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Tangent lines

Problem 2.780 · medium Video

Find an equation of the tangent line to \( \displaystyle y = - x^{2} + 3 x - 4 + \frac{1}{x} \) at \( \displaystyle x = 1 \).

The tangent line passes through the point (a, f(a)) and has slope f'(a).
Reviewed
\[ \left. - x^{2} + 3 x - 4 + 1 \frac{1}{x} \right|_{\substack{ x=1 }} = -1 \]
The point of tangency.✓ Proved
\[ \frac{d}{d x} \left(- x^{2} + 3 x - 4 + \frac{1}{x}\right) = - 2 x + 3 - \frac{1}{x^{2}} \]
Differentiate.✓ Proved
\[ \left. - 2 x + 3 - 1 \frac{1}{x^{2}} \right|_{\substack{ x=1 }} = 0 \]
The slope at the point.✓ Proved
\[ -1 \]
Point-slope form, then simplify.

Answer \( y = -1 \)

The graph of y = -x² + 3x - 4 + 1/x near x = 1. A secant line through the point (1, -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, 1 reviewed, 1 not checked. 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	claude-sonnet-5-5	claude-sonnet-5-5: This is the standard definition of the tangent line at x = a: it passes through (a, f(a)) with slope f'(a). It is a sentence, not an algebraic equivalence.
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 solution claims the final equation is y = -1, but the tangent line with slope 0 passing through (1, -1) is indeed y = -1. However, step 5 is labeled as an equation that was 'proved' by a CAS, yet it represents the final answer derivation rather than a standard algebraic identity check. More critically, the prompt asks for an 'equation of the tangent line', and while y = -1 is correct, the step description 'Point-slope form, then simplify' is vague and the CAS check '-1, -1' is meaningless in this context (it likely checked f(1) == -1 again, not the line equation). But looking closer, the main issue is that step 5 is presented as a 'proved equation' which is just the constant -1. A CAS cannot 'prove' the equation of a line from the previous steps without explicit construction. The step is logically disconnected. However, the result is correct. Let's look for a harder error. Wait, let's re-read carefully. Point: (1, -1). Slope: 0. Equation: y - (-1) = 0(x - 1) => y + 1 = 0 => y = -1. The stated answer is Eq(y, -1). Step 5 says '-1, -1'. This looks like the CAS checked if the y-coordinate of the point is -1, or perhaps it checked if the slope is 0? No, it just lists -1. Actually, the biggest issue is that Step 5 is not a valid derivation step marked as 'proved'. It's just the answer. But is it 'wrong'? Let's look at Step 1. 'The tangent line passes through the point (a, f(a)) and has slope f'(a).' This is correct. Step 2: Point is (1, -1). Correct. Step 3: Derivative is -2x + 3 - 1/x^2. Correct. Step 4: Slope at x=1 is 0. Correct. Step 5: '-1, -1'. This is likely the CAS checking that the y-value is -1. It does NOT show the construction of the line equation y = -1. The solution jumps from 'Slope is 0' and 'Point is (1, -1)' to 'Answer is -1' without explicitly stating the line equation y = -1 in the steps, although the final answer is correct. Is this an error? The solution fails to explicitly state the equation of the line in the steps, only providing the y-value. The 'Stated answer' is Eq(y, -1). Step 5 is just '-1, -1'. This is misleading because it doesn't show the equation y = -1. It just shows the number -1. A student might think the answer is just the number -1, not the equation y = -1. However, usually 'Find an equation' implies the full equation. The step 5 is insufficient to derive the equation. It's a 'misleading' or 'error' in presentation. Let's look at the 'proved' tag. Step 5 is marked proved. What equation was proved? '-1 == -1'? That's trivial and irrelevant to finding the line equation. The step does not demonstrate the formation of the line equation. Therefore, the solution is incomplete/misleading. Verdict: fail. Severity: misleading. The final step does not construct the equation of the line, only re-verifying the y-coordinate, leaving the student without the explicit equation y = -1 in the derivation.

Senior review claude-sonnet-5-5, 2026-10-12: pass — f(1) = -1 and f'(1) = -2+3-1 = 0, so the tangent line is y = -1, matching the stated answer. Line 5 compresses point-slope form (y = -1 + 0*(x-1)) and simplification into one step, which is a minor presentational point, not a defect.

qwen3.6:27b-mlx: dismiss — The objection finds no actual mathematical error and concedes that every value is correct (point (1,-1), f'(x) = -2x+3-1/x^2, slope 0, tangent y = -1). The complaint about the vague label and the displayed '-1 = -1' is cosmetic, and the final line is the right line y = -1.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-12 — f(1) = -1 and f'(1) = -2+3-1 = 0, so the tangent line is y = -1, matching the stated answer. Line 5 compresses point-slope form (y = -1 + 0*(x-1)) and simplification into one step, which is a minor presentational point, not a defect.
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution claims the final equation is y = -1, but the tangent line with slope 0 passing through (1, -1) is indeed y = -1. However, step 5 is labeled as an equation that was 'proved' by a CAS, yet it represents the final answer derivation rather than a standard algebraic identity check. More critically, the prompt asks for an 'equation of the tangent line', and while y = -1 is correct, the step description 'Point-slope form, then simplify' is vague and the CAS check '-1, -1' is meaningless in this context (it likely checked f(1) == -1 again, not the line equation). But looking closer, the main issue is that step 5 is presented as a 'proved equation' which is just the constant -1. A CAS cannot 'prove' the equation of a line from the previous steps without explicit construction. The step is logically disconnected. However, the result is correct. Let's look for a harder error. Wait, let's re-read carefully. Point: (1, -1). Slope: 0. Equation: y - (-1) = 0(x - 1) => y + 1 = 0 => y = -1. The stated answer is Eq(y, -1). Step 5 says '-1, -1'. This looks like the CAS checked if the y-coordinate of the point is -1, or perhaps it checked if the slope is 0? No, it just lists -1. Actually, the biggest issue is that Step 5 is not a valid derivation step marked as 'proved'. It's just the answer. But is it 'wrong'? Let's look at Step 1. 'The tangent line passes through the point (a, f(a)) and has slope f'(a).' This is correct. Step 2: Point is (1, -1). Correct. Step 3: Derivative is -2x + 3 - 1/x^2. Correct. Step 4: Slope at x=1 is 0. Correct. Step 5: '-1, -1'. This is likely the CAS checking that the y-value is -1. It does NOT show the construction of the line equation y = -1. The solution jumps from 'Slope is 0' and 'Point is (1, -1)' to 'Answer is -1' without explicitly stating the line equation y = -1 in the steps, although the final answer is correct. Is this an error? The solution fails to explicitly state the equation of the line in the steps, only providing the y-value. The 'Stated answer' is Eq(y, -1). Step 5 is just '-1, -1'. This is misleading because it doesn't show the equation y = -1. It just shows the number -1. A student might think the answer is just the number -1, not the equation y = -1. However, usually 'Find an equation' implies the full equation. The step 5 is insufficient to derive the equation. It's a 'misleading' or 'error' in presentation. Let's look at the 'proved' tag. Step 5 is marked proved. What equation was proved? '-1 == -1'? That's trivial and irrelevant to finding the line equation. The step does not demonstrate the formation of the line equation. Therefore, the solution is incomplete/misleading. Verdict: fail. Severity: misleading. The final step does not construct the equation of the line, only re-verifying the y-coordinate, leaving the student without the explicit equation y = -1 in the derivation.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The final equation is incorrect. The tangent line has slope 0 and passes through (1, -1), so the equation should be y = -1. The solution lists '-1, -1' which is ambiguous and does not explicitly state the equation y = -1, but more importantly, step 5 is labeled as an equation that was 'proved' by a CAS, yet '-1, -1' is not a standard equation format and likely represents a tuple or a failed simplification step rather than the derived line equation y = -1. A student would be confused by the output '-1, -1' as the final answer.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The final equation is incorrect. The tangent line has slope 0 and passes through (1, -1), so the equation is y = -1. The solution lists '-1, -1' which is not a valid equation format and fails to equate y to the constant.
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.