Tangent lines
Problem 2.1150 · easy
Find an equation of the tangent line to \( \displaystyle y = 3 x^{3} - 3 x^{2} - 2 x + 2 \) at \( \displaystyle x = 4 \).
- The tangent line passes through the point (a, f(a)) and has slope f'(a).
- \[ \left. 3 x^{3} - 3 x^{2} - 2 x + 2 \right|_{\substack{ x=4 }} = 138 \]The point of tangency.✓ Proved
- \[ \frac{d}{d x} \left(3 x^{3} - 3 x^{2} - 2 x + 2\right) = 9 x^{2} - 6 x - 2 \]Differentiate.✓ Proved
- \[ \left. 9 x^{2} - 6 x - 2 \right|_{\substack{ x=4 }} = 118 \]The slope at the point.✓ Proved
- \[ 118 x - 334 \]Point-slope form, then simplify.✓ Proved
Answer \( y = 118 x - 334 \)
Lines: 4 proved, 1 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.
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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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: passqwen3.6:27b-mlx: fail (error) — The solution claims to use point-slope form in step 5, but the resulting equation y = 118x - 334 does not pass through the point of tangency (4, 138). The correct y-intercept should be 138 - 118*4 = -334, wait, 138 - 472 = -334. Let me re-calculate. 118 * 4 = 472. 138 - 472 = -334. The equation is actually correct. Why did I think it was wrong? 118*4 = 472. 472 - 334 = 138. Yes, it is correct. The algebra is correct. The sentence in step 5 is vague but the result is correct. Is there any other error? Step 1 is correct. Step 2 is correct. Step 3 is correct. Step 4 is correct. Step 5 is correct. The verdict should be pass.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution claims to use point-slope form in step 5, but the resulting equation y = 118x - 334 does not pass through the point of tangency (4, 138). The correct y-intercept should be 138 - 118*4 = -334, wait, 138 - 472 = -334. Let me re-calculate. 118 * 4 = 472. 138 - 472 = -334. The equation is actually correct. Why did I think it was wrong? 118*4 = 472. 472 - 334 = 138. Yes, it is correct. The algebra is correct. The sentence in step 5 is vague but the result is correct. Is there any other error? Step 1 is correct. Step 2 is correct. Step 3 is correct. Step 4 is correct. Step 5 is correct. The verdict should be pass.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-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
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 2026-09-28 with SymPy 1.14.0.