Which statement correctly represents the Maclaurin series for cos x truncated to degree 2?

Which statement correctly represents the Maclaurin series for cos x truncated to degree 2?

• A. cos x ≈ 1 - x
• B. cos x ≈ 1 - x^2/2
• C. cos x ≈ 1 - x^2
• D. cos x ≈ 1 - x^3/3

Answer: (B)

The idea is to approximate cos x by a Maclaurin (Taylor at 0) polynomial and keep terms up to degree 2. The Maclaurin series for cos x is 1 − x^2/2! + x^4/4! − …, since cosine is an even function and all odd-power terms vanish. Keeping only terms up to x^2 gives cos x ≈ 1 − x^2/2. This matches the option with 1 − x^2/2. The other forms either miss the x^2 term, use an incorrect coefficient, or include an odd power term that doesn’t appear in the expansion.

The idea is to approximate cos x by a Maclaurin (Taylor at 0) polynomial and keep terms up to degree 2. The Maclaurin series for cos x is 1 − x^2/2! + x^4/4! − …, since cosine is an even function and all odd-power terms vanish. Keeping only terms up to x^2 gives cos x ≈ 1 − x^2/2. This matches the option with 1 − x^2/2. The other forms either miss the x^2 term, use an incorrect coefficient, or include an odd power term that doesn’t appear in the expansion.