GATE AG 2022 • Q13 MCQ • 1 Mark

Section 1: Engineering Mathematics

Calculus

Subtopic: Fourier, Taylor and MacLaurin series

Question Statement:

Function f(x) by Maclaurin's series (as an infinite series) can be expressed as

Options:

  • (A) f(x) = f(1) + x f'(1) + (x²/2!) f''(1) + (x³/3!) f'''(1) + ⋯ + ∞
  • (B) f(x) = f(0) + x f'(0) + (x²/2!) f''(0) + (x³/3!) f'''(0) + ⋯ + ∞
  • (C) f(x) = f(1) − x f'(1) + (x²/2!) f''(1) − (x³/3!) f'''(1) + ⋯ + ∞
  • (D) f(x) = f(0) − x f'(0) + (x²/2!) f''(0) − (x³/3!) f'''(0) + ⋯ + ∞

Official Answer Key:

B

Step-by-Step Solution & Explanation

• The Maclaurin series is the Taylor series expanded about x = 0. • General Taylor series about x=a: f(x) = f(a) + (x−a)f'(a) + [(x−a)²/2!]f''(a) + ⋯. Setting a=0 gives the Maclaurin series with all '+' signs. • Distractor logic: (A),(C) expand about x=1, not x=0. (C),(D) incorrectly alternate signs.

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