Nonuniform Composite Trapezoids, Affine Exactness and Signed Reference Error
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batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 3] and aligned endpoint values=[0, 1, 3]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 3], [0, 1, 3], 2, 3, [1, 2], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 3], [0, 1, 3], 2, 3, [1, 2], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
2. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 4] and aligned endpoint values=[1, 2, 4]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 4], [1, 2, 4], 2, 3, [1, 3], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 4], [1, 2, 4], 2, 3, [1, 3], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
3. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 5] and aligned endpoint values=[2, 3, 5]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 5], [2, 3, 5], 2, 3, [1, 4], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 5], [2, 3, 5], 2, 3, [1, 4], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
4. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 6] and aligned endpoint values=[3, 4, 6]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 6], [3, 4, 6], 2, 3, [1, 5], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 6], [3, 4, 6], 2, 3, [1, 5], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
5. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 7] and aligned endpoint values=[4, 5, 7]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 7], [4, 5, 7], 2, 3, [1, 6], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 7], [4, 5, 7], 2, 3, [1, 6], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
6. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 8] and aligned endpoint values=[5, 6, 8]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 8], [5, 6, 8], 2, 3, [1, 7], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 8], [5, 6, 8], 2, 3, [1, 7], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
7. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 9] and aligned endpoint values=[6, 7, 9]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 9], [6, 7, 9], 2, 3, [1, 8], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 9], [6, 7, 9], 2, 3, [1, 8], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
8. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 10] and aligned endpoint values=[7, 8, 10]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 10], [7, 8, 10], 2, 3, [1, 9], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 10], [7, 8, 10], 2, 3, [1, 9], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
9. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 11] and aligned endpoint values=[8, 9, 11]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 11], [8, 9, 11], 2, 3, [1, 10], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 11], [8, 9, 11], 2, 3, [1, 10], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
10. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 12] and aligned endpoint values=[9, 10, 12]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 12], [9, 10, 12], 2, 3, [1, 11], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 12], [9, 10, 12], 2, 3, [1, 11], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
11. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 13] and aligned endpoint values=[10, 11, 13]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 13], [10, 11, 13], 2, 3, [1, 12], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 13], [10, 11, 13], 2, 3, [1, 12], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
12. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 14] and aligned endpoint values=[11, 12, 14]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 14], [11, 12, 14], 2, 3, [1, 13], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 14], [11, 12, 14], 2, 3, [1, 13], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
13. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 15] and aligned endpoint values=[12, 13, 15]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 15], [12, 13, 15], 2, 3, [1, 14], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 15], [12, 13, 15], 2, 3, [1, 14], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
14. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 16] and aligned endpoint values=[13, 14, 16]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 16], [13, 14, 16], 2, 3, [1, 15], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 16], [13, 14, 16], 2, 3, [1, 15], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
15. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 17] and aligned endpoint values=[14, 15, 17]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 17], [14, 15, 17], 2, 3, [1, 16], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 17], [14, 15, 17], 2, 3, [1, 16], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
16. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 18] and aligned endpoint values=[15, 16, 18]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 18], [15, 16, 18], 2, 3, [1, 17], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 18], [15, 16, 18], 2, 3, [1, 17], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
17. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 19] and aligned endpoint values=[16, 17, 19]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 19], [16, 17, 19], 2, 3, [1, 18], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 19], [16, 17, 19], 2, 3, [1, 18], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
18. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 20] and aligned endpoint values=[17, 18, 20]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 20], [17, 18, 20], 2, 3, [1, 19], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 20], [17, 18, 20], 2, 3, [1, 19], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
19. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 21] and aligned endpoint values=[18, 19, 21]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 21], [18, 19, 21], 2, 3, [1, 20], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 21], [18, 19, 21], 2, 3, [1, 20], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
20. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 22] and aligned endpoint values=[19, 20, 22]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 22], [19, 20, 22], 2, 3, [1, 21], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 22], [19, 20, 22], 2, 3, [1, 21], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
21. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 23] and aligned endpoint values=[20, 21, 23]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 23], [20, 21, 23], 2, 3, [1, 22], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 23], [20, 21, 23], 2, 3, [1, 22], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
22. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 24] and aligned endpoint values=[21, 22, 24]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 24], [21, 22, 24], 2, 3, [1, 23], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 24], [21, 22, 24], 2, 3, [1, 23], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
23. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 25] and aligned endpoint values=[22, 23, 25]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 25], [22, 23, 25], 2, 3, [1, 24], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 25], [22, 23, 25], 2, 3, [1, 24], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
24. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 26] and aligned endpoint values=[23, 24, 26]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 26], [23, 24, 26], 2, 3, [1, 25], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 26], [23, 24, 26], 2, 3, [1, 25], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
25. n Subintervals Require n+1 Aligned Strictly Increasing Nodes
Use ordered nodes=[0, 1, 27] and aligned endpoint values=[24, 25, 27]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).
View answer and explanation
([0, 1, 27], [24, 25, 27], 2, 3, [1, 26], Yes, n_subintervals_have_n_plus_one_nodes)
Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 27], [24, 25, 27], 2, 3, [1, 26], Yes, n_subintervals_have_n_plus_one_nodes).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-partition-qualification · candidate
26. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 3], aligned endpoint values=[0, 1, 3]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 3], [0, 1, 3], [1/2, 4], 9/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 3], [0, 1, 3], [1/2, 4], 9/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
27. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 4], aligned endpoint values=[1, 2, 4]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 4], [1, 2, 4], [3/2, 9], 21/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 4], [1, 2, 4], [3/2, 9], 21/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
28. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 5], aligned endpoint values=[2, 3, 5]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 5], [2, 3, 5], [5/2, 16], 37/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 5], [2, 3, 5], [5/2, 16], 37/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
29. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 6], aligned endpoint values=[3, 4, 6]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 6], [3, 4, 6], [7/2, 25], 57/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 6], [3, 4, 6], [7/2, 25], 57/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
30. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 7], aligned endpoint values=[4, 5, 7]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 7], [4, 5, 7], [9/2, 36], 81/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 7], [4, 5, 7], [9/2, 36], 81/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
31. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 8], aligned endpoint values=[5, 6, 8]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 8], [5, 6, 8], [11/2, 49], 109/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 8], [5, 6, 8], [11/2, 49], 109/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
32. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 9], aligned endpoint values=[6, 7, 9]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 9], [6, 7, 9], [13/2, 64], 141/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 9], [6, 7, 9], [13/2, 64], 141/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
33. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 10], aligned endpoint values=[7, 8, 10]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 10], [7, 8, 10], [15/2, 81], 177/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 10], [7, 8, 10], [15/2, 81], 177/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
34. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 11], aligned endpoint values=[8, 9, 11]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 11], [8, 9, 11], [17/2, 100], 217/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 11], [8, 9, 11], [17/2, 100], 217/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
35. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 12], aligned endpoint values=[9, 10, 12]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 12], [9, 10, 12], [19/2, 121], 261/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 12], [9, 10, 12], [19/2, 121], 261/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
36. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 13], aligned endpoint values=[10, 11, 13]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 13], [10, 11, 13], [21/2, 144], 309/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 13], [10, 11, 13], [21/2, 144], 309/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
37. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 14], aligned endpoint values=[11, 12, 14]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 14], [11, 12, 14], [23/2, 169], 361/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 14], [11, 12, 14], [23/2, 169], 361/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
38. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 15], aligned endpoint values=[12, 13, 15]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 15], [12, 13, 15], [25/2, 196], 417/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 15], [12, 13, 15], [25/2, 196], 417/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
39. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 16], aligned endpoint values=[13, 14, 16]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 16], [13, 14, 16], [27/2, 225], 477/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 16], [13, 14, 16], [27/2, 225], 477/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
40. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 17], aligned endpoint values=[14, 15, 17]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 17], [14, 15, 17], [29/2, 256], 541/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 17], [14, 15, 17], [29/2, 256], 541/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
41. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 18], aligned endpoint values=[15, 16, 18]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 18], [15, 16, 18], [31/2, 289], 609/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 18], [15, 16, 18], [31/2, 289], 609/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
42. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 19], aligned endpoint values=[16, 17, 19]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 19], [16, 17, 19], [33/2, 324], 681/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 19], [16, 17, 19], [33/2, 324], 681/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
43. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 20], aligned endpoint values=[17, 18, 20]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 20], [17, 18, 20], [35/2, 361], 757/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 20], [17, 18, 20], [35/2, 361], 757/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
44. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 21], aligned endpoint values=[18, 19, 21]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 21], [18, 19, 21], [37/2, 400], 837/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 21], [18, 19, 21], [37/2, 400], 837/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
45. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 22], aligned endpoint values=[19, 20, 22]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 22], [19, 20, 22], [39/2, 441], 921/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 22], [19, 20, 22], [39/2, 441], 921/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
46. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 23], aligned endpoint values=[20, 21, 23]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 23], [20, 21, 23], [41/2, 484], 1009/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 23], [20, 21, 23], [41/2, 484], 1009/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
47. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 24], aligned endpoint values=[21, 22, 24]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 24], [21, 22, 24], [43/2, 529], 1101/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 24], [21, 22, 24], [43/2, 529], 1101/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
48. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 25], aligned endpoint values=[22, 23, 25]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 25], [22, 23, 25], [45/2, 576], 1197/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 25], [22, 23, 25], [45/2, 576], 1197/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
49. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 26], aligned endpoint values=[23, 24, 26]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 26], [23, 24, 26], [47/2, 625], 1297/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 26], [23, 24, 26], [47/2, 625], 1297/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
50. Each Trapezoid Uses Its Own Width and Endpoint Average
Use strictly increasing nodes=[0, 1, 27], aligned endpoint values=[24, 25, 27]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).
View answer and explanation
([0, 1, 27], [24, 25, 27], [49/2, 676], 1401/2, each_subinterval_uses_its_own_width)
For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 27], [24, 25, 27], [49/2, 676], 1401/2, each_subinterval_uses_its_own_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-composite-value · candidate
51. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=-3, b=1 and strictly increasing nodes=[0, 1, 3]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(-3, 1, [0, 1, 3], [1, -2, -8], -21/2, -21/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-3, 1, [0, 1, 3], [1, -2, -8], -21/2, -21/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
52. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=-2, b=2 and strictly increasing nodes=[0, 1, 4]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(-2, 2, [0, 1, 4], [2, 0, -6], -8, -8, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-2, 2, [0, 1, 4], [2, 0, -6], -8, -8, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
53. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=-1, b=3 and strictly increasing nodes=[0, 1, 5]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(-1, 3, [0, 1, 5], [3, 2, -2], 5/2, 5/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-1, 3, [0, 1, 5], [3, 2, -2], 5/2, 5/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
54. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=2, b=4 and strictly increasing nodes=[0, 1, 6]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(2, 4, [0, 1, 6], [4, 6, 16], 60, 60, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (2, 4, [0, 1, 6], [4, 6, 16], 60, 60, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
55. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=1, b=5 and strictly increasing nodes=[0, 1, 7]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(1, 5, [0, 1, 7], [5, 6, 12], 119/2, 119/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (1, 5, [0, 1, 7], [5, 6, 12], 119/2, 119/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
56. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=2, b=6 and strictly increasing nodes=[0, 1, 8]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(2, 6, [0, 1, 8], [6, 8, 22], 112, 112, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (2, 6, [0, 1, 8], [6, 8, 22], 112, 112, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
57. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=3, b=7 and strictly increasing nodes=[0, 1, 9]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(3, 7, [0, 1, 9], [7, 10, 34], 369/2, 369/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (3, 7, [0, 1, 9], [7, 10, 34], 369/2, 369/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
58. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
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).
View answer and explanation
(-3, 8, [0, 1, 10], [8, 5, -22], -70, -70, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-3, 8, [0, 1, 10], [8, 5, -22], -70, -70, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
59. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=-2, b=9 and strictly increasing nodes=[0, 1, 11]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(-2, 9, [0, 1, 11], [9, 7, -13], -22, -22, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-2, 9, [0, 1, 11], [9, 7, -13], -22, -22, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
60. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=-1, b=10 and strictly increasing nodes=[0, 1, 12]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(-1, 10, [0, 1, 12], [10, 9, -2], 48, 48, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-1, 10, [0, 1, 12], [10, 9, -2], 48, 48, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
61. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=2, b=11 and strictly increasing nodes=[0, 1, 13]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(2, 11, [0, 1, 13], [11, 13, 37], 312, 312, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (2, 11, [0, 1, 13], [11, 13, 37], 312, 312, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
62. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=1, b=12 and strictly increasing nodes=[0, 1, 14]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(1, 12, [0, 1, 14], [12, 13, 26], 266, 266, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (1, 12, [0, 1, 14], [12, 13, 26], 266, 266, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
63. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=2, b=13 and strictly increasing nodes=[0, 1, 15]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(2, 13, [0, 1, 15], [13, 15, 43], 420, 420, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (2, 13, [0, 1, 15], [13, 15, 43], 420, 420, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
64. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=3, b=14 and strictly increasing nodes=[0, 1, 16]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(3, 14, [0, 1, 16], [14, 17, 62], 608, 608, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (3, 14, [0, 1, 16], [14, 17, 62], 608, 608, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
65. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=-3, b=15 and strictly increasing nodes=[0, 1, 17]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(-3, 15, [0, 1, 17], [15, 12, -36], -357/2, -357/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-3, 15, [0, 1, 17], [15, 12, -36], -357/2, -357/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
66. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=-2, b=16 and strictly increasing nodes=[0, 1, 18]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(-2, 16, [0, 1, 18], [16, 14, -20], -36, -36, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-2, 16, [0, 1, 18], [16, 14, -20], -36, -36, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
67. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=-1, b=17 and strictly increasing nodes=[0, 1, 19]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(-1, 17, [0, 1, 19], [17, 16, -2], 285/2, 285/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-1, 17, [0, 1, 19], [17, 16, -2], 285/2, 285/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
68. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=2, b=18 and strictly increasing nodes=[0, 1, 20]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(2, 18, [0, 1, 20], [18, 20, 58], 760, 760, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (2, 18, [0, 1, 20], [18, 20, 58], 760, 760, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
69. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=1, b=19 and strictly increasing nodes=[0, 1, 21]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(1, 19, [0, 1, 21], [19, 20, 40], 1239/2, 1239/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (1, 19, [0, 1, 21], [19, 20, 40], 1239/2, 1239/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
70. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=2, b=20 and strictly increasing nodes=[0, 1, 22]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(2, 20, [0, 1, 22], [20, 22, 64], 924, 924, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (2, 20, [0, 1, 22], [20, 22, 64], 924, 924, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
71. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=3, b=21 and strictly increasing nodes=[0, 1, 23]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(3, 21, [0, 1, 23], [21, 24, 90], 2553/2, 2553/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (3, 21, [0, 1, 23], [21, 24, 90], 2553/2, 2553/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
72. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=-3, b=22 and strictly increasing nodes=[0, 1, 24]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(-3, 22, [0, 1, 24], [22, 19, -50], -336, -336, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-3, 22, [0, 1, 24], [22, 19, -50], -336, -336, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
73. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=-2, b=23 and strictly increasing nodes=[0, 1, 25]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(-2, 23, [0, 1, 25], [23, 21, -27], -50, -50, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-2, 23, [0, 1, 25], [23, 21, -27], -50, -50, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
74. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=-1, b=24 and strictly increasing nodes=[0, 1, 26]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(-1, 24, [0, 1, 26], [24, 23, -2], 286, 286, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-1, 24, [0, 1, 26], [24, 23, -2], 286, 286, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
75. An Affine Integrand Is Integrated Exactly on Any Qualified Partition
Use f(x)=m*x+b with m=2, b=25 and strictly increasing nodes=[0, 1, 27]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).
View answer and explanation
(2, 25, [0, 1, 27], [25, 27, 79], 1404, 1404, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)
Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (2, 25, [0, 1, 27], [25, 27, 79], 1404, 1404, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-affine-exactness · candidate
76. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=quadratic_reference, complete model={"coefficients":[1,1,1],"nodes":[0,1,3]}. Give (mode, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(quadratic_reference, [1, 1, 1], [0, 1, 3], 18, 33/2, -3/2, 3/2, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (quadratic_reference, [1, 1, 1], [0, 1, 3], 18, 33/2, -3/2, 3/2, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
77. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=nonincreasing_partition, complete model={"nodes":[0,0,2],"values":[1,2,3]}. Give (mode, nodes, endpoint_values, aligned, strictly_increasing, partition_qualification_claim, boundary).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(nonincreasing_partition, [0, 0, 2], [1, 2, 3], Yes, No, No, partition_must_be_strictly_increasing)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (nonincreasing_partition, [0, 0, 2], [1, 2, 3], Yes, No, No, partition_must_be_strictly_increasing).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
78. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=sample_table_scope, complete model={"nodes":[0,1,5],"values":[2,3,5]}. Give (mode, nodes, endpoint_values, trapezoid_value, sum_determined, unique_integrand_or_integral_claim, boundary).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(sample_table_scope, [0, 1, 5], [2, 3, 5], 37/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (sample_table_scope, [0, 1, 5], [2, 3, 5], 37/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
79. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=single_width_confusion, complete model={"nodes":[0,1,6],"values":[3,7,13]}. Give (mode, individual_widths, correct_value, wrong_single_width_value, single_width_method_claim, boundary).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(single_width_confusion, [1, 5], 55, 15, No, nonuniform_partition_cannot_use_one_global_width)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (single_width_confusion, [1, 5], 55, 15, No, nonuniform_partition_cannot_use_one_global_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
80. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=quadratic_reference, complete model={"coefficients":[5,1,1],"nodes":[0,1,7]}. Give (mode, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(quadratic_reference, [5, 1, 1], [0, 1, 7], 210, 1043/6, -217/6, 217/6, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (quadratic_reference, [5, 1, 1], [0, 1, 7], 210, 1043/6, -217/6, 217/6, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
81. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=nonincreasing_partition, complete model={"nodes":[0,0,6],"values":[5,6,7]}. Give (mode, nodes, endpoint_values, aligned, strictly_increasing, partition_qualification_claim, boundary).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(nonincreasing_partition, [0, 0, 6], [5, 6, 7], Yes, No, No, partition_must_be_strictly_increasing)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (nonincreasing_partition, [0, 0, 6], [5, 6, 7], Yes, No, No, partition_must_be_strictly_increasing).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
82. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=sample_table_scope, complete model={"nodes":[0,1,9],"values":[6,7,9]}. Give (mode, nodes, endpoint_values, trapezoid_value, sum_determined, unique_integrand_or_integral_claim, boundary).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(sample_table_scope, [0, 1, 9], [6, 7, 9], 141/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (sample_table_scope, [0, 1, 9], [6, 7, 9], 141/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
83. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=single_width_confusion, complete model={"nodes":[0,1,10],"values":[7,15,25]}. Give (mode, individual_widths, correct_value, wrong_single_width_value, single_width_method_claim, boundary).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(single_width_confusion, [1, 9], 191, 31, No, nonuniform_partition_cannot_use_one_global_width)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (single_width_confusion, [1, 9], 191, 31, No, nonuniform_partition_cannot_use_one_global_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
84. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=quadratic_reference, complete model={"coefficients":[9,1,1],"nodes":[0,1,11]}. Give (mode, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(quadratic_reference, [9, 1, 1], [0, 1, 11], 770, 3619/6, -1001/6, 1001/6, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (quadratic_reference, [9, 1, 1], [0, 1, 11], 770, 3619/6, -1001/6, 1001/6, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
85. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=nonincreasing_partition, complete model={"nodes":[0,0,10],"values":[9,10,11]}. Give (mode, nodes, endpoint_values, aligned, strictly_increasing, partition_qualification_claim, boundary).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(nonincreasing_partition, [0, 0, 10], [9, 10, 11], Yes, No, No, partition_must_be_strictly_increasing)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (nonincreasing_partition, [0, 0, 10], [9, 10, 11], Yes, No, No, partition_must_be_strictly_increasing).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
86. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=sample_table_scope, complete model={"nodes":[0,1,13],"values":[10,11,13]}. Give (mode, nodes, endpoint_values, trapezoid_value, sum_determined, unique_integrand_or_integral_claim, boundary).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(sample_table_scope, [0, 1, 13], [10, 11, 13], 309/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (sample_table_scope, [0, 1, 13], [10, 11, 13], 309/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
87. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=single_width_confusion, complete model={"nodes":[0,1,14],"values":[11,23,37]}. Give (mode, individual_widths, correct_value, wrong_single_width_value, single_width_method_claim, boundary).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(single_width_confusion, [1, 13], 407, 47, No, nonuniform_partition_cannot_use_one_global_width)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (single_width_confusion, [1, 13], 407, 47, No, nonuniform_partition_cannot_use_one_global_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
88. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=quadratic_reference, complete model={"coefficients":[13,1,1],"nodes":[0,1,15]}. Give (mode, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(quadratic_reference, [13, 1, 1], [0, 1, 15], 1890, 2865/2, -915/2, 915/2, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (quadratic_reference, [13, 1, 1], [0, 1, 15], 1890, 2865/2, -915/2, 915/2, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
89. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=nonincreasing_partition, complete model={"nodes":[0,0,14],"values":[13,14,15]}. Give (mode, nodes, endpoint_values, aligned, strictly_increasing, partition_qualification_claim, boundary).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(nonincreasing_partition, [0, 0, 14], [13, 14, 15], Yes, No, No, partition_must_be_strictly_increasing)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (nonincreasing_partition, [0, 0, 14], [13, 14, 15], Yes, No, No, partition_must_be_strictly_increasing).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
90. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=sample_table_scope, complete model={"nodes":[0,1,17],"values":[14,15,17]}. Give (mode, nodes, endpoint_values, trapezoid_value, sum_determined, unique_integrand_or_integral_claim, boundary).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(sample_table_scope, [0, 1, 17], [14, 15, 17], 541/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (sample_table_scope, [0, 1, 17], [14, 15, 17], 541/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
91. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=single_width_confusion, complete model={"nodes":[0,1,18],"values":[15,31,49]}. Give (mode, individual_widths, correct_value, wrong_single_width_value, single_width_method_claim, boundary).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(single_width_confusion, [1, 17], 703, 63, No, nonuniform_partition_cannot_use_one_global_width)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (single_width_confusion, [1, 17], 703, 63, No, nonuniform_partition_cannot_use_one_global_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
92. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=quadratic_reference, complete model={"coefficients":[17,1,1],"nodes":[0,1,19]}. Give (mode, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(quadratic_reference, [17, 1, 1], [0, 1, 19], 3762, 16739/6, -5833/6, 5833/6, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (quadratic_reference, [17, 1, 1], [0, 1, 19], 3762, 16739/6, -5833/6, 5833/6, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
93. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=nonincreasing_partition, complete model={"nodes":[0,0,18],"values":[17,18,19]}. Give (mode, nodes, endpoint_values, aligned, strictly_increasing, partition_qualification_claim, boundary).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(nonincreasing_partition, [0, 0, 18], [17, 18, 19], Yes, No, No, partition_must_be_strictly_increasing)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (nonincreasing_partition, [0, 0, 18], [17, 18, 19], Yes, No, No, partition_must_be_strictly_increasing).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
94. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=sample_table_scope, complete model={"nodes":[0,1,21],"values":[18,19,21]}. Give (mode, nodes, endpoint_values, trapezoid_value, sum_determined, unique_integrand_or_integral_claim, boundary).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(sample_table_scope, [0, 1, 21], [18, 19, 21], 837/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (sample_table_scope, [0, 1, 21], [18, 19, 21], 837/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
95. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=single_width_confusion, complete model={"nodes":[0,1,22],"values":[19,39,61]}. Give (mode, individual_widths, correct_value, wrong_single_width_value, single_width_method_claim, boundary).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(single_width_confusion, [1, 21], 1079, 79, No, nonuniform_partition_cannot_use_one_global_width)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (single_width_confusion, [1, 21], 1079, 79, No, nonuniform_partition_cannot_use_one_global_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
96. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=quadratic_reference, complete model={"coefficients":[21,1,1],"nodes":[0,1,23]}. Give (mode, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(quadratic_reference, [21, 1, 1], [0, 1, 23], 6578, 28819/6, -10649/6, 10649/6, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (quadratic_reference, [21, 1, 1], [0, 1, 23], 6578, 28819/6, -10649/6, 10649/6, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
97. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=nonincreasing_partition, complete model={"nodes":[0,0,22],"values":[21,22,23]}. Give (mode, nodes, endpoint_values, aligned, strictly_increasing, partition_qualification_claim, boundary).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(nonincreasing_partition, [0, 0, 22], [21, 22, 23], Yes, No, No, partition_must_be_strictly_increasing)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (nonincreasing_partition, [0, 0, 22], [21, 22, 23], Yes, No, No, partition_must_be_strictly_increasing).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
98. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=sample_table_scope, complete model={"nodes":[0,1,25],"values":[22,23,25]}. Give (mode, nodes, endpoint_values, trapezoid_value, sum_determined, unique_integrand_or_integral_claim, boundary).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(sample_table_scope, [0, 1, 25], [22, 23, 25], 1197/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (sample_table_scope, [0, 1, 25], [22, 23, 25], 1197/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
99. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=single_width_confusion, complete model={"nodes":[0,1,26],"values":[23,47,73]}. Give (mode, individual_widths, correct_value, wrong_single_width_value, single_width_method_claim, boundary).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(single_width_confusion, [1, 25], 1535, 95, No, nonuniform_partition_cannot_use_one_global_width)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (single_width_confusion, [1, 25], 1535, 95, No, nonuniform_partition_cannot_use_one_global_width).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
100. Quadratic Reference Error Is Not a General Quadrature Error Theorem
Use mode=quadratic_reference, complete model={"coefficients":[25,1,1],"nodes":[0,1,27]}. Give (mode, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope).
The four modes have9/7/7/6 fields. Signed error is I−T. A sample table fixes the quadrature sum only. A complete quadratic reference still does not supply a general second-derivative error theorem.
View answer and explanation
(quadratic_reference, [25, 1, 1], [0, 1, 27], 10530, 15201/2, -5859/2, 5859/2, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)
The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (quadratic_reference, [25, 1, 1], [0, 1, 27], 10530, 15201/2, -5859/2, 5859/2, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound).
Reviewed step-by-step derivation
Composite trapezoidal quadrature · qf-mod50-trap-error-boundary · candidate
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