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GATE 2022 AG – Question 13

Engineering Mathematics · Calculus · 1 mark · Multiple choice

Which expression is the Maclaurin expansion of f(x), assuming the series converges to the function?

  1. $f(1)+xf^{\prime}(1)+\frac{x^2}{2!}f^{\prime\prime}(1)+\frac{x^3}{3!}f^{\prime\prime\prime}(1)+\cdots$
  2. $f(0)+xf^{\prime}(0)+\frac{x^2}{2!}f^{\prime\prime}(0)+\frac{x^3}{3!}f^{\prime\prime\prime}(0)+\cdots$
  3. $f(1)-xf^{\prime}(1)+\frac{x^2}{2!}f^{\prime\prime}(1)-x^3f^{\prime\prime\prime}(1)+\cdots$
  4. $f(0)-xf^{\prime}(0)+\frac{x^2}{2!}f^{\prime\prime}(0)-x^3f^{\prime\prime\prime}(0)+\cdots$

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Correct answer: (B) $f(0)+xf^{\prime}(0)+\frac{x^2}{2!}f^{\prime\prime}(0)+\frac{x^3}{3!}f^{\prime\prime\prime}(0)+\cdots$

Explanation

Maclaurin means Taylor expansion about zero: $f(x)=\sum_{n=0}^{\infty}f^{(n)}(0)x^n/n!$. Hence the constant and every derivative are evaluated at zero, with the displayed positive powers and factorial denominators. This is **B**.

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