Taylor’s theorem

This argument tacitly assumes that J"M h t) kdt < oo for all positive mtegers A more careful argument, based on Taylor s theorem with the remainder, shows that it is sufficient that this condition be fulfilled for k a= 1,2. 

The conditions which must be satisfied at the plait point may be deduced as follows Expand by Taylor s theorem the expressions on the right of (9) and (10), omitting terms of higher orders than the second ... 

From (9.27), we see that this approach will work nicely if the variance is always small Taylor s theorem with remainder tells us that the error of the first-derivative - mean-field - contribution is proportional to the second derivative evaluated at an intermediate A. That second derivative can be identified with the variance as in (9.27). If that variance is never large, then this approach should be particularly effective. For further discussion, see Chap. 4 on thermodynamic integration, and Chap. 6 on error analysis in free energy calculations. 

Similarly, many different types of functions can be used. Arden discusses, for example, the use of Chebyshev polynomials, which are based on trigonometric functions (sines and cosines). But these polynomials have a major limitation they require the data to be collected at uniform -intervals throughout the range of X, and real data will seldom meet that criterion. Therefore, since they are also by far the simplest to deal with, the most widely used approximating functions are simple polynomials they are also convenient in that they are the direct result of applying Taylor s theorem, since Taylor s theorem produces a description of a polynomial that estimates the function being reproduced. Also, as we shall see, they lead to a procedure that can be applied to data having any distribution of the X-values. 

The corresponding expression for X. = u is obtained by putting ft = A + e, where e is small, using Taylor s theorem and then letting e tend to zero. We find that... 

Taylor s theorem with remainder, taken to the 1st derivative, is written ... 

Taylor s theorem permits the expansion of certain functions, often in the form of a polynomial. Only the terms which contribute in a significant way to the response are utilized, in this way facilitating the mathematical... 

The topological analysis of p(r, X) then proceeds through the search for and identification of its critical points. In the neighbourhood of a critical point, the field p(r, X) is expanded by Taylor s theorem, the first non-trivial terms being those quadratic in the variables r. The collection of the nine second derivatives of p(r, X) constitute the so-called Hessian matrix A of p(r, X) at the critical point. 

We can apply Taylor s theorem to the left-hand side we also rewrite the right-hand side as... 

Taylor s theorem, published in 1715, and its accompanying series, which was not developed until more than a century later, has a variety of uses in the numerical analysis of functions, including ... 

Using either elementary algebra or Taylor s theorem we can expand this about any point P, giving... 

There are several methods for the development of functions in series, depending on algebraic, trigonometrical, or other processes. The one of greatest utility is known as Taylor s theorem. Mac-laurin s1 theorem is but a special case of Taylor s. We shall work from the special to the general. 

The came is here a historical misnomer. Taylor published his series in 1715. In 1717, Stirling showed that the series under consideration was a special case of Taylor s. Twenty-five years after this Maclaurin independently published Stirling s series. But then both Maclaurin and Stirling, adds De Morgan, would have been astonished to know that a particular case of Taylor s theorem would be called by either of their names... 

The development by Maclaurin s series cannot be used if the function or any of its derivatives becomes infinite or discontinuous when x is equated to zero. For example, the first differential coefficient of f(x) = >Jx, is x which is infinite for x = 0, in other words, the series is no longer convergent. The same thing will be found with the functions log a , cot a , 1/x, a1,x and sec lx. Some of these functions may, however, be developed as a fractional or some other simple function of x, or we may use Taylor s theorem. 

Taylor s theorem determines the law for the expansion of a function of the sum, or difference of two variables into a series of ascending powers of one of the variables. Now let... 

While Maclaurin s theorem evaluates the series upon the assumption that the variable becomes zero, Taylor s theorem deduces a value for the series when x = a. Let z = a, then y = 0, and we get... 

Taylor s theorem is used in tabulating the values of a function for different values of the variable. Suppose we want the value of y = a>(24 - x2) for values of x ranging from 2 7 to 3 3. First draw up a set of values of the successive differential coefficients of y. 

Let the abscissa of each curve at any given point, be increased by a small amount h, then, by Taylor s theorem,... 

Taylor s theorem may be extended so as to include the expansion of functions of two or more independent variables. Let... 

It is required to find particular values of x in order that y may be a maximum or a minimum. If x changes by a small amount h, Taylor s theorem tells us that... 

These indeterminate functions may often be evalued by algebraic or trigonometrical methods, but not always. Taylor s theorem furnishes a convenient means of dealing with many of these functions. The most important case for discussion is since this form most frequently occurs and most of the other forms can be referred to it by some special artifice. 

If the reader is able to develop a function in terms of Taylor s series, this method of integration will require but few words of explanation. One illustration will suffice. By division, or by Taylor s theorem,... 

The expression, e Sy

symbolic form of Taylor s theorem. Having had considerable practice in the use of the symbol of... 

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