NEWTON METHOD and HP-48G

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2 NEWTON METHOD an HP-48G DE TING WU DEPART. of MATH. MOREHOUSE COLLEGE I. Introuction Newton metho is an often-use proceure to fin the approximate values of the solutions of an equation. Now, it is covere by most textbooks of Calculus as an application of erivative. Its basic iea is to use the intersection point of the tangent line with X-axis to approximate the solution of the equation. This iea is escribe by following iterative formula: Initial guess: x 0 Iterative formula: f x f x f 0 n=0,,,... When using Newton metho, usually we follow steps: Step : Locate the solution of the equation an make a goo initial guess; Step : Compute the erivative of f(x) an set up the iterative formula; Step : Perform the calculation to fin approximate values of the solution up to the require accuracy. It shoul be emphasize that the Step is ifficult an significant since it is har to etermine where the solutions are an to make a goo guess from an equation an moreover the sequence of approximate values may not converge if the initial guess is sellect blinly. An, sometime the Step is formiable if f(x) is complicate. Moreover, the Step is always a time-consuming job.

3 f(x) is com licate. Moreover, the Ste is always a time consuming job. However, the graphing calculator will help us to overcome the ifficulty which exits in Newton metho. II.Approach of graphic calculator(hp-48g) for Newton metho The approach of graphic calculator for Newton metho is as follows:.use "Plot application" of HP-48G to graph the equation. The graph of the equation provies us with a viewable image so that it is easy to locate solutions of the equation an the "trace" function of HP-48g helps us to get a goo initial guess of the esire solution..use the program "ITF" to set up the iterative formula. This program can be prouce by following keys in HP-48G. Left Shift << >>, Left Shifi Stack NXT DUP, x, /x, /, x, Left Shift SWAP, -, N F, STO. This program will take a function from the first stack an fin its erivative an then set up the iterative formula an store it in the variable NF..Use the program "APV" to compute the approximate values up to esire accuracy. This program can be prouce by following keys in HP-48G. Left Shift << >>, N F, Left Shift ->NUM, Left Shift Stack NXT DUP, x, STO. This program will take a number store in the variable x to compute the successive approximate value an put it the first stack an store in x" for " computing next approximate value. Now, Let s see an example to illustrate this approach. Fin: first 6 approximate values of positive solution of x. sin( x )

4 Consier the function: fx ( ) x. sin( x, ) zeros of f(x) are solutions of equation. From the graph below, it is easy to see that f(x) has zeros: x 0.9 r root( f( x), x) r = x 0 r root( f( x), x) r = x 0.6 r root( f( x), x) r = x,.99.. Graph of f(x) fx ( ) x It is obvious that the approximate values of r are our esire." Then, use the program "ITF" to set up its iterative formula. f( x) x x. sin( x ) f( x). x. cos( x ) 4

5 f x n x n sin f x x n n. x. n cos Finally, take initial guess x an use the program "APV" to fin 6 approximate values of r., The results are : To check the accuracy, efine the error : E n n = = E n = III. Approach of graphic calculator for graing When graing stuents paper about Newton metho, the work loa becomes very heavy because stuents take ifferent initial guess an get ifferent approximate values.now, this approach can save our time an labor.simplely store stuents initial guess in variable x" an press APV key to get approximate values. Then, we can use them to check stuents calculation. For example, if a stuent selects his initial guess t then we get : t t n t n t n. sin t n. t. n cos t n t n =

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