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Derive the approximation cos(x) ≈ 1 - (1/2)x².

a) cos(x) = 1 - x²
b) cos(x) ≈ 1 - (1/2)x²
c) cos(x) = 1 + x²
d) cos(x) ≈ 1 + (1/2)x²

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Final answer:

The correct approximation of the cosine function near x=0 is 'cos(x) ≈ 1 - (1/2)x²', which is a second-order Taylor series expansion.

Step-by-step explanation:

The approximation cos(x) ≈ 1 - (1/2)x² is a second-order Taylor series expansion of the cosine function around x=0. The Taylor series provides a way to approximate functions using a sum of terms based on the function's derivatives at a single point. Looking at the options provided:

  • a) cos(x) = 1 - x² is incorrect because the coefficient of the x² term is not -(1/2).
  • b) cos(x) ≈ 1 - (1/2)x² is correct as it reflects the second-order Taylor approximation.
  • c) cos(x) = 1 + x² is incorrect because it suggests a positive quadratic term and a cosine function decreases as x moves away from zero, not increases.
  • d) cos(x) ≈ 1 + (1/2)x² is incorrect because the quadratic term should subtract from the constant, not add to it.

Therefore, the correct approximation that fits the question is b) cos(x) ≈ 1 - (1/2)x².

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