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Tangent lines

Problem 2.775 · easy Video

Find an equation of the tangent line to \( \displaystyle y = - 3 x^{3} + 2 x^{2} - 2 x - 3 \) at \( \displaystyle x = 4 \).

The tangent line passes through the point (a, f(a)) and has slope f'(a).
Reviewed
\[ \left. - 3 x^{3} + 2 x^{2} - 2 x - 3 \right|_{\substack{ x=4 }} = -171 \]
The point of tangency.✓ Proved
\[ \frac{d}{d x} \left(- 3 x^{3} + 2 x^{2} - 2 x - 3\right) = - 9 x^{2} + 4 x - 2 \]
Differentiate.✓ Proved
\[ \left. - 9 x^{2} + 4 x - 2 \right|_{\substack{ x=4 }} = -130 \]
The slope at the point.✓ Proved
\[ 349 - 130 x \]
Point-slope form, then simplify.Reviewed

Answer \( y = 349 - 130 x \)

The graph of y = -3x³ + 2x² - 2x - 3 near x = 4. A secant line through the point (4, -171) and a second point on the curve turns into the tangent line as the second point slides in; its slope settles at -130, giving y = 349 - 130x.

✓ Nihil obstat Lines: 3 proved, 2 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly identifies the point of tangency and the slope, then derives the correct linear equation. The steps are logically sound and algebraically verified.

Senior review claude-sonnet-5-5, 2026-10-12: pass — f(4) = -171 and f'(4) = -130 are correct, and -171 - 130(x - 4) = 349 - 130x matches the stated answer. Line 5 is a tautology that skips writing the point-slope form y = -171 - 130(x - 4), and its label bundles two steps. This is only a minor style issue, not an error.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-12 — f(4) = -171 and f'(4) = -130 are correct, and -171 - 130(x - 4) = 349 - 130x matches the stated answer. Line 5 is a tautology that skips writing the point-slope form y = -171 - 130(x - 4), and its label bundles two steps. This is only a minor style issue, not an error.
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the point of tangency and the slope, then derives the correct linear equation. The steps are logically sound and algebraically verified.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution claims to use point-slope form but fails to show the substitution of the point (4, -171) and slope -130 into the formula y - y1 = m(x - x1). The resulting equation y = 349 - 130x is incorrect; the correct y-intercept is 349 only if the calculation -171 - (-130)*4 = 349 is performed, but the step skips the derivation entirely, making it impossible to verify the arithmetic or the method application. Specifically, y = -130(x-4) - 171 = -130x + 520 - 171 = -130x + 349. While the final answer is numerically correct, the sentence 'Point-slope form, then simplify' is misleading because it implies a direct transformation that isn't shown, and typically 'point-slope form' refers to y - y1 = m(x - x1), not the simplified slope-intercept form. More critically, without showing the substitution, a student cannot learn how to apply the formula.
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 point of tangency is calculated as (4, -171), but the final equation y = 349 - 130x yields y = -171 at x=4. However, the constant term 349 impli
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.