Mathematica has a command that generates the Taylor polynomial about x = a of any desired degree n for sufficiently differentiable functions f. To see how closely a Taylor polynomial such as (7) actually approximates its corresponding function f, plot the graphs of f and its for x near a.
From (6) then, the 8th-degree Taylor polynomial of f about x = 0 is The first step is to compute the first eight derivatives of the cosine function f, and evaluate them and f at x = 0:į ''( x) = - cos x -> f ''(0) = - 1įrom these calculations, it is clear that the next four derivatives of f cycle through the same sequence as the first four, so that in particular at 0 they have the respective values 0, -1, 0, and 1. Find ( x) for the cosine function, if the base point is 0 and n = 8. To illustrate, consider the cosine function about the origin x = 0.Įxample. This intuitive idea is underscored by the graphical behavior of f and the approximating Taylor polynomials.
The nth-degree Taylor polynomial of f about x = a should be a very close approximation to f, because it has the same value as f at x = a, as well as the same slope, concavity, rate of change of the concavity, etc. The Maclaurin polynomial of f is the Taylor polynomial of f about x = 0 (so involves powers of x rather than x - a). The n th - degree Taylor polynomial of f (expanded) about x = a is Suppose that a is in the domain of f and its first n derivatives. Ī similar discussion in case f is differentiable n times at x = a leads to the following formula for the unique nth-degree polynomial ( x) that agrees with f at x = a and whose first n derivatives have the same values at x = a as do the first n derivatives of f.ĭefinition. Together, (3), (4) and (5) determine a unique formula for ( x), namely It follows from (2) that 2 = f ''( a), that is, How? From the condition that ( a) = f( a), it follows that This is enough to determine a unique polynomial ( x) = + (x - a) +. To produce a polynomial ( x) of degree 2 that approximates f near x = a more closely than does T( x) = ( x), require not only that ( a) = f( a) and '( a) = f '( a) but also that have the same concavity as f at x = a: To generate, first impose additional conditions like (1). It takes its name from the English mathematician Brook Taylor (1685-1731). The nth-degree Taylor polynomial approximation generally is a better approximating polynomial. If f has derivatives of order higher than 1 at x = a then the answer is yes. Is it possible to approximate f more closely by a polynomial of degree > 1?.T not only agrees with f at x = a, but also has the same slope at that point:įor any value of x, including of course x = a. The simplest approximation to a (perhaps complicated) differentiable function f is the tangent approximation, which approximates f( x) by the first-degree (linear) polynomial Keep in mind that Mathematica isn't available for quizzes or exams, so it is important to become adept at generating the polynomials for simple functions f by hand!ġ. Once you have mastered that process, you can use this notebook to check such work or to generate Taylor polynomials for more complicated functions. For many important functions, it is actually feasible to calculate Taylor polynomials by hand. Originally, Taylor polynomials were the means for generating tables of values of such transcendental functions as the sine, cosine and natural logarithm. Graphing the Taylor polynomials illustrates how close to f(x) the values of actually are near x = a. Mathematica can generate Taylor polynomials to approximate a function f that has derivatives of every order at a point x = a. Hurley, University of Connecticut, Department of Mathematics, Unit 3009, Storrs CT 06269-3009. Taylor Polynomials and Series in MathematicaĬopyright © 1995, 1997, 2001, 2002 by James F. TayPolyn.nb TayPolyn: A Mathematica Notebook
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