Newton’s Method

Solved Problems

Solution.

The iterative formula for Newton's method is given as

Find out the derivative:

The first iteration is equal to

Next, we perform two more iterations:

After $$ iterations we've got the approximate solution with an accuracy of $$ decimal places.

Answer: $$

Solution.

Consider the function

and apply Newton's method to find zero of the function.

Find the derivative by the product rule:

Calculate the first approximation:

Continue the iterative process until we reach an accuracy of $$ decimal places.

As you can see, we have obtained the required accuracy after only $$ steps.

Answer: $$

Solution.

We choose $$ and compute the first approximation:

Continue the process until we get the result with the required accuracy.

We see that we've already got a stable result to $$ decimal places, so the approximate solution is $$

Solution.

First we rewrite this equation in the form

Suppose that the initial value of the root is $$ Let's get the first approximation using Newton's method:

Here and further we write approximate values with 6 decimal places to track the convergence of the result.

In the next step, we have

The third approximation gives the following value of the root:

Continue computations:

The $$th iteration preserves the first $$ decimal places. This means that the required accuracy of $$ decimal places was reached at the $$rd step, so the answer is

Solution.

We apply Newton's method with the initial guess $$

Consider the function

The derivative is written as

Then the first approximation is given by

Continue the iteration process until we get an accuracy of $$ decimal places.

As you can see, the process converges slowly enough, so it took $$ steps to obtain a stable result to $$ decimal places.

The answer is $$

Solution.

Take the derivative:

Notice that $$ $$ so we choose $$ as the initial guess.

Using the recurrent relation

we compute several successive approximations:

You can see that we've got the solution accurate up to $$ decimal places in the $$th step. Therefore, we can write the answer as $$