Newtons Method problem

Question

The number 6^(1/5) can be thought of as the solution of the equation x^5−6=0. (by the way the 6^(1/5) is supposed to be in square root form but I cant write it in that form here)

1.To find 6^(1/5) the  iteration function is g(x)= __________

2.With the starting value x0=1, the first few iterates are:

x1=____

x2=____

x3=____

x4=____

x5=____

Answer 1

Newton's Method typically uses the equation:

where you iterate beginning at n = 0 so, the first iteration is x₁ = x₀ - f(x₀)/f '(x₀), then, after x1 is calculated, you use it to calculate x2 in the second iteration using x₂ = x₁ - f(x₁)/f '(x₁) and so forth.

So, you could call the iteration function:

g(x) = x - f(x)/f '(x) or, even more specifically for this problem:

g(x) = x - (x⁵ - 6)/(5x⁴) since f(x) = x⁵ - 6 and f '(x) = 5x⁴ (by using the power rule)

First Iteration:

x₁ = x₀ - f(x₀)/f '(x₀)

x₁ = 1 - f(1)/f '(1) = 2

Second Iteration:

x₂ = x₁ - f(x₁)/f '(x₁)

x₂ = 2 - f(2)/f '(2) = 1.675

Third Iteration:

x₃ = x₂ - f(x₂)/f '(x₂)

x₃ = 1.675 - f(1.675)/f '(1.675) = 1.492448095

Fourth Iteration:

x₄ = x₃ - f(x₃)/f '(x₃)

x₄ = 1.492448095 - f(1.492448095)/f '(1.492448095) = 1.435829756

And so forth