Question

4.8.39 Questione The values of various roots can be approximated using Newton's method. For example, to approximate the value of V10, x V10 and cube both sides of the equation to obtain 22-10, rx-100. Therefore, 10 is a root of pix)x-10, which can be approximated by applying Newton's method. Approximate the following value of by first finding a polynomial with integer colors that has a roof Use an appropriate value of and stop calculating approximations when two c i approximations agree to five digits to the right of the decimal point for rounding TR Type an integer or decimal rounded to five decimal places as needed)

Answer 1

giren that Prky = 2 f182) So plr) = -2 + f(n)= x3-2 Mow using to = 2 for initial value. P'(x) = 3x² 3 diteration! f(22) = 0.00482 f'(2)= 4.76985 x3 = 2 - Flas) Blog: 1.26093-00 4176985 [x2= 1,25992 4th iteration : f(x) = 0 f'(X) = 4.76221 X = X3 - flus) = 1.2.5492_o f'(X) ( 34= 1,259921 што | |2 here fll) = -1 <0 & f(2)= 670 i noot lies between 142 20 = 1+2 = 1.5 20 = 1.5 est iteration f(0) = f(.5) - 1.375 f(x) = 6.75 x= xo - H2) = 1.5 - 1.375 f'(2) 6.75 [24 = 1.2963) and iteration! here x3 = 24 So Approximate noot of the equation x3_r=o using mewton raphson method is 125992 f(x) = 0.17828 f'144) = 5:04115 $2= 24- f(x) f(11) 1.2963-0.14828 5.06115 [x = 1.26093