Approximate the zero(s) of the function. Use Newton’s Method and continue the process until two successive approximations ................

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. Show all work. 

 

 

1. f(x) = x^3 +4

 

 

 

2. f(x) = x^3 + x - 1

Answer 1

Problem 1.

Note that the exact root is: x = - ³√4

 

Machine precision: -1.5874010519681994

 

Using seed x₀ = -2:

 

f(x) = x³ + 4 so f'(x) = 3x².

 

Thus we iterate: x ← x - (x³ + 4) / (3x²) = 2x/3 - 4/(3x²)

 

x₀ = -2

x₁ =  -1.6666666666666665

x₂ = -1.5911111111111111

x₃ = -1.5874096961416333

x₄ = -1.587401052015271 (|This - Previous| < 0.001, so you can stop here)

x₅ = -1.5874010519681994 (Machine precision reached)

 

Problem 2.
Note that the exact root is: x = (³√(108 + 12√93) + ³√(108 - 12√93))/6

Machine precision: 0.68232780382801927

Using seed x₀ = 1:

f(x) = x³ + x - 1 so f'(x) = 3x² + 1.

Thus we iterate: x ← x - (x³ + x - 1) / (3x² + 1)

x₀ = 1
x₁ =  0.75
x₂ = 0.68604651162790697
x₃ = 0.6823395825973142
x₄ = 0.68232780394651271 (|This - Previous| < 0.001, so you can stop here)
x₅ = 0.68232780382801939

x₆ = 0.68232780382801927 (Machine precision reached)