Generate the Taylor Polynomial

Question

Generate the Taylor Polynomial of order 3, by definition, using f(x)=sin(2x) at x=pi/4

Answer 1

The 3rd degree Taylor Polynomial is:

f(π/4) + f'(π/4)(x - π/4) + [f"(π/4) / 2!](x-π/4)² + [f'''(π/4) / 3!]( x - π/4)³

f(x) = sin(2x), so f(π/4) = sin(π/2) = 1

f'(x) = 2cos(2x), so f'(π/4) = 0

f"(x) = -4sin(2x), so f"(π/4) = -4

f'''(x) = -8cos(2x), so f'''(π/4) = 0

Taylor Polynomial: 1 - 2(x - π/4)²

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