Tangent lines
Problem 2.915 · easy
Find an equation of the tangent line to \( \displaystyle y = - x^{3} + 3 x^{2} - 2 x - 2 \) at \( \displaystyle x = 1 \).
- The tangent line passes through the point (a, f(a)) and has slope f'(a).
- \[ \left. - x^{3} + 3 x^{2} - 2 x - 2 \right|_{\substack{ x=1 }} = -2 \]The point of tangency.✓ Proved
- \[ \frac{d}{d x} \left(- x^{3} + 3 x^{2} - 2 x - 2\right) = - 3 x^{2} + 6 x - 2 \]Differentiate.✓ Proved
- \[ \left. - 3 x^{2} + 6 x - 2 \right|_{\substack{ x=1 }} = 1 \]The slope at the point.✓ Proved
- \[ x - 3 \]Point-slope form, then simplify.✓ Proved
Answer \( y = x - 3 \)
Lines: 4 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 | 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 | ✓ 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 | the line meets the curve at x = a, and its slope matches a central difference quotient of f there |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The final equation is incorrect. The point-slope form using point (1, -2) and slope 1 is y - (-2) = 1(x - 1), which simplifies to y = x - 3. The stated answer 'Eq(y, x - 3)' implies y = x - 3, but the solution line 5 just shows 'x - 3' without explicitly equating it to y, and more importantly, the previous steps establish the point (1, -2) and slope 1. Let's re-verify: f(1) = -1 + 3 - 2 - 2 = -2. f'(x) = -3x^2 + 6x - 2. f'(1) = -3 + 6 - 2 = 1. Tangent line: y - (-2) = 1(x - 1) => y + 2 = x - 1 => y = x - 3. The stated answer is correct. However, line 5 says 'x - 3, x - 3' with the comment 'Point-slope form, then simplify'. This is not an equation, it's just an expression. The problem asks for an 'equation of the tangent line'. The solution fails to provide the equation y = x - 3 explicitly in the final step, merely providing the RHS. While the stated answer is correct, the solution trace is incomplete/misleading because it doesn't show the equation formation. Actually, looking closer, line 5 is marked [equation, proved] but the content is 'x - 3, x - 3'. This is not an equation. It's just an identity. The solution fails to state the final equation y = x - 3. It just outputs the expression for y. This is a significant omission in a 'find an equation' problem.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The final equation is incorrect. The point-slope form using point (1, -2) and slope 1 is y - (-2) = 1(x - 1), which simplifies to y = x - 3. The stated answer 'Eq(y, x - 3)' implies y = x - 3, but the solution line 5 just shows 'x - 3' without explicitly equating it to y, and more importantly, the previous steps establish the point (1, -2) and slope 1. Let's re-verify: f(1) = -1 + 3 - 2 - 2 = -2. f'(x) = -3x^2 + 6x - 2. f'(1) = -3 + 6 - 2 = 1. Tangent line: y - (-2) = 1(x - 1) => y + 2 = x - 1 => y = x - 3. The stated answer is correct. However, line 5 says 'x - 3, x - 3' with the comment 'Point-slope form, then simplify'. This is not an equation, it's just an expression. The problem asks for an 'equation of the tangent line'. The solution fails to provide the equation y = x - 3 explicitly in the final step, merely providing the RHS. While the stated answer is correct, the solution trace is incomplete/misleading because it doesn't show the equation formation. Actually, looking closer, line 5 is marked [equation, proved] but the content is 'x - 3, x - 3'. This is not an equation. It's just an identity. The solution fails to state the final equation y = x - 3. It just outputs the expression for y. This is a significant omission in a 'find an equation' problem.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: fail (error) 2026-09-26 — The final equation is incorrect. Using point-slope form with point (1, -2) and slope 1 yields y - (-2) = 1(x - 1), which simplifies to y = x - 3. The stated answer 'Eq(y, x - 3)' is syntactically ambiguous or incorrect notation for y = x - 3, but more critically, the solution skips the derivation step from point-slope to slope-intercept, making the jump from line 4 to line 5 unjustified without showing the substitution. However, looking closer at the 'Stated answer' vs the solution line 5: Line 5 says 'x - 3, x - 3'. This implies the solution claims the equation is y = x - 3. Let's re-verify the math. f(1) = -1 + 3 - 2 - 2 = -2. f'(x) = -3x^2 + 6x - 2. f'(1) = -3 + 6 - 2 = 1. Point-slope: y - (-2) = 1(x - 1) => y + 2 = x - 1 => y = x - 3. The math is correct. The issue is likely the format of the 'Stated answer' or the lack of explicit derivation in line 5. But wait, the prompt asks to review the *solution*. Line 5 is marked [equation, unchecked] with result 'x - 3, x - 3'. This looks like a CAS output for simplifying an expression, not an equation. The sentence 'Point-slope form, then simplify' is the sentence being reviewed. It claims the result of simplifying the point-slope form is 'x - 3'. This is misleading because 'x - 3' is an expression, not an equation. The tangent line equation is y = x - 3. The solution presents 'x - 3' as the final object, which is technically just the RHS. If the problem asks for an 'equation', providing just the expression is an error. Furthermore, line 5 is labeled [equation] but the content 'x - 3, x - 3' suggests it's just simplifying the RHS. The sentence says 'Point-slope form, then simplify'. It fails to explicitly state the equation y = x - 3. This is a style/clarity issue at best, but since the 'Stated answer' is 'Eq(y, x - 3)', the solution line 5 is incomplete/wrongly formatted as an equation. Let's look at the options. Is it an error? The math is right. Is it misleading? A student might think the answer is just 'x - 3'. Is it style? It's poor style to not write 'y = ...'. However, often in these systems, 'find an equation' might accept the RHS if the LHS is implied, but strictly speaking, an equation requires an equals sign. The sentence 'Point-slope form, then simplify' leads to 'x - 3'. This is not an equation. Therefore, the solution fails to provide an equation.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/tangent_line, checked 2026-09-26 with SymPy 1.14.0.