Use Newton's Method and continue the process until two successive approximations differ by less than 0.001

Question

Approximate the zero(s) of the function. Use Newton's Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results.

f(x) = x³ + x − 3

Newton's method:	Graphing utility:
x =	x =

Answer 1

You got 21 digits of accuracy after 6 iterations!

Here are the first two iterations:

y = x^3+x-3

y' = 3x^2+1

Xn+1 = Xn - f(xn)/f'(xn)

x0=1

f(1) = -1 and f'(1) = 4

x1 = 1 - -1/4 = 5/4

f(5/4) = 125/64 + 5/4 - 3

= 125/64 + 80/64 - 192/64

= 13/64

f'(5/4) = 3(25/16) + 1

= 75/16 + 16/16

= 91/16

X2 = X1 - F(5/4)/f'(5/4)

= 5/4 - 13/64*16/91

= 5/4 - 13/(4*91)

= (5*91) - 13 / 364

= (455-13)/364

= 442/364

= 221/182 = 1.2142857142857142857142857142857

n x accuracy(x) f(x)

1 1.25 1 -1

2 1.214285714285714285714 1.2 0.203125

3 1.213412175782824889128 1.213 0.00473760932944606414

4 1.21341166276240649498 1.213412 2.7790866663646996E-6

5 1.213411662762229634132 1.213411662762 9.580735334E-13

6 1.213411662762229634132 1.213411662762229634132 1.E-22

Desmos says 1.213