Question

find the real root of the equation 3x+ sinx= e^x correct to 4d placed by Newton raphson method​

Answer 1

Answer:

the root is 0.59375 (approx)

Step-by-step explanation:

let, f(x) as 3x + sinx = eˣ

            => f(x) = 3x + sin x − eˣ

now consider

f(0) = 3(0) + sin 0 - e⁰ = 0+0-1  = -1     (so  f(0) < 0 )

and

f(1) = = 3(1) + sin 1 - e¹ = 3+0.017452-2.71828 = 0.299172  (so  f(0) > 0 )

so the answer lies between 0 and 1

take average of 0 and 1

=> (0+1) / 2 = 0.5

f(0.5) = 3(0.5) + sin 0.5 - $e^{0.5}$  = 1.5+0.00873 - 1.64872 = -0.13999              (so  f(0) < 0 )

=> answer lies between 0.5 and 1

average of 0.5 and 1 = 0.5+1 / 2 = 0.75

so

f(0.75) = 3(0.75) + sin 0.75 - $e^{0.75}$ = 2.25 + 0.01309 - 2.117 = 0.146 (so  f(0) > 0 )

=> answer lies between 0.5 and 0.75

consider average of 0.5 & 0.75 = 0.5+0.75/2 = 0.625

f(0.625) = 3(0.625) + sin 0.625 - $e^{0.625}$ = 1.875 + 0.01091 - 1.86825 =0.01766 ( so f(0.625) > 0 )

=> the answer lies between 0.5 & 0.625

take average of 0.5 and 0.625 = (0.5+0.625)/2 = 0.5625

f(0.5625) = 3(0.5625) + sin 0.5625 - $e^{0.5625}$

                = 1.6875 + 0.00982 - 1.75505

                = -0.05773 (f(0.5625)<0)

=> answer lies between 0.625 and 0.5625

average of 0.625 and 0.5625 = (0.625+0.5625)/2 = 0.59375

=> the root is 0.59375 (approx)