Question

Use Newton's method to find all roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. (Do this on paper. Your instructor may ask you to turn in this graph.) 4e-** sin(x) = x2 - x + 1 0.219164 X (smaller value) 1.084225 X (larger value)

Answer 1

Given f(x) = 4€th sin(n) = x²ut1 soli- -ut fom aué e sincal-n²tu-1201 & cueurs Sriw -**7*-1) flom a d an и d 4. en 2 de 2011 sincast 4 sinn dé" da an t' (nl= Чес- -22 tl Cash it usina (En. -2x) teno = 40" Losn - fx sinn en 2011 t'cm -2x-&xsinu ē"274& Cosm ti tlema -21-8e usinn tué nh cosnti ry ut N=0 ast-iliralian HCMO) = f(0) = 4 e sinco) -060 -1 +col= -1 A se to sinco) +4 e esc \fCm) 'colti f'(x) = f'(o) a 1) = -2100-8e co flcosa 4+1=5 Hno) I'cnol il CO2 upt na 10- H, = 0.2 2nd, loration M, f(",) -0.07648242 : +()(0-2) f'(M, 2 = f'(0.2) -t'60.2) : 4.06114744 0.07648242 1₂ = 0.2 t'cn, 4.06114344 0:21483271 grikeralim f(Vx) = f(0.21883273 ) = -0.0012938 Hw+'(0.21 34 32 +5)= 3.91208939 My = 1₂. (10) tina)

-0.219.16357 3.92208139 13 = 0.21883273– -0.00129758 "3=0.21916357 No al - Approximate root of the -22 ut equation 4e "sin(n)=x² tn-lois 0 2/916352 -ML e sincnfn-1-o 3th itration ful-n2tue f(x)=f(11= -1 44 é since141-1=0.2382395 t'cnol = t'lll = -2411-set (1) sinun thé Cos Cilt - --2.681414561 ni a no 0.2382395 1- (Phot (w), -268141456 a, a 1.08884844 Put iteralim H142 - 711.088848447 = -0.01270357 t'(0,2of'(1108884844) = -2.96970141 -0.01370357 (H) 2 2 1.0888 4844 - 'x -2.9697014 film) Na = 1.08423398 3rd itralin final · H11.08423398):0.0000277 f'(12) EH11.08423398) = -2.9576477 fln) z U₂ - - 1.08427998 -0.0000272 Eu - 209576471 y = 1.05422462 -30% he sinh anh th=720 equation . 1.08422462 Approximate root of the using Newton's meubod The two roots are: 1,20-21916357 and 12 =1.08422462 is