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Tangent lines

Problem 2.772 · medium Video

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

The tangent line passes through the point (a, f(a)) and has slope f'(a).
Reviewed
\[ \left. - x^{2} - 2 x - 4 - \frac{3}{x} \right|_{\substack{ x=2 }} = - \frac{27}{2} \]
The point of tangency.✓ Proved
\[ \frac{d}{d x} \left(- x^{2} - 2 x - 4 - \frac{3}{x}\right) = - 2 x - 2 + \frac{3}{x^{2}} \]
Differentiate.✓ Proved
\[ \left. - 2 x - 2 + \frac{3}{x^{2}} \right|_{\substack{ x=2 }} = - \frac{21}{4} \]
The slope at the point.✓ Proved
\[ - \frac{21 x}{4} - 3 \]
Point-slope form, then simplify.Reviewed

Answer \( y = - \frac{21 x}{4} - 3 \)

The graph of y = -x² - 2x - 4 - 3/x near x = 2. A secant line through the point (2, -27/2) and a second point on the curve turns into the tangent line as the second point slides in; its slope settles at -21/4, giving y = -21x/4 - 3.

✓ 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, and the final equation is algebraically correct.

Every verdict on record (6)

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 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the point of tangency and the slope, and the final equation is algebraically correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the point of tangency and the slope, and the final equation is algebraically equivalent to the point-slope form derived from those values.
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.