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Tangent lines

Problem 2.1267 · easy Video

Find an equation of the tangent line to \( \displaystyle y = - 2 x^{3} - 3 \) at \( \displaystyle x = 0 \).

The tangent line passes through the point (a, f(a)) and has slope f'(a).
\[ \left. - 2 x^{3} - 3 \right|_{\substack{ x=0 }} = -3 \]
The point of tangency.✓ Proved
\[ \frac{d}{d x} \left(- 2 x^{3} - 3\right) = - 6 x^{2} \]
Differentiate.✓ Proved
\[ \left. - 6 x^{2} \right|_{\substack{ x=0 }} = 0 \]
The slope at the point.✓ Proved
\[ -3 \]
Point-slope form, then simplify.

Answer \( y = -3 \)

The graph of y = -2x³ - 3 near x = 0. A secant line through the point (0, -3) and a second point on the curve turns into the tangent line as the second point slides in; its slope settles at 0, giving y = -3.

Lines: 3 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see the receipt. How we verify

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	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: fail (misleading) — The solution never writes the equation of the tangent line; it only shows a trivial equality. A student would be left without the required line equation y = -3.
qwen3.6:27b-mlx: fail (error) — Step 5 is mathematically incorrect and logically incoherent. It claims that '-3 = -3' represents the point-slope form of the line, but the point-slope form is 'y - y1 = m(x - x1)', which simplifies to 'y = -3'. The equation '-3 = -3' is a tautology that does not define the line y = -3.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 5 is mathematically incorrect and logically incoherent. It claims that '-3 = -3' represents the point-slope form of the line, but the point-slope form is 'y - y1 = m(x - x1)', which simplifies to 'y = -3'. The equation '-3 = -3' is a tautology that does not define the line y = -3.
gpt-oss:20b: fail (misleading) 2026-10-03 — The solution never writes the equation of the tangent line; it only shows a trivial equality. A student would be left without the required line equation y = -3.
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 5 claims to use point-slope form but provides no derivation or equation, jumping directly to the constant -3. It fails to show the application of y - y1 = m(x - x1) with the calculated slope and point.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The solution fails to explicitly state the point-slope formula or the equation of the line (y = mx + b) before concluding. Step 5 jumps from the slope and y-intercept values directly to the final answer without showing the construction of the linear equation, which is the core of the problem.
gpt-oss:20b: pass 2026-09-29

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.