# Question 28666: Average Rate between Explicit Function Endpoints [Return to catalogue page 144](https://ocme.evemisslab.com/en/questions/page/144/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261005-advance-20261006-r03-rate-of-change-foundations/#q-20261005-advance-20261006-r03-rate-of-change-0016) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0056-28666 · q-20261005-advance-20261006-r03-rate-of-change-0016 ## Original question and givens For the same specified function, the permitted endpoints are (2, 21) and (0, 37), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers. - G001 — For the same specified function, the permitted endpoints are (2, 21) and (0, 37), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers. ## 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-rate-of-change](https://ocme.evemisslab.com/en/mko/mko-rate-of-change/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-rate-endpoint-quotient, endpoint_quotient] After derived = [exact_oriented_difference_quotient] Conditions: none Use the declared scalar/set/Yes-No/fraction answer role;integer fractions retain/1.Do not add an undeclared domain or change original bilingual data,formulas or questions. Step identifiers and dependenciesRule: R56_source_answer_role_and_EXACT_family_interface · Uses: G001 S002 · calculationBefore Rate ORIGINAL x1 minus x0 := sub(0, 2) After Rate ORIGINAL x1 minus x0 = -2 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Rate ORIGINAL y1 minus y0 := sub(37, 21) After Rate ORIGINAL y1 minus y0 = 16 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Rate ORIGINAL dy over dx := div(16, -2) After Rate ORIGINAL dy over dx = -8 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore source roles = [2, 0, 21, 37, -2, 16] After derived = [-8/1] Conditions: none Keep the SAME endpoint orientation:(y1-y0)/(x1-x0).Both differences may reverse together,not just one.The result may be negative or0;normalize a positive denominator and retain/1.This finite endpoint rate is not a general derivative certificate. Step identifiers and dependenciesRule: Rate_oriented_quotient_reverse_BOTH_differences · Uses: G001, S004 ## Reverse checks - S005 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance. - S004 · premise_and_result_audit — Recompute Rate ORIGINAL dy over dx with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Rate ORIGINAL y1 minus y0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Rate ORIGINAL x1 minus x0 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance. ## Result -8/1 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 536a36082f84ba344af9e06f567538a77a682f70ca43537cfc6cad3b8c8e6a4e - candidate record SHA-256: 3c68ec7c6bf9a46eff488da03058a6f2d58e012bba2e16fa6ba8b309e75e0fee - semantic review receipt SHA-256: 8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8 - question-first witness SHA-256: 3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759f - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261005-advance-20261006-r03-rate-of-change-0016.json)