Calculus 1 - Newton's Method

Question

Use Newton’s method to find all roots of the equation (𝑥−2)²=ln𝑥, correct to six decimal places.

 

 

Please show a step-by-step solution to this problem. Thanks.

 

Answer 1

If you are having trouble with Newton's method, I reccomend this site:http://tutorial.math.lamar.edu/Classes/CalcI/NewtonsMethod.aspx.

 

Paul's Notes is great for many other math topics as well!

 

Using the equation for Newton's method:

If X_n+1 = X_n - [f(x_n) / f`(x_n)], what is f(x) ???

 

Let's take your equation and set it equal to zero, so (x-2)² - ln(x) = 0. This equation will be our f(x)

 

So X_n+1 = X_n - [(X_n - 2)² - ln(X_n)] / [f`(X_n)]

 

Now calculate and plug in the derivative of f(x), f`(x)!

 

Once you plug in the derivative and simplify that equation in terms of X_n, let's start finding the roots by setting Xn = X₀ = 1, since ln(0) does not exist. Continue plugging in values for X (e.g. 2, 3, 4, 5) to find the roots!

 

Good luck!