find the two nonzero terms of the maclaurin expansion of the given function f(x) =2xe4sinx , the first nonzero term of maclaurin expansion is? the second nonzero maclaurin expansion is?

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Hi

Here first we find f(0) = 2(0) e^0 = 0

f'(x) = 2e^4sinx + 8x e^4sinx

f'(0) = 2e^0 + 0 = 2

so the first non zero term is f'(0) x = 2x

to find the second term, we find f"(x) = 8 e^4sinx + 8 e^4sinx + 32 x e^4sinx

f"(0) = 8 e^0 + 8 e^0 + 0 = 8 + 8 = 16

so the second non zero term is [f"(0) / 2! ] x^2 = (16/2!) x^2 = 8 x^2.

Hence the first two non zero terms of this series are 2x + 8x^2

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