Average rate of change and secant lines
Problem 2.2211 · easy
Find the average rate of change of \( \displaystyle f(x) = x^{2} - 4 x + 2 \) on \( \displaystyle [2, 4] \), and the equation of the secant line through the endpoints.
- \[ \left. x^{2} - 4 x + 2 \right|_{\substack{ x=2 }} = -2 \]The value at the left endpoint.✓ Proved
- \[ \left. x^{2} - 4 x + 2 \right|_{\substack{ x=4 }} = 2 \]The value at the right endpoint.✓ Proved
- \[ 2 \]Change in output over change in input.✓ Proved
- \[ 2 x - 6 \]Point-slope form through the left endpoint.✓ Proved
Answer \( \frac{\Delta f}{\Delta x} = 2,\quad y = 2 x - 6 \)
Every line of this solution was proved by the computer algebra system SymPy. 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | recomputed in floating point |
Reviewers
gpt-oss:20b: fail (error) — Step 3 incorrectly states the average rate of change as "2 = 2" without showing the actual computation (Δf = 2 – (–2) = 4, Δx = 4 – 2 = 2, so Δf/Δx = 2). The conclusion is correct, but the justification is missing, which could mislead a student about how the average rate is derived.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the formula for average rate of change or the calculation of the secant line equation, relying on unproven algebraic identities that do not demonstrate the method. Specifically, line 3 is a tautology that does not show the division (2 - (-2)) / (4 - 2), and line 4 is a tautology that does not show the derivation of the slope-intercept form from the point-slope formula.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-13 — Step 3 incorrectly states the average rate of change as "2 = 2" without showing the actual computation (Δf = 2 – (–2) = 4, Δx = 4 – 2 = 2, so Δf/Δx = 2). The conclusion is correct, but the justification is missing, which could mislead a student about how the average rate is derived.qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution fails to explicitly state the formula for average rate of change or the calculation of the secant line equation, relying on unproven algebraic identities that do not demonstrate the method. Specifically, line 3 is a tautology that does not show the division (2 - (-2)) / (4 - 2), and line 4 is a tautology that does not show the derivation of the slope-intercept form from the point-slope formula.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution lacks the necessary sentences to define the average rate of change formula and the point-slope form, making the derivation opaque. Specifically, it fails to explicitly state the calculation (2 - (-2)) / (4 - 2) = 2, jumping directly to the result.
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/secant_slope, checked 2026-10-13 with SymPy 1.14.0.