Question

Find the approximate values of : tan^–1 (1.001)

Answer 1

Let f(x) = tan^–1x

∴ f'(x) = `d/dx(tan^-1x) = (1)/(1 + x^2)`

Take a = 1 and h = 0.001

Then f(a) = f(1) = tan^–11 = `pi/(4)`

and f'(a) = f'(1) = `(1)/(1 + 1^2) = (1)/(2)`
The formula for approximation is
f(a + h) ≑ f(a) + h.f'(a)
∴ tan^–11 (1.001)
= f(1.001)
= f(1 + 0.001)
= f(1) + (0.001).f'(1)

≑ `pi/(4) + (0.001) xx (1)/(2)`

= `pi/(4) + 0.0005`

∴ tan^–1 (1.001) ≑ `pi/(4) + 0.0005`.
Remark: the answer can also be given as :
tan^–1 (1.001) ≑ f(1) + (0.001).f'(1)

≑ `pi/(4) + (0.001) xx (1)/(2)`

≑ `(3.1416)/(4) + 0.0005`

≑ 0.7854 + 0.0005
= 0.7859.