OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 28108: An Affine Integrand Is Integrated Exactly on Any Qualified Partition

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

Return to catalogue page 141

← Question derivation index · Original question batch

R0055-28108 · q-20261015-pulse-20261119-r02-0058

Original question and givens

Use f(x)=m*x+b with m=-3, b=8 and strictly increasing nodes=[0, 1, 10]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).

  • G001 — Use f(x)=m*x+b with m=-3, b=8 and strictly increasing nodes=[0, 1, 10]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).

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.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-mod50-trap-affine-exactness, trap-affine-exactness]

After

derived = [[slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope]]

Conditions: none

List this scalar/label/set/point answer role or mode-specific ordered fields,not an imposed universal tuple.Retain legacy domain/underscore operation and integer fractions with/1;original bilingual snapshots stay unchanged.

Step identifiers and dependencies

Rule: source_answer_roles_or_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

Trapezoid affine ORIGINAL value product0 := mul(-3, 0)

After

Trapezoid affine ORIGINAL value product0 = 0

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Trapezoid affine ORIGINAL value plus intercept0 := sum(0, 8)

After

Trapezoid affine ORIGINAL value plus intercept0 = 8

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Trapezoid affine ORIGINAL value product1 := mul(-3, 1)

After

Trapezoid affine ORIGINAL value product1 = -3

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Trapezoid affine ORIGINAL value plus intercept1 := sum(-3, 8)

After

Trapezoid affine ORIGINAL value plus intercept1 = 5

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Trapezoid affine ORIGINAL value product2 := mul(-3, 10)

After

Trapezoid affine ORIGINAL value product2 = -30

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

Trapezoid affine ORIGINAL value plus intercept2 := sum(-30, 8)

After

Trapezoid affine ORIGINAL value plus intercept2 = -22

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Trapezoid affine partition ORIGINAL width0 := sub(1, 0)

After

Trapezoid affine partition ORIGINAL width0 = 1

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Trapezoid affine partition ORIGINAL width1 := sub(10, 1)

After

Trapezoid affine partition ORIGINAL width1 = 9

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Trapezoid affine partition ORIGINAL width over2 interval0 := div(1, 2)

After

Trapezoid affine partition ORIGINAL width over2 interval0 = 1/2

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

Trapezoid affine partition ORIGINAL endpoint sum interval0 := sum(8, 5)

After

Trapezoid affine partition ORIGINAL endpoint sum interval0 = 13

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

Trapezoid affine partition ORIGINAL individual trapezoid interval0 := mul(1/2, 13)

After

Trapezoid affine partition ORIGINAL individual trapezoid interval0 = 13/2

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · implies

Before

source roles = [Trapezoid affine partition, 0, 0, 1, 8, 5]

After

derived = [1, 1/2, 13, 13/2]

Conditions: none

Use the own width and aligned endpoints of this ORIGINAL interval times the endpoint average.Widths may differ and values may be negative;do not sort or replace all widths with the first.

Step identifiers and dependencies

Rule: Trapezoid_EACH_original_interval_own_width_and_aligned_endpoints · Uses: G001, S012

S014 · calculation

Before

Trapezoid affine partition ORIGINAL width over2 interval1 := div(9, 2)

After

Trapezoid affine partition ORIGINAL width over2 interval1 = 9/2

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

Trapezoid affine partition ORIGINAL endpoint sum interval1 := sum(5, -22)

After

Trapezoid affine partition ORIGINAL endpoint sum interval1 = -17

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Trapezoid affine partition ORIGINAL individual trapezoid interval1 := mul(9/2, -17)

After

Trapezoid affine partition ORIGINAL individual trapezoid interval1 = -153/2

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · implies

Before

source roles = [Trapezoid affine partition, 1, 1, 10, 5, -22]

After

derived = [9, 9/2, -17, -153/2]

Conditions: none

Use the own width and aligned endpoints of this ORIGINAL interval times the endpoint average.Widths may differ and values may be negative;do not sort or replace all widths with the first.

Step identifiers and dependencies

Rule: Trapezoid_EACH_original_interval_own_width_and_aligned_endpoints · Uses: G001, S016

S018 · calculation

Before

Trapezoid affine partition ORIGINAL trapezoid running total0 := sum(0, 13/2)

After

Trapezoid affine partition ORIGINAL trapezoid running total0 = 13/2

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

Trapezoid affine partition ORIGINAL trapezoid running total1 := sum(13/2, -153/2)

After

Trapezoid affine partition ORIGINAL trapezoid running total1 = -70

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

source roles = [Trapezoid affine partition, [[0, 1], [1, 1], [10, 1]], [[8, 1], [5, 1], [-22, 1]]]

After

derived = [[0, 1, 10], [8, 5, -22], [1, 9], [13/2, -153/2], -70]

Conditions: none

A complete partition is STRICTLY increasing in original order,with n+1 aligned nodes/values for n intervals;the24-node cap is not the vector8 cap.The table determines this sum,not a unique integral without a full integrand.

Step identifiers and dependencies

Rule: Trapezoid_COMPLETE_partition_n_intervals_nplus1_nodes_no_unique_integrand · Uses: G001, S019

S021 · calculation

Before

Trapezoid affine ORIGINAL slope over2 := div(-3, 2)

After

Trapezoid affine ORIGINAL slope over2 = -3/2

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

Trapezoid affine ORIGINAL right square := mul(10, 10)

After

Trapezoid affine ORIGINAL right square = 100

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

Trapezoid affine ORIGINAL left square := mul(0, 0)

After

Trapezoid affine ORIGINAL left square = 0

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

Trapezoid affine ORIGINAL square difference := sub(100, 0)

After

Trapezoid affine ORIGINAL square difference = 100

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

Trapezoid affine ORIGINAL quadratic integral part := mul(-3/2, 100)

After

Trapezoid affine ORIGINAL quadratic integral part = -150

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

Trapezoid affine ORIGINAL whole width := sub(10, 0)

After

Trapezoid affine ORIGINAL whole width = 10

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

Trapezoid affine ORIGINAL intercept integral part := mul(8, 10)

After

Trapezoid affine ORIGINAL intercept integral part = 80

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

Trapezoid affine ORIGINAL exact integral := sum(-150, 80)

After

Trapezoid affine ORIGINAL exact integral = -70

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · implies

Before

source roles = [-3, 8, [0, 1, 10], [8, 5, -22]]

After

derived = [-70, -70, Yes]

Conditions: none

The WHOLE affine law makes each trapezoid equal to the antiderivative difference over the same interval.Internal endpoints telescope,hence exactness on any qualified partition;not for arbitrary functions.

Step identifiers and dependencies

Rule: Trapezoid_WHOLE_affine_exactness_each_interval_antiderivative_telescopes · Uses: G001, S028

Reverse checks

  • S029 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S028 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL exact integral with exact rational arithmetic, not rounded decimals.
  • S027 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL intercept integral part with exact rational arithmetic, not rounded decimals.
  • S026 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL whole width with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL quadratic integral part with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL square difference with exact rational arithmetic, not rounded decimals.
  • S023 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL left square with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL right square with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL slope over2 with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S019 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL trapezoid running total1 with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL trapezoid running total0 with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S016 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL individual trapezoid interval1 with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL endpoint sum interval1 with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL width over2 interval1 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S012 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL individual trapezoid interval0 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL endpoint sum interval0 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL width over2 interval0 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL width1 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL width0 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL value plus intercept2 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL value product2 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL value plus intercept1 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL value product1 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL value plus intercept0 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL value product0 with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.

Result

(-3, 8, [0, 1, 10], [8, 5, -22], -70, -70, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)

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: ba27a02f94a6a84fd76b53dfd343d613a7c4f8d3a92f78103681998a35d89be8
  • candidate record SHA-256: 9627cbdb5b1036b9c44bb1dd8b954301e34b7fb76ae917f0cd7723ffe84f2d9d
  • semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683
  • question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON