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R0056-28621 · q-20261005-advance-20261006-r03-rate-of-change-0071
On the full real domain, f(x)=(8)x+(53). For distinct inputs 0 and 21, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.
G001 — On the full real domain, f(x)=(8)x+(53). For distinct inputs 0 and 21, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.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.
Before
source roles = [qf-rate-affine-average, affine_average]
After
derived = [exact_average_rate_fraction]
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.
Rule: R56_source_answer_role_and_EXACT_family_interface · Uses: G001
Before
Rate ORIGINAL x1 minus x0 := sub(21, 0)
After
Rate ORIGINAL x1 minus x0 = 21
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
Rate affine ORIGINAL m times x0 := mul(8, 0)
After
Rate affine ORIGINAL m times x0 = 0
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Rate affine ORIGINAL endpoint0 plus b := sum(0, 53)
After
Rate affine ORIGINAL endpoint0 plus b = 53
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
Rate affine ORIGINAL m times x1 := mul(8, 21)
After
Rate affine ORIGINAL m times x1 = 168
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
Rate affine ORIGINAL endpoint1 plus b := sum(168, 53)
After
Rate affine ORIGINAL endpoint1 plus b = 221
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
source roles = [8, 53, 0, 21]
After
derived = [[53, 221], 8]
Conditions: none
Substitute both endpoints in the WHOLE law f(x)=mx+b.The intercept cancels,so dy=m*dx;m=0 is allowed.This is an endpoint secant computation,not a substituted derivative.
Rule: Rate_WHOLE_affine_endpoint_subtraction · Uses: G001, S006
Before
Rate ORIGINAL y1 minus y0 := sub(221, 53)
After
Rate ORIGINAL y1 minus y0 = 168
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
Rate ORIGINAL dy over dx := div(168, 21)
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.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
source roles = [0, 21, 53, 221, 21, 168]
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.
Rule: Rate_oriented_quotient_reverse_BOTH_differences · Uses: G001, S009
S010 · 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.S009 · premise_and_result_audit — Recompute Rate ORIGINAL dy over dx with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute Rate ORIGINAL y1 minus y0 with exact rational arithmetic, not rounded decimals.S007 · 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.S006 · premise_and_result_audit — Recompute Rate affine ORIGINAL endpoint1 plus b with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute Rate affine ORIGINAL m times x1 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute Rate affine ORIGINAL endpoint0 plus b with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute Rate affine ORIGINAL m times x0 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.8/1
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
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