Tangent lines
Problem 2.1153 · medium Video
Find an equation of the tangent line to \( \displaystyle y = 2 x - 3 \sin{\left(x \right)} - 2 \) at \( \displaystyle x = \pi \).
- The tangent line passes through the point (a, f(a)) and has slope f'(a).
- \[ \left. 2 x - 3 \sin{\left(x \right)} - 2 \right|_{\substack{ x=\pi }} = -2 + 2 \pi \]The point of tangency.✓ Proved
- \[ \frac{d}{d x} \left(2 x - 3 \sin{\left(x \right)} - 2\right) = 2 - 3 \cos{\left(x \right)} \]Differentiate.✓ Proved
- \[ \left. 2 - 3 \cos{\left(x \right)} \right|_{\substack{ x=\pi }} = 5 \]The slope at the point.✓ Proved
- \[ 5 x - 3 \pi - 2 \]Point-slope form, then simplify.
Answer \( y = 5 x - 3 \pi - 2 \)
Lines: 3 proved, 2 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. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | 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 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the line meets the curve at x = a, and its slope matches a central difference quotient of f there |
Reviewers
gpt-oss:20b: fail (error) — The final line does not produce the equation of the tangent line; it merely repeats an identity and omits the variable y. The point‑slope form should be written as y - y₀ = m(x - x₀) and then simplified to y = 5x - 3π - 2.qwen3.6:27b-mlx: pass — The solution correctly identifies the point of tangency and the slope, and the final equation matches the stated answer. The logic is sound.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies the point of tangency and the slope, and the final equation matches the stated answer. The logic is sound.gpt-oss:20b: fail (error) 2026-10-03 — The final line does not produce the equation of the tangent line; it merely repeats an identity and omits the variable y. The point‑slope form should be written as y - y₀ = m(x - x₀) and then simplified to y = 5x - 3π - 2.qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the point of tangency and the slope, and the final equation is algebraically correct.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-28 — The final equation is incorrect. The point-slope form y - f(a) = m(x - a) yields y - (2pi - 2) = 5(x - pi), which simplifies to y = 5x - 3pi - 2. However, the stated answer is Eq(y, 5*x - 3*pi - 2), which implies y = 5x - 3pi - 2. Wait, let me re-calculate. f(pi) = 2pi - 3sin(pi) - 2 = 2pi - 2. Slope m = 2 - 3cos(pi) = 2 - 3(-1) = 5. Equation: y - (2pi - 2) = 5(x - pi) => y = 5x - 5pi + 2pi - 2 => y = 5x - 3pi - 2. The stated answer matches this. Why did I think it was wrong? Let me check the solution steps again. Step 5 says '5*x - 3*pi - 2, 5*x - 3*pi - 2'. This is just an identity, not a derivation. The sentence in step 5 is missing or implied. The prompt asks to review the solution. Step 5 is marked as an equation and 'unchecked' in the prompt description but the prompt says 'equation, unchecked' for steps 2-5. Actually, looking closely at the provided text: '5. [equation, unchecked] 5*x - 3*pi - 2, 5*x - 3*pi - 2 -- Point-slope form, then simplify.' This line is nonsensical as a derivation step. It asserts an identity rather than showing the work from point-slope form to the final equation. A student would not learn how to get from the point and slope to the final equation. It skips the actual algebraic manipulation. Is this an error or misleading? It's misleading because it presents the final answer without showing the connection to the previous steps (point and slope). It looks like a copy-paste error in the solution generation. The step does not follow from the previous lines logically; it just states the answer. Therefore, the solution is incomplete/misleading.gpt-oss:20b: pass 2026-09-28
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/tangent_line, checked 2026-10-13 with SymPy 1.14.0.