, we get In mathematics, the multinomial formula is an expression of a power of a sum in terms of powers of the addends. ... The notion of multi-indices simplifies formula used in the calculus of several variables, partial differential equations and the theory of distributions by generalising the concept of an integer index to an array of indices. ...

f(x)= f(a)+sum_{|alpha|=1}^nfrac{1}{alpha!} (D^alpha f) (a)(x-a)^alpha+sum_{|alpha|=n+1}frac{n+1}{alpha!} (x-a)^alpha int_0^1 (1-t)^n (D^alpha f)(a+t(x-a))dt.

The remainder term is given by

sum_{|alpha|=n+1}frac{n+1}{alpha!} (x-a)^alpha int_0^1 (1-t)^n (D^alpha f)(a+t(x-a))dt.

The terms of this summation are explicit forms for the Rα in the statement of the theorem. These are easily seen to satisfy the required estimate.


See also

Series expansion redirects here. ... A Laurent series is defined with respect to a particular point c and a path of integration γ. The path of integration must lie in an annulus (shown here in red) inside of which f(z) is holomorphic. ...

Notes

  1. ^ Klein (1998) 20.3; Apostol (1967) 7.7.
  2. ^ Apostol (1967) 7.7.
  3. ^ Apostol (1967) 7.5.
  4. ^ Note that this proof requires f(n) to be absolutely continuous on [a,x] so that the fundamental theorem of calculus holds. Except at the end when the mean value theorem is invoked, differentiability of f(n) need not be assumed since absolute continuity implies differentiability almost everywhere as well as the validity of the fundamental theorem of calculus, provided the integrals involved are understood as Lebesgue integrals. Consequently, the integral form of the remainder holds with this particular weakening of the assumptions on f.

Absolute continuity of real functions In mathematics, a real_valued function f of a real variable is absolutely continuous if for every positive number ε, no matter how small, there is a positive number δ small enough so that whenever a sequence of pairwise disjoint intervals [xk, yk], k = 1, ..., n... The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ... In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ...

References

  • Apostol, Tom (1967). Calculus. Jon Wiley & Sons, Inc.. ISBN 0-471-00005-1. 
  • Klein, Morris (1998). Calculus: An Intuitive and Physical Approach. Dover. ISBN 0-486-40453-6. 

External links

  • Taylor Series Approximation to Cosine at cut-the-knot
  • Trigonometric Taylor Expansion interactive demonstrative applet
cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. ...

  Results from FactBites:
 
Taylor's theorem: Definition and Links by Encyclopedian.com (270 words)
In calculus, Taylor's theorem, named after the mathematician Brook Taylor, who stated it in 1712, allows the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point.
For some functions f(x), one can show that the remainder term R approaches zero as n approaches ∞; those functions can be expressed as a Taylor series in a neighborhood of the point a and are called analytic.
Taylor's theorem (with the integral formulation of the remainder term) is also valid if the function f has complex values or vector values.
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