Exact Polynomial Remainders, Lagrange Witnesses and Qualified Bounds
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batch-20261015-advance-20261116-r03-taylor-remainder-theorem-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.
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1. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[2, 3] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([2, 3], 0, 1/2, 1, [2, 3, 0], [2, 3/2], 7/2, 7/2, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([2, 3], 0, 1/2, 1, [2, 3, 0], [2, 3/2], 7/2, 7/2, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
2. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[3, 5, 7] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([3, 5, 7], 0, 1/2, 2, [3, 5, 14, 0], [3, 5/2, 7/4], 29/4, 29/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([3, 5, 7], 0, 1/2, 2, [3, 5, 14, 0], [3, 5/2, 7/4], 29/4, 29/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
3. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[4, 7, 10, 13] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([4, 7, 10, 13], 0, 1/2, 3, [4, 7, 20, 78, 0], [4, 7/2, 5/2, 13/8], 93/8, 93/8, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([4, 7, 10, 13], 0, 1/2, 3, [4, 7, 20, 78, 0], [4, 7/2, 5/2, 13/8], 93/8, 93/8, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
4. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[5, 9, 13, 17, 4] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=4; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([5, 9, 13, 17, 4], 0, 1/2, 4, [5, 9, 26, 102, 96, 0], [5, 9/2, 13/4, 17/8, 1/4], 121/8, 121/8, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([5, 9, 13, 17, 4], 0, 1/2, 4, [5, 9, 26, 102, 96, 0], [5, 9/2, 13/4, 17/8, 1/4], 121/8, 121/8, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
5. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[6, 11, 16, 4, 9, 14] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([6, 11, 16, 4, 9, 14], 0, 1/2, 5, [6, 11, 32, 24, 216, 1680, 0], [6, 11/2, 4, 1/2, 9/16, 7/16], 17, 17, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([6, 11, 16, 4, 9, 14], 0, 1/2, 5, [6, 11, 32, 24, 216, 1680, 0], [6, 11/2, 4, 1/2, 9/16, 7/16], 17, 17, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
6. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[7, 13] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([7, 13], 0, 1/2, 1, [7, 13, 0], [7, 13/2], 27/2, 27/2, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([7, 13], 0, 1/2, 1, [7, 13, 0], [7, 13/2], 27/2, 27/2, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
7. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[8, 15, 5] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([8, 15, 5], 0, 1/2, 2, [8, 15, 10, 0], [8, 15/2, 5/4], 67/4, 67/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([8, 15, 5], 0, 1/2, 2, [8, 15, 10, 0], [8, 15/2, 5/4], 67/4, 67/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
8. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[9, 17, 8, 16] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([9, 17, 8, 16], 0, 1/2, 3, [9, 17, 16, 96, 0], [9, 17/2, 2, 2], 43/2, 43/2, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([9, 17, 8, 16], 0, 1/2, 3, [9, 17, 16, 96, 0], [9, 17/2, 2, 2], 43/2, 43/2, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
9. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[10, 2, 11, 3, 12] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=4; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([10, 2, 11, 3, 12], 0, 1/2, 4, [10, 2, 22, 18, 288, 0], [10, 1, 11/4, 3/8, 3/4], 119/8, 119/8, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([10, 2, 11, 3, 12], 0, 1/2, 4, [10, 2, 22, 18, 288, 0], [10, 1, 11/4, 3/8, 3/4], 119/8, 119/8, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
10. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[11, 4, 14, 7, 17, 10] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([11, 4, 14, 7, 17, 10], 0, 1/2, 5, [11, 4, 28, 42, 408, 1200, 0], [11, 2, 7/2, 7/8, 17/16, 5/16], 75/4, 75/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([11, 4, 14, 7, 17, 10], 0, 1/2, 5, [11, 4, 28, 42, 408, 1200, 0], [11, 2, 7/2, 7/8, 17/16, 5/16], 75/4, 75/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
11. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[12, 6] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([12, 6], 0, 1/2, 1, [12, 6, 0], [12, 3], 15, 15, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([12, 6], 0, 1/2, 1, [12, 6, 0], [12, 3], 15, 15, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
12. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[13, 8, 3] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([13, 8, 3], 0, 1/2, 2, [13, 8, 6, 0], [13, 4, 3/4], 71/4, 71/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([13, 8, 3], 0, 1/2, 2, [13, 8, 6, 0], [13, 4, 3/4], 71/4, 71/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
13. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[14, 10, 6, 2] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([14, 10, 6, 2], 0, 1/2, 3, [14, 10, 12, 12, 0], [14, 5, 3/2, 1/4], 83/4, 83/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([14, 10, 6, 2], 0, 1/2, 3, [14, 10, 12, 12, 0], [14, 5, 3/2, 1/4], 83/4, 83/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
14. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[15, 12, 9, 6, 3] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=4; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([15, 12, 9, 6, 3], 0, 1/2, 4, [15, 12, 18, 36, 72, 0], [15, 6, 9/4, 3/4, 3/16], 387/16, 387/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([15, 12, 9, 6, 3], 0, 1/2, 4, [15, 12, 18, 36, 72, 0], [15, 6, 9/4, 3/4, 3/16], 387/16, 387/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
15. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[16, 14, 12, 10, 8, 6] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([16, 14, 12, 10, 8, 6], 0, 1/2, 5, [16, 14, 24, 60, 192, 720, 0], [16, 7, 3, 5/4, 1/2, 3/16], 447/16, 447/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([16, 14, 12, 10, 8, 6], 0, 1/2, 5, [16, 14, 24, 60, 192, 720, 0], [16, 7, 3, 5/4, 1/2, 3/16], 447/16, 447/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
16. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[17, 16] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([17, 16], 0, 1/2, 1, [17, 16, 0], [17, 8], 25, 25, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([17, 16], 0, 1/2, 1, [17, 16, 0], [17, 8], 25, 25, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
17. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[1, 1, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([1, 1, 1], 0, 1/2, 2, [1, 1, 2, 0], [1, 1/2, 1/4], 7/4, 7/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([1, 1, 1], 0, 1/2, 2, [1, 1, 2, 0], [1, 1/2, 1/4], 7/4, 7/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
18. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[2, 3, 4, 5] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([2, 3, 4, 5], 0, 1/2, 3, [2, 3, 8, 30, 0], [2, 3/2, 1, 5/8], 41/8, 41/8, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([2, 3, 4, 5], 0, 1/2, 3, [2, 3, 8, 30, 0], [2, 3/2, 1, 5/8], 41/8, 41/8, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
19. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[3, 5, 7, 9, 11] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=4; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([3, 5, 7, 9, 11], 0, 1/2, 4, [3, 5, 14, 54, 264, 0], [3, 5/2, 7/4, 9/8, 11/16], 145/16, 145/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([3, 5, 7, 9, 11], 0, 1/2, 4, [3, 5, 14, 54, 264, 0], [3, 5/2, 7/4, 9/8, 11/16], 145/16, 145/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
20. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[4, 7, 10, 13, 16, 2] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([4, 7, 10, 13, 16, 2], 0, 1/2, 5, [4, 7, 20, 78, 384, 240, 0], [4, 7/2, 5/2, 13/8, 1, 1/16], 203/16, 203/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([4, 7, 10, 13, 16, 2], 0, 1/2, 5, [4, 7, 20, 78, 384, 240, 0], [4, 7/2, 5/2, 13/8, 1, 1/16], 203/16, 203/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
21. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[5, 9] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([5, 9], 0, 1/2, 1, [5, 9, 0], [5, 9/2], 19/2, 19/2, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([5, 9], 0, 1/2, 1, [5, 9, 0], [5, 9/2], 19/2, 19/2, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
22. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[6, 11, 16] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([6, 11, 16], 0, 1/2, 2, [6, 11, 32, 0], [6, 11/2, 4], 31/2, 31/2, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([6, 11, 16], 0, 1/2, 2, [6, 11, 32, 0], [6, 11/2, 4], 31/2, 31/2, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
23. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[7, 13, 2, 8] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([7, 13, 2, 8], 0, 1/2, 3, [7, 13, 4, 48, 0], [7, 13/2, 1/2, 1], 15, 15, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([7, 13, 2, 8], 0, 1/2, 3, [7, 13, 4, 48, 0], [7, 13/2, 1/2, 1], 15, 15, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
24. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[8, 15, 5, 12, 2] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=4; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([8, 15, 5, 12, 2], 0, 1/2, 4, [8, 15, 10, 72, 48, 0], [8, 15/2, 5/4, 3/2, 1/8], 147/8, 147/8, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([8, 15, 5, 12, 2], 0, 1/2, 4, [8, 15, 10, 72, 48, 0], [8, 15/2, 5/4, 3/2, 1/8], 147/8, 147/8, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
25. An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses
Coefficients=[9, 17, 8, 16, 7, 15] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).
View answer and explanation
([9, 17, 8, 16, 7, 15], 0, 1/2, 5, [9, 17, 16, 96, 168, 1800, 0], [9, 17/2, 2, 2, 7/16, 15/32], 717/32, 717/32, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)
Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([9, 17, 8, 16, 7, 15], 0, 1/2, 5, [9, 17, 16, 96, 168, 1800, 0], [9, 17/2, 2, 2, 7/16, 15/32], 717/32, 717/32, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-qualification · candidate
26. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[2, 3] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(1, 1, [2, 3/2], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (1, 1, [2, 3/2], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
27. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[3, 5, 7] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(2, 2, [3, 5/2, 7/4], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (2, 2, [3, 5/2, 7/4], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
28. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[4, 7, 10, 13] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(3, 3, [4, 7/2, 5/2, 13/8], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (3, 3, [4, 7/2, 5/2, 13/8], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
29. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[5, 9, 13, 17, 4] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=4; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(4, 4, [5, 9/2, 13/4, 17/8, 1/4], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (4, 4, [5, 9/2, 13/4, 17/8, 1/4], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
30. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[6, 11, 16, 4, 9, 14] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(5, 5, [6, 11/2, 4, 1/2, 9/16, 7/16], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (5, 5, [6, 11/2, 4, 1/2, 9/16, 7/16], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
31. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[7, 13] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(1, 1, [7, 13/2], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (1, 1, [7, 13/2], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
32. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[8, 15, 5] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(2, 2, [8, 15/2, 5/4], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (2, 2, [8, 15/2, 5/4], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
33. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[9, 17, 8, 16] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(3, 3, [9, 17/2, 2, 2], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (3, 3, [9, 17/2, 2, 2], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
34. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[10, 2, 11, 3, 12] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=4; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(4, 4, [10, 1, 11/4, 3/8, 3/4], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (4, 4, [10, 1, 11/4, 3/8, 3/4], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
35. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[11, 4, 14, 7, 17, 10] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(5, 5, [11, 2, 7/2, 7/8, 17/16, 5/16], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (5, 5, [11, 2, 7/2, 7/8, 17/16, 5/16], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
36. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[12, 6] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(1, 1, [12, 3], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (1, 1, [12, 3], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
37. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[13, 8, 3] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(2, 2, [13, 4, 3/4], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (2, 2, [13, 4, 3/4], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
38. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[14, 10, 6, 2] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(3, 3, [14, 5, 3/2, 1/4], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (3, 3, [14, 5, 3/2, 1/4], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
39. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[15, 12, 9, 6, 3] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=4; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(4, 4, [15, 6, 9/4, 3/4, 3/16], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (4, 4, [15, 6, 9/4, 3/4, 3/16], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
40. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[16, 14, 12, 10, 8, 6] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(5, 5, [16, 7, 3, 5/4, 1/2, 3/16], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (5, 5, [16, 7, 3, 5/4, 1/2, 3/16], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
41. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[17, 16] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(1, 1, [17, 8], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (1, 1, [17, 8], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
42. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[1, 1, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(2, 2, [1, 1/2, 1/4], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (2, 2, [1, 1/2, 1/4], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
43. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[2, 3, 4, 5] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(3, 3, [2, 3/2, 1, 5/8], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (3, 3, [2, 3/2, 1, 5/8], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
44. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[3, 5, 7, 9, 11] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=4; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(4, 4, [3, 5/2, 7/4, 9/8, 11/16], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (4, 4, [3, 5/2, 7/4, 9/8, 11/16], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
45. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[4, 7, 10, 13, 16, 2] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(5, 5, [4, 7/2, 5/2, 13/8, 1, 1/16], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (5, 5, [4, 7/2, 5/2, 13/8, 1, 1/16], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
46. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[5, 9] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(1, 1, [5, 9/2], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (1, 1, [5, 9/2], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
47. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[6, 11, 16] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(2, 2, [6, 11/2, 4], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (2, 2, [6, 11/2, 4], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
48. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[7, 13, 2, 8] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(3, 3, [7, 13/2, 1/2, 1], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (3, 3, [7, 13/2, 1/2, 1], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
49. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[8, 15, 5, 12, 2] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=4; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(4, 4, [8, 15/2, 5/4, 3/2, 1/8], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (4, 4, [8, 15/2, 5/4, 3/2, 1/8], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
50. A Polynomial of Degree at Most the Taylor Order Has Zero Remainder
Coefficients=[9, 17, 8, 16, 7, 15] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).
View answer and explanation
(5, 5, [9, 17/2, 2, 2, 7/16, 15/32], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)
Require degree≤order after removing trailing zeros. Retain all terms through the requested order, including zero terms, and the exact zero remainder. This finite polynomial identity does not establish analyticity of an arbitrary function. The exact result, retaining the requested order, is (5, 5, [9, 17/2, 2, 2, 7/16, 15/32], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-zero-remainder · candidate
51. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[2, 3, 2] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(2, 1, 4, 1/4, Yes, 1/2, 1/2, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (2, 1, 4, 1/4, Yes, 1/2, 1/2, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
52. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[3, 5, 7, 3] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(3, 2, 18, 1/4, Yes, 3/8, 3/8, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (3, 2, 18, 1/4, Yes, 3/8, 3/8, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
53. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[4, 7, 10, 13, 4] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(4, 3, 96, 1/4, Yes, 1/4, 1/4, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (4, 3, 96, 1/4, Yes, 1/4, 1/4, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
54. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[5, 9, 13, 17, 4, 5] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=4; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(5, 4, 600, 1/4, Yes, 5/32, 5/32, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (5, 4, 600, 1/4, Yes, 5/32, 5/32, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
55. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[6, 11, 16, 4, 9, 14, 6] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(6, 5, 4320, 1/4, Yes, 3/32, 3/32, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (6, 5, 4320, 1/4, Yes, 3/32, 3/32, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
56. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[7, 13, 7] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(2, 1, 14, 1/4, Yes, 7/4, 7/4, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (2, 1, 14, 1/4, Yes, 7/4, 7/4, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
57. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[8, 15, 5, 8] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(3, 2, 48, 1/4, Yes, 1, 1, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (3, 2, 48, 1/4, Yes, 1, 1, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
58. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[9, 17, 8, 16, 9] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(4, 3, 216, 1/4, Yes, 9/16, 9/16, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (4, 3, 216, 1/4, Yes, 9/16, 9/16, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
59. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[10, 2, 11, 3, 12, 10] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=4; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(5, 4, 1200, 1/4, Yes, 5/16, 5/16, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (5, 4, 1200, 1/4, Yes, 5/16, 5/16, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
60. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[11, 4, 14, 7, 17, 10, 11] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(6, 5, 7920, 1/4, Yes, 11/64, 11/64, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (6, 5, 7920, 1/4, Yes, 11/64, 11/64, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
61. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[12, 6, 12] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(2, 1, 24, 1/4, Yes, 3, 3, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (2, 1, 24, 1/4, Yes, 3, 3, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
62. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[13, 8, 3, 13] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(3, 2, 78, 1/4, Yes, 13/8, 13/8, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (3, 2, 78, 1/4, Yes, 13/8, 13/8, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
63. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[14, 10, 6, 2, 14] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(4, 3, 336, 1/4, Yes, 7/8, 7/8, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (4, 3, 336, 1/4, Yes, 7/8, 7/8, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
64. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[15, 12, 9, 6, 3, 15] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=4; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(5, 4, 1800, 1/4, Yes, 15/32, 15/32, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (5, 4, 1800, 1/4, Yes, 15/32, 15/32, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
65. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[16, 14, 12, 10, 8, 6, 16] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(6, 5, 11520, 1/4, Yes, 1/4, 1/4, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (6, 5, 11520, 1/4, Yes, 1/4, 1/4, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
66. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[17, 16, 17] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(2, 1, 34, 1/4, Yes, 17/4, 17/4, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (2, 1, 34, 1/4, Yes, 17/4, 17/4, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
67. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[1, 1, 1, 18] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(3, 2, 108, 1/4, Yes, 9/4, 9/4, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (3, 2, 108, 1/4, Yes, 9/4, 9/4, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
68. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[2, 3, 4, 5, 19] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(4, 3, 456, 1/4, Yes, 19/16, 19/16, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (4, 3, 456, 1/4, Yes, 19/16, 19/16, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
69. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[3, 5, 7, 9, 11, 20] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=4; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(5, 4, 2400, 1/4, Yes, 5/8, 5/8, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (5, 4, 2400, 1/4, Yes, 5/8, 5/8, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
70. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[4, 7, 10, 13, 16, 2, 21] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(6, 5, 15120, 1/4, Yes, 21/64, 21/64, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (6, 5, 15120, 1/4, Yes, 21/64, 21/64, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
71. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[5, 9, 22] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(2, 1, 44, 1/4, Yes, 11/2, 11/2, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (2, 1, 44, 1/4, Yes, 11/2, 11/2, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
72. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[6, 11, 16, 23] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(3, 2, 138, 1/4, Yes, 23/8, 23/8, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (3, 2, 138, 1/4, Yes, 23/8, 23/8, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
73. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[7, 13, 2, 8, 24] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(4, 3, 576, 1/4, Yes, 3/2, 3/2, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (4, 3, 576, 1/4, Yes, 3/2, 3/2, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
74. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[8, 15, 5, 12, 2, 25] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=4; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(5, 4, 3000, 1/4, Yes, 25/32, 25/32, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (5, 4, 3000, 1/4, Yes, 25/32, 25/32, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
75. A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness
Coefficients=[9, 17, 8, 16, 7, 15, 26] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).
View answer and explanation
(6, 5, 18720, 1/4, Yes, 13/32, 13/32, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)
Require degree=order+1 and x distinct from the center. The next derivative is constant, so the midpoint lies strictly between center and x in either orientation and gives the exact Lagrange remainder. Do not extend this construction to an arbitrary nonconstant next derivative. The exact result, retaining the requested order, is (6, 5, 18720, 1/4, Yes, 13/32, 13/32, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-lagrange-witness · candidate
76. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[2, 3, 2] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=valid_bound, supplied M=4≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(valid_bound, 1/2, 4, 4, 1/2, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (valid_bound, 1/2, 4, 4, 1/2, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
77. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[3, 5, 7, 3] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=undersized_M, supplied M=17≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(undersized_M, 3/8, 18, 17, 17/48, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (undersized_M, 3/8, 18, 17, 17/48, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
78. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[4, 7, 10, 13, 4] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=valid_bound, supplied M=96≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(valid_bound, 1/4, 96, 96, 1/4, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (valid_bound, 1/4, 96, 96, 1/4, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
79. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[5, 9, 13, 17, 4, 5] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=4; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=undersized_M, supplied M=599≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(undersized_M, 5/32, 600, 599, 599/3840, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (undersized_M, 5/32, 600, 599, 599/3840, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
80. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[6, 11, 16, 4, 9, 14, 6] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=valid_bound, supplied M=4320≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(valid_bound, 3/32, 4320, 4320, 3/32, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (valid_bound, 3/32, 4320, 4320, 3/32, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
81. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[7, 13, 7] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=undersized_M, supplied M=13≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(undersized_M, 7/4, 14, 13, 13/8, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (undersized_M, 7/4, 14, 13, 13/8, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
82. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[8, 15, 5, 8] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=valid_bound, supplied M=48≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(valid_bound, 1, 48, 48, 1, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (valid_bound, 1, 48, 48, 1, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
83. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[9, 17, 8, 16, 9] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=undersized_M, supplied M=215≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(undersized_M, 9/16, 216, 215, 215/384, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (undersized_M, 9/16, 216, 215, 215/384, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
84. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[10, 2, 11, 3, 12, 10] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=4; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=valid_bound, supplied M=1200≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(valid_bound, 5/16, 1200, 1200, 5/16, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (valid_bound, 5/16, 1200, 1200, 5/16, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
85. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[11, 4, 14, 7, 17, 10, 11] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=undersized_M, supplied M=7919≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(undersized_M, 11/64, 7920, 7919, 7919/46080, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (undersized_M, 11/64, 7920, 7919, 7919/46080, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
86. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[12, 6, 12] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=valid_bound, supplied M=24≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(valid_bound, 3, 24, 24, 3, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (valid_bound, 3, 24, 24, 3, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
87. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[13, 8, 3, 13] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=undersized_M, supplied M=77≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(undersized_M, 13/8, 78, 77, 77/48, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (undersized_M, 13/8, 78, 77, 77/48, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
88. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[14, 10, 6, 2, 14] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=valid_bound, supplied M=336≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(valid_bound, 7/8, 336, 336, 7/8, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (valid_bound, 7/8, 336, 336, 7/8, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
89. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[15, 12, 9, 6, 3, 15] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=4; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=undersized_M, supplied M=1799≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(undersized_M, 15/32, 1800, 1799, 1799/3840, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (undersized_M, 15/32, 1800, 1799, 1799/3840, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
90. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[16, 14, 12, 10, 8, 6, 16] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=valid_bound, supplied M=11520≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(valid_bound, 1/4, 11520, 11520, 1/4, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (valid_bound, 1/4, 11520, 11520, 1/4, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
91. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[17, 16, 17] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=undersized_M, supplied M=33≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(undersized_M, 17/4, 34, 33, 33/8, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (undersized_M, 17/4, 34, 33, 33/8, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
92. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[1, 1, 1, 18] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=valid_bound, supplied M=108≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(valid_bound, 9/4, 108, 108, 9/4, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (valid_bound, 9/4, 108, 108, 9/4, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
93. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[2, 3, 4, 5, 19] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=undersized_M, supplied M=455≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(undersized_M, 19/16, 456, 455, 455/384, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (undersized_M, 19/16, 456, 455, 455/384, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
94. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[3, 5, 7, 9, 11, 20] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=4; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=valid_bound, supplied M=2400≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(valid_bound, 5/8, 2400, 2400, 5/8, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (valid_bound, 5/8, 2400, 2400, 5/8, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
95. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[4, 7, 10, 13, 16, 2, 21] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=undersized_M, supplied M=15119≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(undersized_M, 21/64, 15120, 15119, 15119/46080, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (undersized_M, 21/64, 15120, 15119, 15119/46080, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
96. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[5, 9, 22] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=valid_bound, supplied M=44≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(valid_bound, 11/2, 44, 44, 11/2, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (valid_bound, 11/2, 44, 44, 11/2, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
97. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[6, 11, 16, 23] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=undersized_M, supplied M=137≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(undersized_M, 23/8, 138, 137, 137/48, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (undersized_M, 23/8, 138, 137, 137/48, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
98. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[7, 13, 2, 8, 24] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=valid_bound, supplied M=576≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(valid_bound, 3/2, 576, 576, 3/2, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (valid_bound, 3/2, 576, 576, 3/2, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
99. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[8, 15, 5, 12, 2, 25] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=4; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=undersized_M, supplied M=2999≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(undersized_M, 25/32, 3000, 2999, 2999/3840, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (undersized_M, 25/32, 3000, 2999, 2999/3840, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
100. An Actual Error Inequality Is Different from a Qualified Derivative Bound
Coefficients=[9, 17, 8, 16, 7, 15, 26] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=valid_bound, supplied M=18720≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).
actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound.
View answer and explanation
(valid_bound, 13/32, 18720, 18720, 13/32, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)
The degree=order+1 polynomial has a constant next derivative, whose absolute value is the exact interval supremum D. valid_bound requires M≥D; undersized_M requires M<D. Test the actual absolute error against M*|x−center|^(order+1)/(order+1)! separately. At x=center an undersized M can still satisfy0≤0 without qualifying the derivative bound. Neither a finite jet nor a few samples prove the general remainder theorem or analyticity. The exact result, retaining the requested order, is (valid_bound, 13/32, 18720, 18720, 13/32, Yes, Yes, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity).
Reviewed step-by-step derivation
Taylor remainder theorem · qf-mod47-taylor-bound-boundary · candidate
Canonical question JSONL · English JSONL · Validation report