(Calculus 1 class) Use Newton's method to approximate the indicated root of the following equation: the root of tanx=x in the interval (pi/2, 3pi/2)

Answer 1

From tan x = x, write f(x) = tan x − x and differentiate tan x − x, rewritten as sin x/cos x − x.

Then f'(x) equals [cos x(cos x) − sin x(-sin x)]/cos²x − 1 or 1/cos²x − 1 or sec²x − 1.

With the aid of computer or calculator graphing, note that the graph of tan x − x crosses the

x-axis (or "zeroes") only once in the interval (π/2,3π/2) at a point close to (4.5,0).

Next, construct the expression x − [tan x − x]/[sec²x − 1]. Using the radian as the unit of angle

measurement and starting with x = 4.5, compute the first value of x − [tan x − x]/[sec²x − 1] which

will give 4.493613903. Feed 4.493613903 back into x − [tan x − x]/[sec²x − 1] to obtain 4.493409655

and then feed 4.493409655 into x − [tan x − x]/[sec²x − 1] to obtain 4.493409458.

Finally, feed 4.493409458 into x − [tan x − x]/[sec²x − 1] and note that the result returned is

also 4.493409458. This indicates that tan x = x is solved to high accuracy by x = 4.493409458.

For x = 4.493409458, my calculator gives tan x equal to 4.493409460 as opposed to 4.493409458,

which amounts to a difference of two billionths.