Taylor's theorem
The Taylor formula (also Taylor's Theorem ) is a result from the mathematical subfield of Analysis. It is named after the mathematician Brook Taylor. You can use this formula to approximate functions in the neighborhood of a point by polynomials, the so-called Taylor polynomials. This is known as the Taylor approximation. The Taylor formula has become because of their relatively simple applicability and usefulness of a tool in many engineering and natural sciences. Thus, a more complicated analytical expression of a Taylor polynomial of small degree (often fine ) can be approximated, for example, in physics and in geodetic adjustment of networks: For example, the small angle approximation is often used in the sine wave is canceled by the first term of the Taylor series function.
Closely related to the Taylor formula is known as the Taylor series ( Taylor series expansion ).
- 3.1 Taylor polynomial
- 3.2 Integral remainder
- 3.3 Theorem ( Taylor formula with integral remainder term )
- 5.1 Schlomilch - residual limb and its derivation
- 5.2 Special cases of Schlomilch - residual limb 5.2.1 Peano remainder term
- 8.1 Multidimensional Taylor polynomial 8.1.1 Schmiegquadrik
Motivation
Approximation by tangent
An approximation of a differentiable function at a point by a straight line, ie by a 1st order polynomial is given by the tangent to the equation
They can be characterized by that at the point the function values and the values of the first derivative ( = slope ) of and match.
If one defines the rest, applies. The function is approximated in the vicinity of the location in the sense that it applies to the rest
An interesting property is the following: Since, show the invoice
Combined with that of the rest of the derivation in is zero.
Approximation by osculating parabola
One can suspect that one obtains for twice differentiable even better approximation, if one used a quadratic polynomial, of which one required in addition that still applies. The approach leads to through calculation of discharges and, therefore.
This approximation function is also called the osculating parabola.
We define now to the appropriate rest so again. Then we obtain that the osculating parabola at the given function in fact better approximated as now ( with the rule of L' Hospital):
Applies.
N approximation by polynomials of degree
This approach can be easily generalized to polynomials of degree now: Here is to apply
There arises.
With the rule of L'Hospital, we also find:
It therefore follows by induction over that for the following applies:
Qualitative Taylor formula
Is - times differentiable, it follows immediately from the above consideration that
Where stands for the Landau notation. This formula is called "qualitative Taylor formula ".
Thus, at the closer the better approximated ( the so-called Taylor polynomial, see below) at the location of the function.
Definitions and sentence
In the following, the Taylor formula with integral remainder term is presented. The Taylor formula also exists in variants with different residual limb; However, these formulas follow from the Taylor formula with integral remainder term. You are in the section below remainder formulas.
Be an interval and a - times continuously differentiable function. In the following formulas stand for the first, second, ..., nth derivative of the function.
Taylor polynomial
The Taylor polynomial -th in the development station is defined by:
Integral remainder term
The integral -th residual limb is defined by:
Rate ( Taylor formula with integral remainder term )
For all and from the following applies:
Evidence
The proof of the Taylor formula with integral remainder term is carried out by induction on.
The base case corresponds exactly to the Fundamental Theorem of Calculus applied to the once continuously differentiable function:
The induction step ( is to show that the formula always also applies if she applies for a ) by partial integration. For times continuously differentiable follows:
And thus
Remainder formulas
There are in addition to the integral formula, other representations of the residual limb.
Schlomilch - residual limb and its derivation
By the mean value theorem of integral calculus arises with for each natural number, that there is a between and are such that:
This follows the Schlömilchsche residual limb shape:
For between and.
Special cases of the Schlomilch - residual limb
A particular case, namely with, the shape according to the Cauchy:
For between and.
In the special case we obtain the Lagrange remainder term:
For between and. In this illustration, the -th derivative of need not be continuous.
Peano remainder term
Using the Taylor formula with Lagrange remainder term is obtained for times continuously differentiable also:
That's why you can as residual limb also
Use, only times must be continuously differentiable here. This remainder is called Peano remainder term.
Other views
Is one, that is, the Lagrangian representation takes the form of
The Schlömilchsche
And the Cauchy
For one between 0 and 1
Remainder estimate
If the interval ( the domain of ), you can take the remainder of Lagrange (see the section remainder formulas ) for all and derive a between and (and hence ) the following estimate:
Applies to all, so therefore applies to the remainder estimate
Approximation formulas for sine and cosine
An application of the Taylor formula are approximate formulas presented here using the example of sine and cosine ( where the argument is in radians ).
For true, so is the fourth Taylor polynomial of the sine function in the development of site 0
From the result for the remainder of Lagrange with between 0 and. Because followed by the remainder term estimate.
Lies between and, then, the relative deviation of less than about 0.5%.
In fact, already sufficient for the approximation of the sine on this accuracy even the Taylor polynomial 3rd order, since, and therefore. This also provides the following additional assessment for third and fourth Taylor polynomial, which is more accurate for very large x:
The following figure shows the graphs of some Taylor polynomials of the sine wave at point 0 for development. The graph to a part of the Taylor series, which corresponds to the sine function.
X is between and, then, the relative deviation of less than 0.05%.
Also for cotangent and tangent can use these formulas, because it is
With a relative deviation of less than 0.5 % for, and with the same relative deviation (this is not a Taylor polynomial of the tangent).
Do you need an even higher accuracy for its approximation formulas, then you can rely on higher Taylor polynomials approximating the functions even better.
Taylor formula in Multidimensional
Now let hereinafter a - times continuously differentiable function and. Further, let, where.
Be also as in the multi- index notation. In the following section, the multi- index notation is used so that you can see immediately that the multi-dimensional case, for actually the same formulas obtained as the one-dimensional case.
Multidimensional Taylor polynomial
With the multidimensional chain rule and induction we obtain that
Where the multinomial coefficient is, see Multinomialtheorem.
If, at point 1 is by a Taylor polynomial with development point 0, we obtain by this formula, the definition of the multidimensional Taylorpolynoms of the development site:
Here one has used that.
Schmiegquadrik
The second Taylor polynomial of a scalar-valued function in more than one variable can be written more compact to second order as:
Here, the gradient and the Hessian matrix at the location of each.
The second Taylor polynomial is also called Schmiegquadrik.
Multidimensional integral remainder term
Similarly, we define the multi-dimensional residual limb using the multi- index notation:
Multidimensional Taylor formula
From the one-dimensional Taylor 's formula it follows that
According to the above definition of we obtain therefore:
Multidimensional remainder formulas
One can generalize the one-dimensional non -integral remainder term formulas using the formula for the multidimensional case.
The Schlomilch - residual limb so to
The Lagrange remainder term to
And the Cauchy's residual limb to
For each.
Qualitative Taylor formula
After the multi-dimensional Taylor equation is derived with the Lagrange remainder term:
Because we also get:
The last part goes to zero, since the partial derivatives, by assumption, all of degree steadily and is located between and and thus even after converges if.
We obtain the following estimate, which is called " (multidimensional ) qualitative Taylor formula ":
For, where is the Landau notation.
Example
It is the function
Be developed to the point.
The partial derivatives of the function are as follows:
It follows with the multidimensional Taylor formula:
If one uses the alternative representation with the help of the Jacobian and the Hessian matrix, we obtain:
With the Jacobian and the Hessian matrix.