Question Detail
Question
Which of the following best suits the Taylors series expansion of a given function f(x) about the point x=b?
Correct Answer
Option A is the correct answer.
Detailed Explanation
The Taylor series expansion of a function f(x) about a point x=b (also called the expansion around x=b or centered at x=b) is defined as:f(x) = f(b) + f'(b)(x-b)/1! + f''(b)(x-b)²/2! + f'''(b)(x-b)³/3! + ...In summation notation, this is:f(x) = Σ [fʳ(b) / r!] * (x-b)ʳ from r=0 to ∞,where fʳ(b) denotes the r-th derivative of f evaluated at b (with f⁰(b) = f(b)).Option A correctly represents this definition.
Hint
Recall the general formula for the Taylor series expansion of a function f(x) centered at a point x=b. Pay attention to the terms in the numerator and denominator, and where the function and its derivatives are evaluated.