Question 22296: The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Original question and givens

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=1, B=10, C=401, a=-22, b=23 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

• G001 — Use the whole real polynomial f(t)=A*t^2+B*t+C with A=1, B=10, C=401, a=-22, b=23 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-mean-value-theorem

Step-by-step derivation

source operation=mvt-secant-slope;original snapshot and mathematics retained

explicit ordered source output fields=[value_at_a, value_at_b, interval_length, secant_slope, certificate]

ORIGINAL WHOLE f(t)=(1)t^2+(10)t+(401);a=-22<b=23;all endpoint[Ax,Axx,Bx,Axx+Bx,Axx+Bx+C] rows=[[-22, 484, -220, 264, 665], [23, 529, 230, 759, 1160]]

English source ALL endpoint intermediate SAFE guards passed;polynomial continuous on CLOSED[a,b] and differentiable on OPEN(a,b),not inferred from finite samples

MVT endpointa ORIGINAL A times x := mul(1, -22)

MVT endpointa ORIGINAL A times x = -22

MVT endpointa A*x times x := mul(-22, -22)

MVT endpointa A*x times x = 484

MVT endpointa ORIGINAL B times x := mul(10, -22)

MVT endpointa ORIGINAL B times x = -220

MVT endpointa quadratic plus linear := sum(484, -220)

MVT endpointa quadratic plus linear = 264

MVT endpointa plus ORIGINAL constant := sum(264, 401)

MVT endpointa plus ORIGINAL constant = 665

MVT endpointb ORIGINAL A times x := mul(1, 23)

MVT endpointb ORIGINAL A times x = 23

MVT endpointb A*x times x := mul(23, 23)

MVT endpointb A*x times x = 529

MVT endpointb ORIGINAL B times x := mul(10, 23)

MVT endpointb ORIGINAL B times x = 230

MVT endpointb quadratic plus linear := sum(529, 230)

MVT endpointb quadratic plus linear = 759

MVT endpointb plus ORIGINAL constant := sum(759, 401)

MVT endpointb plus ORIGINAL constant = 1160

MVT ORIGINAL endpoint f(b) minus f(a) := sub(1160, 665)

MVT ORIGINAL endpoint f(b) minus f(a) = 495

MVT ORIGINAL ordered interval length := sub(23, -22)

MVT ORIGINAL ordered interval length = 45

MVT exact secant slope := div(495, 45)

MVT exact secant slope = 11

Reverse checks

• S015 · premise_and_result_audit — Recompute MVT exact secant slope with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute MVT ORIGINAL ordered interval length with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute MVT ORIGINAL endpoint f(b) minus f(a) with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute MVT endpointb plus ORIGINAL constant with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute MVT endpointb quadratic plus linear with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute MVT endpointb ORIGINAL B times x with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute MVT endpointb A*x times x with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute MVT endpointb ORIGINAL A times x with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute MVT endpointa plus ORIGINAL constant with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute MVT endpointa quadratic plus linear with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute MVT endpointa ORIGINAL B times x with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute MVT endpointa A*x times x with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute MVT endpointa ORIGINAL A times x with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(665, 1160, 45, 11, exact_endpoint_secant_slope)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.