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Question
Find the approximate value of tan−1 (1.001).
Solution
Let f(y) = tan-1y
Differentiating f(y) w.r.t.y, we have
⇒ f'(y) = `1/( 1 + y^2)`
y = 1.001 = x + Δx
Here,
x = 1
Δx = 0.001
Therefore, f(x) = f(1) = tan-1(1) = `π/4`
Similarly, f'(x) = f'(1) = `1/(1 + 1^2) = 1/2`
Now,
f(y) = f( x + Δx ) = f(x) + Δx. f'(x) ...[ ∵ Δx <<< x ]
tan-1y = tan-1( x + Δx ) = tan-1x + Δx.`(1/( 1 + x^2))`
∴ tan-11.001 = tan-1( 1 + 1.001 ) = tan-11 + (0.001). tan-11
⇒ tan-11.001 = `π/4 + 0.001 (1/2)`
⇒ tan-11.001 = `π/4 + 0.0005` ≈ 0.7855
Hence the approxiate value of tan-10.001 will be 0.7855.
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