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Find the approximate values of : tan–1(0.999)

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प्रश्न

Find the approximate values of : tan–1(0.999)

बेरीज
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उत्तर

Let f(x) = tan–1x

Then 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–1 (0.999)
= f(0.999)
= f(1 – 0.001)
≑ f(1) – (0.001).f'(1)

≑ `pi/(4) - 0.001 xx (1)/(2)`

= `pi/(4) - 0.0005`

∴ tan–1 (0999) ≑ `pi/(4) - 0.0005`.
Remark: The answer can also be given as :

tan–1 (0.999) ≑ `(3.1416)/(4) - 0.0005`

≑ 0.7854 – 0.0005
= 0.7849.

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पाठ 2: Applications of Derivatives - Exercise 2.2 [पृष्ठ ७५]

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बालभारती Mathematics and Statistics 2 (Arts and Science) [English] Standard 12 Maharashtra State Board
पाठ 2 Applications of Derivatives
Exercise 2.2 | Q 3.1 | पृष्ठ ७५

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