Question

Using Newton - Rapshion method find the
soat of the equation x tanx + 10
correct to three decimal places.​

Answer 1

Draw the curves y = tan x and y = 4x. The roots of our

equation are the x-coordinates of the places where these two curves

meet.

A glance at the picture shows that (for x ≥ 0) the curves meet at

x = 0, then at a point with x just shy of π/2, and then again at a

point with x just shy of 3π/2 (the pattern continues).

We first find the root that is near π/2. Let f(x) = tan x − 4x. The

f0

(x) = sec2 x − 4, and the Newton Method recurrence is

xn+1 = xn − tan xn − 4xn

sec2 xn − 4 .

Some simplification is possible. For example, we can use the identity

sec2 x = 1 + tan2 x to rewrite the recurrence as

xn+1 = xn − tan xn − 4xn

tan2 xn − 3 .

This trick cuts down on the computational work. This was a particu-

larly important consideration in the old days when computations were

done by hand, with the aid of tables and slide rules.

For the first root, a bit of fooling around suggests taking x0 = 1.4.

Then x1 = 1.393536477, x2 = 1.393249609, and x3 = 1.393249075.

This suggests that to 4 decimal places the root is 1.3932. We can

verify this by the sign change criterion in the usual way.

For the second root, after some work we can for example arrive at the

initial estimate x0 = 4.66. The computation is quite sensitive to the

right choice of initial value. And then we get x1 = 4.658806388 and

x2 = 4.658778278. To 4 decimal places the root is 4.6588. We can

verify that we are close enough by the sign change criterio