Taylor and Maclaurin polynomials
Problem 7.100 · hard
Find the Maclaurin polynomial of degree 5 for \( \displaystyle f(x) = \cos{\left(x \right)} \).
- p_n(x) = Σ f⁽ᵏ⁾(0)/k! · xᵏ, for k = 0 to n.Reviewed
- \[ \left. \cos{\left(x \right)} \right|_{\substack{ x=0 }} = 1 \]f⁽0⁾(0).✓ Proved
- \[ \left. \frac{d}{d x} \cos{\left(x \right)} \right|_{\substack{ x=0 }} = 0 \]f⁽1⁾(0).✓ Proved
- \[ \left. \frac{d^{2}}{d x^{2}} \cos{\left(x \right)} \right|_{\substack{ x=0 }} = -1 \]f⁽2⁾(0).✓ Proved
- \[ \left. \frac{d^{3}}{d x^{3}} \cos{\left(x \right)} \right|_{\substack{ x=0 }} = 0 \]f⁽3⁾(0).✓ Proved
- \[ \left. \frac{d^{4}}{d x^{4}} \cos{\left(x \right)} \right|_{\substack{ x=0 }} = 1 \]f⁽4⁾(0).✓ Proved
- \[ \left. \frac{d^{5}}{d x^{5}} \cos{\left(x \right)} \right|_{\substack{ x=0 }} = 0 \]f⁽5⁾(0).✓ Proved
- \[ x^{4} \cdot 1 \cdot \frac{1}{24} - 1 x^{2} \cdot 1 \cdot \frac{1}{2} + 1 = \frac{x^{4}}{24} - \frac{x^{2}}{2} + 1 \]Assemble the polynomial.✓ Proved
Answer \( p_{5}(x) = \frac{x^{4}}{24} - \frac{x^{2}}{2} + 1 \)
✓ Nihil obstat Lines: 7 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | sympy.series expands f on its own and matches |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the formula for the Maclaurin polynomial, computes the necessary derivatives and their values at zero, and assembles the polynomial of degree 5 correctly.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the formula for the Maclaurin polynomial, computes the necessary derivatives and their values at zero, and assembles the polynomial of degree 5 correctly.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the definition of the Maclaurin polynomial, computes the necessary derivatives and their values at zero, and assembles the polynomial of degree 5. The result matches the known expansion for cos(x).gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the definition of the Maclaurin polynomial, computes the necessary derivatives and their values at zero, and assembles the polynomial of degree 5 correctly.gpt-oss:20b: pass 2026-09-26
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/taylor_polynomial, checked 2026-09-28 with SymPy 1.14.0.