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Tangent lines

Problem 2.1265 · easy Video

Find an equation of the tangent line to \( \displaystyle y = 2 x^{3} - 3 x^{2} - 1 \) at \( \displaystyle x = 0 \).

The tangent line passes through the point (a, f(a)) and has slope f'(a).
\[ \left. 2 x^{3} - 3 x^{2} - 1 \right|_{\substack{ x=0 }} = -1 \]
The point of tangency.✓ Proved
\[ \frac{d}{d x} \left(2 x^{3} - 3 x^{2} - 1\right) = 6 x \left(x - 1\right) \]
Differentiate.✓ Proved
\[ \left. 6 x \left(x - 1\right) \right|_{\substack{ x=0 }} = 0 \]
The slope at the point.✓ Proved
\[ -1 \]
Point-slope form, then simplify.

Answer \( y = -1 \)

The graph of y = 2x³ - 3x² - 1 near x = 0. A secant line through the point (0, -1) 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 = -1.

Lines: 3 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in 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: pass
qwen3.6:27b-mlx: fail (error) — The final equation y = -1 is incorrect; the correct tangent line is y = -1 + 0(x), which simplifies to y = -1, but the derivation in step 5 is nonsensical ('-1 = -1') and fails to explicitly state the equation of the line. While the final answer happens to be correct numerically, the solution lacks the logical step of forming the line equation y - y1 = m(x - x1). However, looking closer, y = -1 IS the correct answer. Let me re-read the prompt constraints. 'Find what is wrong'. Is there a logical gap? Step 5 says '-1 = -1 -- Point-slope form, then simplify.' This is not a valid derivation. It asserts a tautology instead of deriving the line equation. A student would not learn how to get from the slope and point to the line equation. It is misleading/error in reasoning.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — The final equation y = -1 is incorrect; the correct tangent line is y = -1 + 0(x), which simplifies to y = -1, but the derivation in step 5 is nonsensical ('-1 = -1') and fails to explicitly state the equation of the line. While the final answer happens to be correct numerically, the solution lacks the logical step of forming the line equation y - y1 = m(x - x1). However, looking closer, y = -1 IS the correct answer. Let me re-read the prompt constraints. 'Find what is wrong'. Is there a logical gap? Step 5 says '-1 = -1 -- Point-slope form, then simplify.' This is not a valid derivation. It asserts a tautology instead of deriving the line equation. A student would not learn how to get from the slope and point to the line equation. It is misleading/error in reasoning.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Line 5 claims to use point-slope form but provides no equation, only the y-intercept value. It fails to state the actual equation of the line (y = -1) derived from the point (0, -1) and slope 0.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 5 is mathematically incorrect and misleading. It claims that '-1, -1' represents the point-slope form, but point-slope form is an equation (e.g., y - y1 = m(x - x1)), not a pair of numbers. Furthermore, the derivation from the previous steps to the final answer is missing the actual equation construction.
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.