Question

X-tonx XGL6, 4.50 Selve the equation with bisection method work with 3 in 3 integers oftor commal &=103 12 tcos(ex-2)-ex=0 X€ [05, in solve this equation with regula false method. Work with 3 integers after comma. And take nodian for cos (-) function

Answer 1

Solution: Here tan(x) - x = 0 Solution (1) by Bisection Method Let fx) =tan(x) - X 1st iteration : Here 4) = -2.842 <0 and 74.5) = 0.137 > 0 :: Now, Root lies between 4 and 4.5 4 + 4.5 Xo = = 4.25 f(x) = R4.25) = tan(4.25) - 4.25 = -2.244 <0 2nd iteration : Here 4.25) = -2.244 <0 and 14.5) = 0.137> 0 : Now. Root lies between 4.25 and 4.5 4.25 +4.5 2 = 4.375 (x1) = 24.375) =tan(4.375) - 4.375 = - 1.524 < 0 3d iteration Here 84.375) = -1.524 <0 and 114.5) = 0.137>0 : Now, Root lies between 4.375 and 4.5 :: Now, Root lies between 4.375 and 4.5 4.375 -4.5 X, = 4.438 (x2) = R4,438) = tan(4.438) - 4.438 = -0.892 < 0 4 iteration : Here 14.438) = -0.892 <0 and 14.5) = 0.137 > 0 : Now, Root lies between 4.438 and 4.5 4.438 - 4.5 X; = 4.469 (x3) = 14.469) = tan(4.469) - 4.469 = -0.446 < 0 5 iteration : Here 84.469) = -0.446 <0 and 74.5) = 0.137>0 Now, Root lies between 4.469 and 4.5 4.469 + 4.5 X4 = 4.484 (x4) = 24.484) = tan(4.484) - 4.484 = -0.175 <0 6th iteration : Here /(4.484) = -0.175 <0 and 14.5) = 0.137>0 :: Now, Root lies between 4.484 and 4.5 : Now, Root lies between 4.484 and 4.5 4.484 - 4.5 X3 = 4.492 M(xs) – 14.492) = tan(4.492) - 4.492 = -0.025 <0 7th iteration : Here 14.492) = -0.025 <0 and 74.5) = 0.137 > 0 : Now, Root lies between 4.492 and 4.5 4.492 -4.5 x= = 4.496 f(x) = 14.496) = tan(4.496) - 4.496 = 0.055 > 0 8th iteration : Here f(4.492) = -0.025 <0 and 14.496) = 0.055 > 0 : Now, Root lies between 4.492 and 4.496 4.492-4.496 X- = 4.494 2 Ax-) = 14.494) = tan(4.494) - 4.494 = 0.015 > 0 9 iteration : Here 114.492) = -0.025 <0 and 14.494) = 0.015 > 0 . Now, Root lies between 4.492 and 4.494 4.492 + 4.494 x's = 4.493 4.492 - 4.494 X's = 4.493 F(x) = R4.493) = tan(4.493) - 4.493 = -0.005 < 0 10+ iteration : Here 14.493) = -0.005 <0 and 4.494) = 0.015 > 0 .: Now. Root lies between 4.493 and 4.494 4.493 -4.494 = 4.494 X9 f(x) = 14.4 =tan(4.494) - 4.494 = 0.005 > 0 11th iteration : Here 4.493) = -0.005 <0 and 114.494) = 0.005 > 0 : Now, Root lies between 4.493 and 4.494 4.493 +4.494 110 = = 4.493 (x10) = 14.493) – tan(4.493) - 4.493 = 0 <0 Approximate root of the equation tan(x) - x = 0 using Bisection mehtod is 4.493 a + b PL a fia) b f(b) fic) Update 1 4 4.5 Q=C 0.137 0.137 2 4.5 a=c 3 0.137 a=c 4.5 4.5 4 auc 5 4.5 4.25 4.375 4.438 4.469 4.484 4.492 4.496 4.494 a=C 4.25 4.375 4.438 4.469 4.484 4.492 4.492 4.492 4.493 4.493 -2.842 -2.244 -1.524 -0.892 -0.446 -0.175 -0.025 -0.025 -0.025 -0.005 -0.005 6 -2.244 -1.524 -0.892 -0.446 -0.175 -0.025 0.055 0.015 -0.005 0.005 4.5 4.5 0.137 0.137 0.137 0.137 0.055 0.015 0.015 0.005 a=c b= 7 8 b = c 9 4.496 4.494 4.494 4.493 a=c 10 b=0 4.494 4.493 11 4.494 0 a=c Here cos(e*- 2) -6*+2=0 Solution (2) by Regula Falsi or False Position Method Let (x) = cos(o* - 2)-84+2 1st iteration : Here (0.5) = 1.29 > 0 and 1.5) = -3.272 <0 • Now, Root lies between xo = 0.5 and x1 = 1.5 X1 - Xo x3 = 40-420) 1x:)-f(x0) x2 = 0.5 - 1.29 1.5 -0.5 -3.272 - 1.29 X = 0.783 M(x2) = R0.783) = cos(0.188)-20.783 – 2 = 0.795 > 0 2nd iteration : Here f(0.783) = 0.795 > 0 and 11.5) = -3.2720 · Now, Root lies between Xo = 0.783 and x1 = 1.5 X; -X0 X; = x -Axo) Axi)-Axo) x3 = 0.783 -0.795 1.5 -0.783 -3.272 -0.795 x-xo x3 = X0-4X) x3 = 0.783 -0.795 1.5 -0.78 -3.272 -0.795 X; = 0.923 F(x3) = 80.923) = cos(0.517) - 0.923 + 2 = 0.353 > 0 3d iteration Here (0.923) = 0.353 > 0 and 1.5) = -3.272 <0 :: Now, Root lies between xo = 0.923 and x1 = 1.5 x-xo X = X x = 0.923 -0.353 1.5 -0.923 -3.272 -0.353 x = 0.979 (x4) = R0.979) = cos(0.662) - €0.979 – 2 = 0.127 > 0 4 iteration Here 10.979) = 0.127>0 and 1.5) = -3.272 <0 :: Now, Root lies between xo = 0.979 and xy = 1.5 4 iteration : Here (0.979) = 0.127>0 and 1.5) = -3.272 <0 • Now, Root lies between xo = 0.979 and x1 = 1.5 X-Xo Axz)--(x) x3 = 0.979-0.127 1.5 -0.979 -3.272 -0.127 xs = 0.999 M(xs) = 10.999) = cos(0.716)-20.999 – 2 = 0.041 > 0 5th iteration : Here 0.999) = 0.041 > 0 and 11.5) = -3.272 <0 • Now, Root lies between Xo = 0.999 and x1 = 1.5 x-xo Axı)--(0) 1.5 -0.999 x = 0.999 - 0.041 - -3.272 -0.041 *6 = 1.005 f(x) = R1.005) = cos(0.732) - 22.05 – 2 = 0.013 > 0 Axs) – 11.005) = cos(0.732) - 21.005 – 2 = 0.013 > 0 6 iteration : Here (1.005) = 0.013 > 0 and (1.5) = -3.272 <0 :: Now, Root lies between xo = 1.005 and x1 = 1.5 *1-* x2 = 10 -Axo) 1x:)--(ro) 1.5 - 1.005 X = 1.005 -0.013. -3.272 -0.013 x = 1.007 A(x) – R1.007) = cos(0.737) - 22.007 – 2 = 0.004 > 0 7th iteration : Here / 1.007) = 0.004 > 0 and f1.5) = -3.272 <0 • Now, Root lies between xo - 1.007 and x1 = 1.5 x-xo X = Xo (xo): Axz)-Axo) x3 = 1.007 -0.004 1.5 - 1.007 -3.272 -0.004 x; = 1.007 H(xs) = R1.007) = cos(0.737) - 22.007 – 2 = 0.001 > 0 8th iteration : Here (1.007) = 0.001 > 0 and 61.5) = -3.2720 • Now, Root lies between Xo = 1.007 and xy = 1.5 x1 - x0 xp = -(x) Axz)-1(x0) x = 1.007 -0.001 1.5 - 1.007 -3.272 -0.001 x = 1.008 F(x) = R1.008) = cos(0.74) - 21.003 + 2 = 0 > 0 Approximate root of the equation cos(8+-2) - ** + 2 = 0 using False Position mehtod is 1.008 11 XO (*) X1 X2 Update 1 0.5 1.29 1.5 -3.272 0.783 0.795 2 0.783 0.795 1.5 -3.272 0.923 0.353 * FX3 Xa xa * TY: 3 0.923 0.353 1.5 -3.272 0.979 0.127 4 0.979 0.127 1.5 -3.272 0.999 0.041 * * * 5 0.999 0.041 1.5 -3.272 1.005 0.013 6 1.005 0.013 1.5 0.004 -3.272 -3.272 1.007 1.007 7 1.007 0.004 1.5 0.001 * FX3 X0 = x2 * FX3 8 1.007 0.001 1.5 -3.272 1.008 0