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Tangent lines

Problem 2.729 · easy Video

Find an equation of the tangent line to \( \displaystyle y = - 3 x^{3} + x^{2} + 3 x - 1 \) at \( \displaystyle x = 2 \).

The tangent line passes through the point (a, f(a)) and has slope f'(a).
Reviewed
\[ \left. - 3 x^{3} + x^{2} + 3 x - 1 \right|_{\substack{ x=2 }} = -15 \]
The point of tangency.✓ Proved
\[ \frac{d}{d x} \left(- 3 x^{3} + x^{2} + 3 x - 1\right) = - 9 x^{2} + 2 x + 3 \]
Differentiate.✓ Proved
\[ \left. - 9 x^{2} + 2 x + 3 \right|_{\substack{ x=2 }} = -29 \]
The slope at the point.✓ Proved
\[ 43 - 29 x \]
Point-slope form, then simplify.

Answer \( y = 43 - 29 x \)

The graph of y = -3x³ + x² + 3x - 1 near x = 2. A secant line through the point (2, -15) and a second point on the curve turns into the tangent line as the second point slides in; its slope settles at -29, giving y = 43 - 29x.

Lines: 3 proved, 1 reviewed, 1 not checked. 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	claude-sonnet-5-5	claude-sonnet-5-5: Line 1 is a sentence stating the standard fact that the tangent line at x = a passes through (a, f(a)) with slope f'(a). It is not an algebraic equivalence, so no script applies.
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: pass
qwen3.6:27b-mlx: fail (error) — The point of tangency is (2, -15), but the final equation y = 43 - 29x yields y = -15 at x=2, which is correct numerically, however, the derivation in line 5 is not shown and the label 'Point-slope form, then simplify' is misleading because the standard point-slope form y - (-15) = -29(x - 2) simplifies to y = -29x + 58 - 15 = -29x + 43. Wait, 58 - 15 is 43. So the math is correct. Let me re-read the prompt constraints. 'Check that... each conclusion follows from the lines before it'. Line 5 jumps from the slope to the final equation without showing the point-slope step explicitly, but the label claims it does. Is this an error? The prompt says 'equation... proved'. If the CAS proved 43-29x == 43-29x, that's a tautology. It didn't prove that 43-29x is the tangent line. The sentence claims 'Point-slope form, then simplify' leads to this. The previous lines established point (2, -15) and slope -29. The equation of the line is y - (-15) = -29(x - 2) => y = -29x + 58 - 15 => y = -29x + 43. The result is correct. However, line 5 is marked as an equation that is 'proved'. The equation `43 - 29*x, 43 - 29*x` is a tautology and does not represent the derivation of the tangent line equation from the previous steps. The sentence says 'Point-slope form, then simplify', implying this line is the result of that process. But the line itself is just an identity. A student might think the CAS verified the tangent line equation, but it only verified a tautology. The connection between the previous steps and this line is missing in the 'proved' equation. This is a 'misleading' or 'error' in the solution structure. The sentence claims a derivation happened, but the equation line doesn't reflect the derivation from previous variables. It just states the answer. This is a style/logic error in the solution format. Let's look closer. Line 5 is `43 - 29*x, 43 - 29*x`. This is likely the CAS checking if the expression equals itself. It does not check if it equals the tangent line. The sentence says 'Point-slope form, then simplify'. This implies the step *is* the simplification. But the equation provided doesn't show the simplification from the point-slope form. It just shows the final answer. This is a 'style' issue or 'misleading' because it hides the work. However, usually, these reviews look for mathematical errors. Is there a mathematical error? No. Is it misleading? Yes, it pretends the CAS verified the tangent line construction when it didn't. But wait, looking at line 2: `Subs(..., 2), -15`. This is the y-value. Line 4: `Subs(..., 2), -29`. This is the slope. Line 5: `43 - 29*x, 43 - 29*x`. This is just the final expression. The sentence says 'Point-slope form, then simplify'. This is a description of the mental step. The equation line is just the result. This is standard in some formats. But is it 'wrong'? The prompt asks to find what is wrong. If the math is correct, is it 'pass'? Let's check the math again. y(2) = -3(8) + 4 + 6 - 1 = -24 + 9 = -15. Correct. y'(x) = -9x^2 + 2x + 3. y'(2) = -36 + 4 + 3 = -29. Correct. Line: y - (-15) = -29(x - 2) => y = -29x + 58 - 15 => y = -29x + 43. Correct. The solution is mathematically sound. The only issue is the presentation of line 5. Is that an 'error'? Probably not. It's 'style'. But wait, look at line 2. `Subs(..., 2), -15`. The comment says 'The point of tangency'. The point is (2, -15). The line only gives the y-coordinate. This is slightly imprecise but acceptable. Line 5 is the main suspect. The equation `43 - 29*x, 43 - 29*x` is trivial. It doesn't prove the line is the tangent. It just defines the expression. The sentence claims it comes from point-slope form. This is a valid derivation step described in text. I will mark it as pass because the math is correct and the text explains the step, even if the equation line is a tautology. Wait, let me look for a subtle error. Did I miss a sign? -3(2)^3 = -24. 2^2 = 4. 3(2) = 6. -1. Sum = -15. Slope: -9(4) + 4 + 3 = -36 + 7 = -29. Line: y = -29(x-2) - 15 = -29x + 58 - 15 = -29x + 43. Everything is correct. Why would it fail? Maybe the format of line 2? 'Subs(..., 2), -15'. This is the value. The point is (2, -15). The sentence says 'The point of tangency'. It's a bit loose but not wrong. I'll go with pass.

Senior review claude-sonnet-5-5, 2026-10-12: pass — The values f(2) = -15 and f'(2) = -29 are correct, and the point-slope form -15 - 29(x - 2) simplifies to 43 - 29x, matching the stated answer. Line 5 is terse, since it shows only the final expression, but it is not wrong.

qwen3.6:27b-mlx: dismiss — The objection itself verifies f(2) = -15, f'(2) = -29 and the line y = 43 - 29x, so it finds no false statement. The label 'Point-slope form, then simplify' describes a legitimate step, and the terse presentation of line 5 is at most a style matter.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-12 — The values f(2) = -15 and f'(2) = -29 are correct, and the point-slope form -15 - 29(x - 2) simplifies to 43 - 29x, matching the stated answer. Line 5 is terse, since it shows only the final expression, but it is not wrong.
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The point of tangency is (2, -15), but the final equation y = 43 - 29x yields y = -15 at x=2, which is correct numerically, however, the derivation in line 5 is not shown and the label 'Point-slope form, then simplify' is misleading because the standard point-slope form y - (-15) = -29(x - 2) simplifies to y = -29x + 58 - 15 = -29x + 43. Wait, 58 - 15 is 43. So the math is correct. Let me re-read the prompt constraints. 'Check that... each conclusion follows from the lines before it'. Line 5 jumps from the slope to the final equation without showing the point-slope step explicitly, but the label claims it does. Is this an error? The prompt says 'equation... proved'. If the CAS proved 43-29x == 43-29x, that's a tautology. It didn't prove that 43-29x is the tangent line. The sentence claims 'Point-slope form, then simplify' leads to this. The previous lines established point (2, -15) and slope -29. The equation of the line is y - (-15) = -29(x - 2) => y = -29x + 58 - 15 => y = -29x + 43. The result is correct. However, line 5 is marked as an equation that is 'proved'. The equation `43 - 29*x, 43 - 29*x` is a tautology and does not represent the derivation of the tangent line equation from the previous steps. The sentence says 'Point-slope form, then simplify', implying this line is the result of that process. But the line itself is just an identity. A student might think the CAS verified the tangent line equation, but it only verified a tautology. The connection between the previous steps and this line is missing in the 'proved' equation. This is a 'misleading' or 'error' in the solution structure. The sentence claims a derivation happened, but the equation line doesn't reflect the derivation from previous variables. It just states the answer. This is a style/logic error in the solution format. Let's look closer. Line 5 is `43 - 29*x, 43 - 29*x`. This is likely the CAS checking if the expression equals itself. It does not check if it equals the tangent line. The sentence says 'Point-slope form, then simplify'. This implies the step *is* the simplification. But the equation provided doesn't show the simplification from the point-slope form. It just shows the final answer. This is a 'style' issue or 'misleading' because it hides the work. However, usually, these reviews look for mathematical errors. Is there a mathematical error? No. Is it misleading? Yes, it pretends the CAS verified the tangent line construction when it didn't. But wait, looking at line 2: `Subs(..., 2), -15`. This is the y-value. Line 4: `Subs(..., 2), -29`. This is the slope. Line 5: `43 - 29*x, 43 - 29*x`. This is just the final expression. The sentence says 'Point-slope form, then simplify'. This is a description of the mental step. The equation line is just the result. This is standard in some formats. But is it 'wrong'? The prompt asks to find what is wrong. If the math is correct, is it 'pass'? Let's check the math again. y(2) = -3(8) + 4 + 6 - 1 = -24 + 9 = -15. Correct. y'(x) = -9x^2 + 2x + 3. y'(2) = -36 + 4 + 3 = -29. Correct. Line: y - (-15) = -29(x - 2) => y = -29x + 58 - 15 => y = -29x + 43. Correct. The solution is mathematically sound. The only issue is the presentation of line 5. Is that an 'error'? Probably not. It's 'style'. But wait, look at line 2. `Subs(..., 2), -15`. The comment says 'The point of tangency'. The point is (2, -15). The line only gives the y-coordinate. This is slightly imprecise but acceptable. Line 5 is the main suspect. The equation `43 - 29*x, 43 - 29*x` is trivial. It doesn't prove the line is the tangent. It just defines the expression. The sentence claims it comes from point-slope form. This is a valid derivation step described in text. I will mark it as pass because the math is correct and the text explains the step, even if the equation line is a tautology. Wait, let me look for a subtle error. Did I miss a sign? -3(2)^3 = -24. 2^2 = 4. 3(2) = 6. -1. Sum = -15. Slope: -9(4) + 4 + 3 = -36 + 7 = -29. Line: y = -29(x-2) - 15 = -29x + 58 - 15 = -29x + 43. Everything is correct. Why would it fail? Maybe the format of line 2? 'Subs(..., 2), -15'. This is the value. The point is (2, -15). The sentence says 'The point of tangency'. It's a bit loose but not wrong. I'll go with pass.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution claims the point of tangency is at x=2 with y=-15, but uses the y-intercept 43 in the final equation. The correct y-intercept for a line with slope -29 passing through (2, -15) is 43, but the derivation step 5 is labeled 'Point-slope form, then simplify' without showing the intermediate step y - (-15) = -29(x - 2), making it unclear how 43 was derived from the previous lines. More critically, the label 'The point of tangency' on line 2 only gives the y-value, not the full point, and line 5 jumps to the final answer without explicitly connecting the point (2, -15) and slope -29 to the equation y = 43 - 29x. While the final answer is correct, the logical flow is broken and misleading.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution claims the point of tangency is at x=2 with y=-15, but the final equation 43 - 29*x yields y = 43 - 58 = -15 at x=2, which is consisten
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 2026-10-13 with SymPy 1.14.0.