Taylor series about a point
Problem 7.393 · hard
Find the Taylor polynomial of degree 3 for \( \displaystyle f(x) = e^{2 x} \) centered at \( \displaystyle a = 1 \).
- \[ \left[\begin{matrix}\left. e^{2 x} \right|_{\substack{ x=1 }}\\\left. \frac{d}{d x} e^{2 x} \right|_{\substack{ x=1 }}\\\left. \frac{d^{2}}{d x^{2}} e^{2 x} \right|_{\substack{ x=1 }}\\\left. \frac{d^{3}}{d x^{3}} e^{2 x} \right|_{\substack{ x=1 }}\end{matrix}\right] = \left[\begin{matrix}e^{2}\\2 e^{2}\\4 e^{2}\\8 e^{2}\end{matrix}\right] \]f and its first three derivatives at a.✓ Proved
- T(x) = Σ f⁽ʲ⁾(a)/j! · (x − a)ʲ.
- \[ \lim_{x \to 1^+}\left(\frac{- \frac{4 \left(x - 1\right)^{3} e^{2}}{3} - 2 \left(x - 1\right)^{2} e^{2} - \left(2 x - 2\right) e^{2} + e^{2 x} - e^{2}}{\left(x - 1\right)^{3}}\right) = 0 \]T agrees with f to third order at a.✓ Proved
Answer \( T_3(x) = \frac{4 \left(x - 1\right)^{3} e^{2}}{3} + 2 \left(x - 1\right)^{2} e^{2} + 2 \left(x - 1\right) e^{2} + e^{2} \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | a sentence; read, not computed |
| 3 | ✓ 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's series() expansion about a |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution correctly computes the derivatives and states the formula, but the final stated answer in the problem description has the terms in reverse order of powers (degree 3 first, then 2, 1, 0). While mathematically equivalent, standard convention lists Taylor polynomials in ascending powers of (x-a). More critically, the provided 'Stated answer' is just a string, but the solution steps don't explicitly write out the final polynomial in standard form to match the prompt's request clearly, and step 3 verifies the limit of the error term rather than explicitly constructing the polynomial from step 1 and 2. However, the biggest issue is that the solution does not explicitly write the final polynomial T_3(x) derived from steps 1 and 2. It jumps to a limit check. A student needs to see the assembly of the polynomial. Furthermore, the 'Stated answer' in the prompt is actually correct algebraically, but the solution text provided is incomplete as it doesn't show the final assembly step explicitly, relying on a limit check which is a verification, not a derivation. Wait, looking closer at the prompt structure: The 'Stated answer' is given. The 'Solution' is what I review. The solution consists of 3 lines. Line 1 computes derivatives. Line 2 states formula. Line 3 checks the limit. It never actually writes down the polynomial T_3(x) = ... explicitly. It assumes the reader can combine 1 and 2. But more importantly, is the stated answer correct? f(1)=e^2, f'(1)=2e^2, f''(1)=4e^2, f'''(1)=8e^2. T_3(x) = e^2 + 2e^2(x-1) + (4e^2/2)(x-1)^2 + (8e^2/6)(x-1)^3 = e^2 + 2e^2(x-1) + 2e^2(x-1)^2 + (4/3)e^2(x-1)^3. The stated answer matches this. The solution steps are technically correct but incomplete as a 'solution' because it doesn't explicitly state the final polynomial. However, usually, if the algebra is checked, the limit check confirms the polynomial is correct. Is there an error? No. Is it misleading? It's a bit weird to verify via limit instead of just writing it out. But let's look for a hard error. There isn't one. The limit check proves the polynomial is the correct Taylor polynomial. The setup is correct. The conclusion follows. It's just a strange way to present the final answer. But wait, the prompt asks to review the *solution*. The solution does not explicitly write the final polynomial. It just checks a limit. This is not a standard way to 'Find' the polynomial. It's a verification. A student might think 'finding' it involves taking a limit. That is misleading. You find it by computing coefficients. You verify it with a limit. The solution presents the verification as the final step without explicitly stating the result derived from 1 and 2. This is misleading pedagogy.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution correctly computes the derivatives and states the formula, but the final stated answer in the problem description has the terms in reverse order of powers (degree 3 first, then 2, 1, 0). While mathematically equivalent, standard convention lists Taylor polynomials in ascending powers of (x-a). More critically, the provided 'Stated answer' is just a string, but the solution steps don't explicitly write out the final polynomial in standard form to match the prompt's request clearly, and step 3 verifies the limit of the error term rather than explicitly constructing the polynomial from step 1 and 2. However, the biggest issue is that the solution does not explicitly write the final polynomial T_3(x) derived from steps 1 and 2. It jumps to a limit check. A student needs to see the assembly of the polynomial. Furthermore, the 'Stated answer' in the prompt is actually correct algebraically, but the solution text provided is incomplete as it doesn't show the final assembly step explicitly, relying on a limit check which is a verification, not a derivation. Wait, looking closer at the prompt structure: The 'Stated answer' is given. The 'Solution' is what I review. The solution consists of 3 lines. Line 1 computes derivatives. Line 2 states formula. Line 3 checks the limit. It never actually writes down the polynomial T_3(x) = ... explicitly. It assumes the reader can combine 1 and 2. But more importantly, is the stated answer correct? f(1)=e^2, f'(1)=2e^2, f''(1)=4e^2, f'''(1)=8e^2. T_3(x) = e^2 + 2e^2(x-1) + (4e^2/2)(x-1)^2 + (8e^2/6)(x-1)^3 = e^2 + 2e^2(x-1) + 2e^2(x-1)^2 + (4/3)e^2(x-1)^3. The stated answer matches this. The solution steps are technically correct but incomplete as a 'solution' because it doesn't explicitly state the final polynomial. However, usually, if the algebra is checked, the limit check confirms the polynomial is correct. Is there an error? No. Is it misleading? It's a bit weird to verify via limit instead of just writing it out. But let's look for a hard error. There isn't one. The limit check proves the polynomial is the correct Taylor polynomial. The setup is correct. The conclusion follows. It's just a strange way to present the final answer. But wait, the prompt asks to review the *solution*. The solution does not explicitly write the final polynomial. It just checks a limit. This is not a standard way to 'Find' the polynomial. It's a verification. A student might think 'finding' it involves taking a limit. That is misleading. You find it by computing coefficients. You verify it with a limit. The solution presents the verification as the final step without explicitly stating the result derived from 1 and 2. This is misleading pedagogy.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The stated answer has a coefficient of 4/3 for the cubic term, but the correct coefficient is 4/6 = 2/3. The solution's verification step (line 3) checks a polynomial with a coefficient of -4/3, which confirms the wrong answer rather than the correct one.gpt-oss:20b: pass 2026-10-06
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_series_center, checked 2026-10-06 with SymPy 1.14.0.