Problem

Use Newton's Method to approximate to 6 decimal places the solution to the equation

Solution

Let

As with the previous problem, there is more than one solution to the equation, since tan(x/4) – 1 has infinitely many roots. Which one we find will depend on the initial guess x₀.

We need the derivative t'(x).

Choose an initial value.

Then for each x_n,

Here are the first 10 iterations.

From the table, the 10th iteration of Newton's Method gives

as the solution; we may recognize that this is 5p (the exact solution) given to 3 decimal places of accuracy. We can check this answer by remembering what we know about the tangent function. The equation

we can solve by inspection; tan(p/4) = 1, so x = p. tan(x/4) is periodic, with period 4p, so whenever x is a solution, so is x + 4p or x – 4p. Therefore, the family of solutions to the equation can be expressed as

where n is an integer.