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Description
This programme demonstrates that for the function whose derivatives are known at x=0 it is possible to construct a series of approximating polynomials called Taylor polynomials. Under certain condi...tions these polynomials provide approximations to any desired degree of accuracy.
Module code and title: MS283, An introduction to calculus MS283; 11 06-11-1978 1978 00:23:53 + Show more... John Jaworski Louis Leithold; Graham Read BBC Open University Sine; Function synthesising machine Graham Read introduces the programme by suggesting that students use the sine formula for working out the value of sine 0.5 and the value of the integral. The method involves taking polynomials as progressive approximations of sine, the Taylor polynomials. Using still and animated graphics, Louis Leithold demonstrates what Taylor approximations of the sine curve look like. He goes as far as fifth degree Taylor polynomials. Continuing to use still and animated graphics, Leithold shows Taylor approximations of the sine curve from the seventh to the seventeenth degree. At this point the Taylor approximation is accurate to within three decimal places. Graham Read, writing on a drawing board, explains how the formula for sin (x) is derived. Graham Read goes on to generalise the technique for deriving the polynomials for sin (x) to any function for which values of the derivative are known at x = O. Graham Read, operating a "function synthesising machine", illustrates the contribution of each term in the polynomials of a function to the shape of the graph. The "machine" generates computer animations. Louis Leithold points to some still graphics while he works out the derivative pattern for tie function (cos x + sin x). Graham Read, still operating his "function synthesising machine", tries out the derivative pattern worked out by Leithold and generates an approximation of the (cos x + sin x) curve up to a sixth degree polynomial. To illustrate the difficulties which may arise when using Taylor polynomials to approximate functions, Leithold considers the function (...). He works out the derivative pattern for this function. Graham Read tries the derivative pattern for this function on his "function synthesising machine". No close approximation is achieved. 6HT/73042 00525_4320 455 no