GATE 2017 EC – Question 36

Engineering Mathematics · Calculus · 2 marks · Multiple choice

Let $f(x) = e^{x + x^2}$ for real $x$. From among the following, choose the Taylor series approximation of $f(x)$ around $x = 0$, which includes all powers of $x$ less than or equal to 3.

$1 + x + x^2 + x^3$
$1 + x + \frac{3}{2}x^2 + x^3$
$1 + x + \frac{3}{2}x^2 + \frac{7}{6}x^3$
$1 + x + 3x^2 + 7x^3$

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Correct answer: (C) $1 + x + \frac{3}{2}x^2 + \frac{7}{6}x^3$

Explanation

Write $e^{x + x^2} = e^x \cdot e^{x^2} = \left(1 + x + \frac{x^2}{2} + \frac{x^3}{6}\right)(1 + x^2)$. Multiplying out and keeping terms up to $x^3$ gives $1 + x + \frac{x^2}{2} + x^2 + \frac{x^3}{6} + x^3 = 1 + x + \frac{3}{2}x^2 + \frac{7}{6}x^3$.