Use Newton's method with the specified initial approximation x1 to find x3, the third approximation to the root of the given equation. (Round your answer to four decimal places.)

Question

Use Newton's method with the specified initial approximation x1 to find x3, the third approximation to the root of the given equation. (Round your answer to four decimal places.) 1 3 x3 + 1 2 x2 + 7 = 0, x1 = −3 x3 =

1

3

x3 + 1

2

x² + 7 = 0,    x₁ = −3

x₃ =

Answer 1

Using the online calculator, here is the output of the Newton's method:

n x f(x)

1 -6.777777777777777777778 34

2 -5.279164410311951295558 -375.8189300411522633745

3 -4.473941374742286165375 -99.94931259190931025231

4 -4.179050348847141635926 -21.45944696971801180027

5 -4.137191513695245021004 -2.38105394292857287859

6 -4.136375958512183621917 -0.04465538662092258231

7 -4.136375651773045738559 -1.678275084794465E-5

8 -4.136375651773002355842 -2.37361666E-12

9 -4.136375651773002355842 0

the approximate solution is:

-4.136375651773002355842