Home›Calculus 1›Tangent lines›Problem 2.733

Tangent lines

Problem 2.733 · medium Video

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

The tangent line passes through the point (a, f(a)) and has slope f'(a).
Reviewed
\[ \left. - 2 x^{2} + x + 3 - \frac{3}{x} \right|_{\substack{ x=-1 }} = 3 \]
The point of tangency.✓ Proved
\[ \frac{d}{d x} \left(- 2 x^{2} + x + 3 - \frac{3}{x}\right) = - 4 x + 1 + \frac{3}{x^{2}} \]
Differentiate.✓ Proved
\[ \left. - 4 x + 1 + \frac{3}{x^{2}} \right|_{\substack{ x=-1 }} = 8 \]
The slope at the point.✓ Proved
\[ 8 x + 11 \]
Point-slope form, then simplify.Reviewed

Answer \( y = 8 x + 11 \)

The graph of y = -2x² + x + 3 - 3/x near x = -1. A secant line through the point (-1, 3) and a second point on the curve turns into the tangent line as the second point slides in; its slope settles at 8, giving y = 8x + 11.

✓ 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 derives the correct tangent line equation. The steps are logically sound and algebraically verified.

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 derives the correct tangent line equation. The steps are logically sound and algebraically verified.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26
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.