Question

Consider the function xtan x -1 defined over all x. Sketch the function to get an idea of the roots 1 find the first couple of roots using bisection to a precision of machine epsilon 2 after straddling a root, find its value using the Newton-Raphson method. 3 after straddling a root, find its value using the secant method 4 after straddling a root, find its value using the false position method. Determine the order of the methods and comment upon the usefulness of each.

Answer 1

=x Tan [x]-1 ] f [ x I [4]:= Out4 -1-xTan[x] Inlej Plot[f[x], (x, 10, 10)] 20 10 10 Outl -10 10 20

using Bisetien Methed f(0)= f(1 ) = o.tan 0-1 = 0-1ニー1Zo | | tan 1-1-0. 55 7408 o-5。-75 root Lies betiuun 041 aut bi 015 2. 2 . oot Lies beatueen 5 a: 0.5 b 0.5+1-1.5 ろこ 2 2.

a nd 0.075 A4二0:75 by-o-875 , 24 :75t0.075 o 812 5 2 fruo. 14222140 oot Lie betuuuen ο.01 25 101875, flZg)--0.059 1106 . 0:557400 f(H): ˊ-' Similia ret-->o 0 4 utl can d th secand host usiv methsal de a pecisiem

using Neunten Pabhden Method -- Let initial leet fand uing 0isectien utueol) uaal) Methed pin) f(to)--0.0591106 f'(0.84125) f[To ) = 3.01202 0.04125,-ー (-0.0591106) 一ㄨㄧ一 3. 0120 2

o-860074 χ- 0001722 59 f '1%) = f (0-860074) 3.19ol f,(%) 0.. 0 60074-0.00172253 3.190 X2二0.060334 f(x2) o, o o o o o l30898 こ χ: 0.060334 is an sing false positien methad f(: 0 557408 >o

bi-I Ox o 5 74o-1x(-1) o. 557408 + 1 O 6420925 f(X1)ニー0.51004310 o 6 420921 a2fb2)-b(a2) 64 2092 X0.s5 7408 5574 08+ 0 519843 0.135804 O tn thuis wa fid teet using bositi em methad

Positien bisecti em muthed aru lesed braekel metod methedi faster than bisectem hethad Neuton Rothsen tethod w het--dexd-_-bracket mettaed buut