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Linear approximation

Problem 3.305 · easy

Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 49 \) to estimate \( \displaystyle f(\frac{487}{10}) \).

The linearization at a is L(x) = f(a) + f'(a)(x − a).
\[ \left. \sqrt{x} \right|_{\substack{ x=49 }} = 7 \]
f(a).✓ Proved
\[ \left. \frac{d}{d x} \sqrt{x} \right|_{\substack{ x=49 }} = \frac{1}{14} \]
f'(a).✓ Proved
\[ \frac{x}{14} + \frac{7}{2} \]
The linearization.
\[ \frac{977}{140} \]
Evaluate at x = \frac{487}{10}.

Answer \( L(\frac{487}{10}) = \frac{977}{140} \approx 6.97857 \)

Lines: 2 proved, 3 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	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
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 estimate is within Taylor's error bound of the true value

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The linearization formula in line 4 is incorrect. The correct linearization is L(x) = 7 + (1/14)(x - 49) = x/14 + 7/2, but the solution presents it as x/14 + 7/2 without showing the derivation from the standard form, and more critically, the constant term 7/2 is correct for the simplified form, but the setup implies a direct substitution error or lack of clarity. Wait, let's re-evaluate. L(x) = 7 + 1/14(x-49) = 7 + x/14 - 49/14 = 7 + x/14 - 3.5 = x/14 + 3.5 = x/14 + 7/2. The equation in line 4 is algebraically correct. However, the sentence in line 1 is generic. The issue is likely that the solution skips the explicit construction of L(x) from the definition in line 1 using the values in 2 and 3. Line 4 just states an identity. It doesn't explicitly link the values from 2 and 3 to the expression in 4. A student might not see how 7 and 1/14 become x/14 + 7/2. But is it an error? The prompt says 'equation... checked'. So the algebra is right. Is the sentence misleading? Line 1 is correct. Line 4 is an identity. It doesn't state 'Therefore L(x) = ...'. It just states an equality. This is a style/clarity issue, but arguably 'misleading' because it doesn't explicitly claim that the RHS is the linearization. However, looking closer at line 4: `x/14 + 7/2 = x/14 + 7/2`. This is a tautology. It does not assert that L(x) equals this expression. It just asserts that an expression equals itself. This fails to connect the previous steps to the final result. It is a logical gap. The solution does not state that L(x) simplifies to x/14 + 7/2. It just shows an identity. This is a 'style' or 'misleading' issue because it doesn't perform the logical step of defining L(x) with the calculated values. Let's look for a harder error. Did I calculate correctly? f(49)=7. f'(49)=1/14. L(x) = 7 + 1/14(x-49). L(48.7) = 7 + 1/14(48.7 - 49) = 7 + 1/14(-0.3) = 7 - 0.3/14 = 7 - 3/140 = 980/140 - 3/140 = 977/140. The math is correct. The issue is purely presentational/logical flow. Line 4 is a tautology that doesn't explicitly define L(x). This is 'style' or 'misleading'. I will call it 'style' as the math is sound, but the step is vacuous.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — The linearization formula in line 4 is incorrect. The correct linearization is L(x) = 7 + (1/14)(x - 49) = x/14 + 7/2, but the solution presents it as x/14 + 7/2 without showing the derivation from the standard form, and more critically, the constant term 7/2 is correct for the simplified form, but the setup implies a direct substitution error or lack of clarity. Wait, let's re-evaluate. L(x) = 7 + 1/14(x-49) = 7 + x/14 - 49/14 = 7 + x/14 - 3.5 = x/14 + 3.5 = x/14 + 7/2. The equation in line 4 is algebraically correct. However, the sentence in line 1 is generic. The issue is likely that the solution skips the explicit construction of L(x) from the definition in line 1 using the values in 2 and 3. Line 4 just states an identity. It doesn't explicitly link the values from 2 and 3 to the expression in 4. A student might not see how 7 and 1/14 become x/14 + 7/2. But is it an error? The prompt says 'equation... checked'. So the algebra is right. Is the sentence misleading? Line 1 is correct. Line 4 is an identity. It doesn't state 'Therefore L(x) = ...'. It just states an equality. This is a style/clarity issue, but arguably 'misleading' because it doesn't explicitly claim that the RHS is the linearization. However, looking closer at line 4: `x/14 + 7/2 = x/14 + 7/2`. This is a tautology. It does not assert that L(x) equals this expression. It just asserts that an expression equals itself. This fails to connect the previous steps to the final result. It is a logical gap. The solution does not state that L(x) simplifies to x/14 + 7/2. It just shows an identity. This is a 'style' or 'misleading' issue because it doesn't perform the logical step of defining L(x) with the calculated values. Let's look for a harder error. Did I calculate correctly? f(49)=7. f'(49)=1/14. L(x) = 7 + 1/14(x-49). L(48.7) = 7 + 1/14(48.7 - 49) = 7 + 1/14(-0.3) = 7 - 0.3/14 = 7 - 3/140 = 980/140 - 3/140 = 977/140. The math is correct. The issue is purely presentational/logical flow. Line 4 is a tautology that doesn't explicitly define L(x). This is 'style' or 'misleading'. I will call it 'style' as the math is sound, but the step is vacuous.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the linearization formula, computes the derivative and function value at the point of tangency, constructs the linear approximation, and evaluates it at the specified point. All steps are mathematically sound.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the formula for linearization, computes the function value and derivative at the point of tangency, constructs the linear approximation, and evaluates it at the specified point. The arithmetic is correct.
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/linear_approximation, checked 2026-10-13 with SymPy 1.14.0.