Question Part Points Submissions Used Find the Maclaurin series for f(x) using t

Question

Question Part Points Submissions Used Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) ? 0.] f(x) = sin(?x/2)

Explanation / Answer

We see that: f(x) = cos(3x/8) ==> f(0) = 1 f'(x) = (-3/8)sin(3x/8) ==> f'(0) = 0 f''(x) = (-9/64)cos(3x/8) ==> f''(0) = -9/64 f'''(x) = (27/512)sin(3x/8) ==> f'''(0) = 0 f^4(x) = (81/4096)cos(3x/8) ==> f^4(0) = 81/4096. So: f(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)^2/2! + f'''(a)(x - a)^3/3! + f^4(a)(x - a)^4/4! + ... = 1 + (0)x - (9/64)x^2/2! + (0)x^3/3! + (81/4096)x^4/4! + ... .

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