Find the approximate value of $tan^{-1}(1.001)$
- $0.6655$
- $0.9385$
- $0.5555$
- $0.7855$
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Correct answer
Explanation
Using the Taylor expansion for tan^-1(1 + x) near x=0, tan^-1(1 + x) is approximately tan^-1(1) + (1 / (1 + 1^2)) * x = pi/4 + x/2. With x = 0.001, this is 0.785398 + 0.0005 = 0.785898, which is closest to 0.7855.
AI explanation
Using the linear approximation formula, the inverse tangent of 1 plus x is approximately pi divided by 4 plus x divided by 2 for a very small x. Substituting 0.001 for x gives pi divided by 4 plus 0.0005, and using the standard value of pi divided by 4 as 0.785 results in 0.7855.