GATE 2026 CE (CE1) – Question 48
Starting with the first approximation as $x = 0.5$, the second approximation for the root of the following function by the Newton-Raphson method is ______ (*rounded off to two decimal places*).
$$f(x) = e^{-x} - x$$
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Correct answer: 0.56 to 0.58
Explanation
The derivative is $f'(x) = -e^{-x} - 1$. At $x_0 = 0.5$, $f = 0.60653 - 0.5 = 0.10653$ and $f' = -1.60653$. So $x_1 = 0.5 - \frac{0.10653}{-1.60653} = 0.5663$, which is 0.57.