GATE 2024 ES – Question 42
For $f(x)=\ln(\sin x)$,which Taylor-expansion statements for f(x+h) are correct?
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Correct answer: (B) The first term is $\ln(\sin x)$.; (C) The $h^2$ term is $-h^2/(2\sin^2x)$.
Explanation
Taylor’s formula gives $f(x+h)=f(x)+hf^{\prime}(x)+h^2f^{\prime\prime}(x)/2+\cdots$. Here $f^{\prime}=\cot x$ and $f^{\prime\prime}=-\csc^2x$,so the first term is ln(sin x),the linear term h cot x, and the quadratic term $-h^2/(2\sin^2x)$. Select **B, C**,where the real logarithm is defined.