# Complete Centered Jets, Exact Factorial Coefficients and Finite Taylor Boundaries

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261112-r02-taylor-polynomial-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[1, 2, 3] at the fixed center=-12, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(-12, 2, [1, 2, 3], 3, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-12, 2, [1, 2, 3], 3, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0001/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 2. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[2, 4, 6, 8] at the fixed center=-11, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(-11, 3, [2, 4, 6, 8], 4, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-11, 3, [2, 4, 6, 8], 4, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0002/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 3. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[3, 6, 9, 12, 15] at the fixed center=-10, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(-10, 4, [3, 6, 9, 12, 15], 5, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-10, 4, [3, 6, 9, 12, 15], 5, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0003/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 4. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[4, 8, 12, 16, 20, 24] at the fixed center=-9, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(-9, 5, [4, 8, 12, 16, 20, 24], 6, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-9, 5, [4, 8, 12, 16, 20, 24], 6, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0004/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 5. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[5, 10, 15] at the fixed center=-8, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(-8, 2, [5, 10, 15], 3, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-8, 2, [5, 10, 15], 3, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0005/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 6. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[6, 12, 18, 24] at the fixed center=-7, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(-7, 3, [6, 12, 18, 24], 4, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-7, 3, [6, 12, 18, 24], 4, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0006/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 7. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[7, 14, 21, 28, 35] at the fixed center=-6, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(-6, 4, [7, 14, 21, 28, 35], 5, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-6, 4, [7, 14, 21, 28, 35], 5, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0007/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 8. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[8, 16, 24, 32, 40, 48] at the fixed center=-5, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(-5, 5, [8, 16, 24, 32, 40, 48], 6, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-5, 5, [8, 16, 24, 32, 40, 48], 6, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0008/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 9. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[9, 18, 27] at the fixed center=-4, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(-4, 2, [9, 18, 27], 3, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-4, 2, [9, 18, 27], 3, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0009/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 10. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[10, 20, 30, 40] at the fixed center=-3, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(-3, 3, [10, 20, 30, 40], 4, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-3, 3, [10, 20, 30, 40], 4, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0010/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 11. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[11, 22, 33, 44, 55] at the fixed center=-2, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(-2, 4, [11, 22, 33, 44, 55], 5, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-2, 4, [11, 22, 33, 44, 55], 5, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0011/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 12. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[12, 24, 36, 48, 60, 72] at the fixed center=-1, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(-1, 5, [12, 24, 36, 48, 60, 72], 6, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-1, 5, [12, 24, 36, 48, 60, 72], 6, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0012/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 13. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[13, 26, 39] at the fixed center=0, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(0, 2, [13, 26, 39], 3, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (0, 2, [13, 26, 39], 3, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0013/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 14. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[14, 28, 42, 56] at the fixed center=1, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(1, 3, [14, 28, 42, 56], 4, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (1, 3, [14, 28, 42, 56], 4, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0014/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 15. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[15, 30, 45, 60, 75] at the fixed center=2, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(2, 4, [15, 30, 45, 60, 75], 5, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (2, 4, [15, 30, 45, 60, 75], 5, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0015/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 16. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[16, 32, 48, 64, 80, 96] at the fixed center=3, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(3, 5, [16, 32, 48, 64, 80, 96], 6, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (3, 5, [16, 32, 48, 64, 80, 96], 6, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0016/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 17. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[17, 34, 51] at the fixed center=4, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(4, 2, [17, 34, 51], 3, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (4, 2, [17, 34, 51], 3, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0017/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 18. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[18, 36, 54, 72] at the fixed center=5, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(5, 3, [18, 36, 54, 72], 4, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (5, 3, [18, 36, 54, 72], 4, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0018/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 19. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[19, 38, 57, 76, 95] at the fixed center=6, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(6, 4, [19, 38, 57, 76, 95], 5, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (6, 4, [19, 38, 57, 76, 95], 5, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0019/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 20. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[20, 40, 60, 80, 100, 120] at the fixed center=7, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(7, 5, [20, 40, 60, 80, 100, 120], 6, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (7, 5, [20, 40, 60, 80, 100, 120], 6, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0020/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 21. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[21, 42, 63] at the fixed center=8, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(8, 2, [21, 42, 63], 3, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (8, 2, [21, 42, 63], 3, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0021/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 22. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[22, 44, 66, 88] at the fixed center=9, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(9, 3, [22, 44, 66, 88], 4, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (9, 3, [22, 44, 66, 88], 4, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0022/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 23. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[23, 46, 69, 92, 115] at the fixed center=10, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(10, 4, [23, 46, 69, 92, 115], 5, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (10, 4, [23, 46, 69, 92, 115], 5, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0023/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 24. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[24, 48, 72, 96, 120, 144] at the fixed center=11, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(11, 5, [24, 48, 72, 96, 120, 144], 6, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (11, 5, [24, 48, 72, 96, 120, 144], 6, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0024/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 25. A Taylor Jet Includes Every Derivative Order from Zero through n

Use the derivative values=[25, 50, 75] at the fixed center=12, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

View answer and explanation(12, 2, [25, 50, 75], 3, Yes, complete_zero_through_n_derivative_jet)

Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (12, 2, [25, 50, 75], 3, Yes, complete_zero_through_n_derivative_jet).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0025/)

Taylor polynomial · qf-mod43-taylor-jet-qualification · candidate





## 26. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[1, 2, 3] at the fixed center=-12, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(-12, 2, [1, 2, 3/2], [[0, 1], [1, 2], [2, 3/2]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-12, 2, [1, 2, 3/2], [[0, 1], [1, 2], [2, 3/2]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0026/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 27. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[2, 4, 6, 8] at the fixed center=-11, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(-11, 3, [2, 4, 3, 4/3], [[0, 2], [1, 4], [2, 3], [3, 4/3]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-11, 3, [2, 4, 3, 4/3], [[0, 2], [1, 4], [2, 3], [3, 4/3]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0027/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 28. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[3, 6, 9, 12, 15] at the fixed center=-10, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(-10, 4, [3, 6, 9/2, 2, 5/8], [[0, 3], [1, 6], [2, 9/2], [3, 2], [4, 5/8]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-10, 4, [3, 6, 9/2, 2, 5/8], [[0, 3], [1, 6], [2, 9/2], [3, 2], [4, 5/8]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0028/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 29. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[4, 8, 12, 16, 20, 24] at the fixed center=-9, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(-9, 5, [4, 8, 6, 8/3, 5/6, 1/5], [[0, 4], [1, 8], [2, 6], [3, 8/3], [4, 5/6], [5, 1/5]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-9, 5, [4, 8, 6, 8/3, 5/6, 1/5], [[0, 4], [1, 8], [2, 6], [3, 8/3], [4, 5/6], [5, 1/5]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0029/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 30. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[5, 10, 15] at the fixed center=-8, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(-8, 2, [5, 10, 15/2], [[0, 5], [1, 10], [2, 15/2]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-8, 2, [5, 10, 15/2], [[0, 5], [1, 10], [2, 15/2]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0030/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 31. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[6, 12, 18, 24] at the fixed center=-7, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(-7, 3, [6, 12, 9, 4], [[0, 6], [1, 12], [2, 9], [3, 4]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-7, 3, [6, 12, 9, 4], [[0, 6], [1, 12], [2, 9], [3, 4]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0031/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 32. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[7, 14, 21, 28, 35] at the fixed center=-6, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(-6, 4, [7, 14, 21/2, 14/3, 35/24], [[0, 7], [1, 14], [2, 21/2], [3, 14/3], [4, 35/24]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-6, 4, [7, 14, 21/2, 14/3, 35/24], [[0, 7], [1, 14], [2, 21/2], [3, 14/3], [4, 35/24]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0032/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 33. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[8, 16, 24, 32, 40, 48] at the fixed center=-5, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(-5, 5, [8, 16, 12, 16/3, 5/3, 2/5], [[0, 8], [1, 16], [2, 12], [3, 16/3], [4, 5/3], [5, 2/5]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-5, 5, [8, 16, 12, 16/3, 5/3, 2/5], [[0, 8], [1, 16], [2, 12], [3, 16/3], [4, 5/3], [5, 2/5]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0033/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 34. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[9, 18, 27] at the fixed center=-4, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(-4, 2, [9, 18, 27/2], [[0, 9], [1, 18], [2, 27/2]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-4, 2, [9, 18, 27/2], [[0, 9], [1, 18], [2, 27/2]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0034/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 35. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[10, 20, 30, 40] at the fixed center=-3, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(-3, 3, [10, 20, 15, 20/3], [[0, 10], [1, 20], [2, 15], [3, 20/3]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-3, 3, [10, 20, 15, 20/3], [[0, 10], [1, 20], [2, 15], [3, 20/3]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0035/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 36. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[11, 22, 33, 44, 55] at the fixed center=-2, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(-2, 4, [11, 22, 33/2, 22/3, 55/24], [[0, 11], [1, 22], [2, 33/2], [3, 22/3], [4, 55/24]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-2, 4, [11, 22, 33/2, 22/3, 55/24], [[0, 11], [1, 22], [2, 33/2], [3, 22/3], [4, 55/24]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0036/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 37. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[12, 24, 36, 48, 60, 72] at the fixed center=-1, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(-1, 5, [12, 24, 18, 8, 5/2, 3/5], [[0, 12], [1, 24], [2, 18], [3, 8], [4, 5/2], [5, 3/5]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-1, 5, [12, 24, 18, 8, 5/2, 3/5], [[0, 12], [1, 24], [2, 18], [3, 8], [4, 5/2], [5, 3/5]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0037/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 38. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[13, 26, 39] at the fixed center=0, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(0, 2, [13, 26, 39/2], [[0, 13], [1, 26], [2, 39/2]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (0, 2, [13, 26, 39/2], [[0, 13], [1, 26], [2, 39/2]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0038/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 39. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[14, 28, 42, 56] at the fixed center=1, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(1, 3, [14, 28, 21, 28/3], [[0, 14], [1, 28], [2, 21], [3, 28/3]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (1, 3, [14, 28, 21, 28/3], [[0, 14], [1, 28], [2, 21], [3, 28/3]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0039/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 40. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[15, 30, 45, 60, 75] at the fixed center=2, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(2, 4, [15, 30, 45/2, 10, 25/8], [[0, 15], [1, 30], [2, 45/2], [3, 10], [4, 25/8]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (2, 4, [15, 30, 45/2, 10, 25/8], [[0, 15], [1, 30], [2, 45/2], [3, 10], [4, 25/8]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0040/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 41. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[16, 32, 48, 64, 80, 96] at the fixed center=3, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(3, 5, [16, 32, 24, 32/3, 10/3, 4/5], [[0, 16], [1, 32], [2, 24], [3, 32/3], [4, 10/3], [5, 4/5]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (3, 5, [16, 32, 24, 32/3, 10/3, 4/5], [[0, 16], [1, 32], [2, 24], [3, 32/3], [4, 10/3], [5, 4/5]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0041/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 42. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[17, 34, 51] at the fixed center=4, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(4, 2, [17, 34, 51/2], [[0, 17], [1, 34], [2, 51/2]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (4, 2, [17, 34, 51/2], [[0, 17], [1, 34], [2, 51/2]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0042/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 43. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[18, 36, 54, 72] at the fixed center=5, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(5, 3, [18, 36, 27, 12], [[0, 18], [1, 36], [2, 27], [3, 12]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (5, 3, [18, 36, 27, 12], [[0, 18], [1, 36], [2, 27], [3, 12]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0043/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 44. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[19, 38, 57, 76, 95] at the fixed center=6, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(6, 4, [19, 38, 57/2, 38/3, 95/24], [[0, 19], [1, 38], [2, 57/2], [3, 38/3], [4, 95/24]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (6, 4, [19, 38, 57/2, 38/3, 95/24], [[0, 19], [1, 38], [2, 57/2], [3, 38/3], [4, 95/24]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0044/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 45. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[20, 40, 60, 80, 100, 120] at the fixed center=7, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(7, 5, [20, 40, 30, 40/3, 25/6, 1], [[0, 20], [1, 40], [2, 30], [3, 40/3], [4, 25/6], [5, 1]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (7, 5, [20, 40, 30, 40/3, 25/6, 1], [[0, 20], [1, 40], [2, 30], [3, 40/3], [4, 25/6], [5, 1]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0045/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 46. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[21, 42, 63] at the fixed center=8, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(8, 2, [21, 42, 63/2], [[0, 21], [1, 42], [2, 63/2]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (8, 2, [21, 42, 63/2], [[0, 21], [1, 42], [2, 63/2]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0046/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 47. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[22, 44, 66, 88] at the fixed center=9, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(9, 3, [22, 44, 33, 44/3], [[0, 22], [1, 44], [2, 33], [3, 44/3]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (9, 3, [22, 44, 33, 44/3], [[0, 22], [1, 44], [2, 33], [3, 44/3]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0047/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 48. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[23, 46, 69, 92, 115] at the fixed center=10, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(10, 4, [23, 46, 69/2, 46/3, 115/24], [[0, 23], [1, 46], [2, 69/2], [3, 46/3], [4, 115/24]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (10, 4, [23, 46, 69/2, 46/3, 115/24], [[0, 23], [1, 46], [2, 69/2], [3, 46/3], [4, 115/24]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0048/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 49. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[24, 48, 72, 96, 120, 144] at the fixed center=11, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(11, 5, [24, 48, 36, 16, 5, 6/5], [[0, 24], [1, 48], [2, 36], [3, 16], [4, 5], [5, 6/5]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (11, 5, [24, 48, 36, 16, 5, 6/5], [[0, 24], [1, 48], [2, 36], [3, 16], [4, 5], [5, 6/5]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0049/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 50. Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Use the derivative values=[25, 50, 75] at the fixed center=12, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

View answer and explanation(12, 2, [25, 50, 75/2], [[0, 25], [1, 50], [2, 75/2]], coefficient_k_equals_derivative_k_over_k_factorial)

Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (12, 2, [25, 50, 75/2], [[0, 25], [1, 50], [2, 75/2]], coefficient_k_equals_derivative_k_over_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0050/)

Taylor polynomial · qf-mod43-taylor-coefficient-construction · candidate





## 51. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[1, 2, 3] of the explicit polynomial sum c_k*(x−a)^k, center a=-12, evaluation point=-14. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([1, 2, 3], [1, 2, 6], [1, 2, 3], -14, 9, 9, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([1, 2, 3], [1, 2, 6], [1, 2, 3], -14, 9, 9, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0051/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 52. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[2, 4, 6, 8] of the explicit polynomial sum c_k*(x−a)^k, center a=-11, evaluation point=-12. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([2, 4, 6, 8], [2, 4, 12, 48], [2, 4, 6, 8], -12, -4, -4, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([2, 4, 6, 8], [2, 4, 12, 48], [2, 4, 6, 8], -12, -4, -4, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0052/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 53. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[3, 6, 9, 12, 15] of the explicit polynomial sum c_k*(x−a)^k, center a=-10, evaluation point=-10. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([3, 6, 9, 12, 15], [3, 6, 18, 72, 360], [3, 6, 9, 12, 15], -10, 3, 3, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([3, 6, 9, 12, 15], [3, 6, 18, 72, 360], [3, 6, 9, 12, 15], -10, 3, 3, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0053/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 54. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[4, 8, 12, 16, 20, 24] of the explicit polynomial sum c_k*(x−a)^k, center a=-9, evaluation point=-8. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([4, 8, 12, 16, 20, 24], [4, 8, 24, 96, 480, 2880], [4, 8, 12, 16, 20, 24], -8, 84, 84, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([4, 8, 12, 16, 20, 24], [4, 8, 24, 96, 480, 2880], [4, 8, 12, 16, 20, 24], -8, 84, 84, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0054/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 55. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[5, 10, 15] of the explicit polynomial sum c_k*(x−a)^k, center a=-8, evaluation point=-6. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([5, 10, 15], [5, 10, 30], [5, 10, 15], -6, 85, 85, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([5, 10, 15], [5, 10, 30], [5, 10, 15], -6, 85, 85, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0055/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 56. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[6, 12, 18, 24] of the explicit polynomial sum c_k*(x−a)^k, center a=-7, evaluation point=-9. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([6, 12, 18, 24], [6, 12, 36, 144], [6, 12, 18, 24], -9, -138, -138, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([6, 12, 18, 24], [6, 12, 36, 144], [6, 12, 18, 24], -9, -138, -138, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0056/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 57. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[7, 14, 21, 28, 35] of the explicit polynomial sum c_k*(x−a)^k, center a=-6, evaluation point=-7. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([7, 14, 21, 28, 35], [7, 14, 42, 168, 840], [7, 14, 21, 28, 35], -7, 21, 21, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([7, 14, 21, 28, 35], [7, 14, 42, 168, 840], [7, 14, 21, 28, 35], -7, 21, 21, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0057/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 58. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[8, 16, 24, 32, 40, 48] of the explicit polynomial sum c_k*(x−a)^k, center a=-5, evaluation point=-5. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([8, 16, 24, 32, 40, 48], [8, 16, 48, 192, 960, 5760], [8, 16, 24, 32, 40, 48], -5, 8, 8, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([8, 16, 24, 32, 40, 48], [8, 16, 48, 192, 960, 5760], [8, 16, 24, 32, 40, 48], -5, 8, 8, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0058/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 59. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[9, 18, 27] of the explicit polynomial sum c_k*(x−a)^k, center a=-4, evaluation point=-3. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([9, 18, 27], [9, 18, 54], [9, 18, 27], -3, 54, 54, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([9, 18, 27], [9, 18, 54], [9, 18, 27], -3, 54, 54, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0059/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 60. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[10, 20, 30, 40] of the explicit polynomial sum c_k*(x−a)^k, center a=-3, evaluation point=-1. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([10, 20, 30, 40], [10, 20, 60, 240], [10, 20, 30, 40], -1, 490, 490, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([10, 20, 30, 40], [10, 20, 60, 240], [10, 20, 30, 40], -1, 490, 490, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0060/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 61. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[11, 22, 33, 44, 55] of the explicit polynomial sum c_k*(x−a)^k, center a=-2, evaluation point=-4. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([11, 22, 33, 44, 55], [11, 22, 66, 264, 1320], [11, 22, 33, 44, 55], -4, 627, 627, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([11, 22, 33, 44, 55], [11, 22, 66, 264, 1320], [11, 22, 33, 44, 55], -4, 627, 627, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0061/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 62. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[12, 24, 36, 48, 60, 72] of the explicit polynomial sum c_k*(x−a)^k, center a=-1, evaluation point=-2. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([12, 24, 36, 48, 60, 72], [12, 24, 72, 288, 1440, 8640], [12, 24, 36, 48, 60, 72], -2, -36, -36, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([12, 24, 36, 48, 60, 72], [12, 24, 72, 288, 1440, 8640], [12, 24, 36, 48, 60, 72], -2, -36, -36, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0062/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 63. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[13, 26, 39] of the explicit polynomial sum c_k*(x−a)^k, center a=0, evaluation point=0. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([13, 26, 39], [13, 26, 78], [13, 26, 39], 0, 13, 13, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([13, 26, 39], [13, 26, 78], [13, 26, 39], 0, 13, 13, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0063/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 64. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[14, 28, 42, 56] of the explicit polynomial sum c_k*(x−a)^k, center a=1, evaluation point=2. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([14, 28, 42, 56], [14, 28, 84, 336], [14, 28, 42, 56], 2, 140, 140, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([14, 28, 42, 56], [14, 28, 84, 336], [14, 28, 42, 56], 2, 140, 140, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0064/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 65. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[15, 30, 45, 60, 75] of the explicit polynomial sum c_k*(x−a)^k, center a=2, evaluation point=4. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([15, 30, 45, 60, 75], [15, 30, 90, 360, 1800], [15, 30, 45, 60, 75], 4, 1935, 1935, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([15, 30, 45, 60, 75], [15, 30, 90, 360, 1800], [15, 30, 45, 60, 75], 4, 1935, 1935, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0065/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 66. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[16, 32, 48, 64, 80, 96] of the explicit polynomial sum c_k*(x−a)^k, center a=3, evaluation point=1. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([16, 32, 48, 64, 80, 96], [16, 32, 96, 384, 1920, 11520], [16, 32, 48, 64, 80, 96], 1, -2160, -2160, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([16, 32, 48, 64, 80, 96], [16, 32, 96, 384, 1920, 11520], [16, 32, 48, 64, 80, 96], 1, -2160, -2160, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0066/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 67. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[17, 34, 51] of the explicit polynomial sum c_k*(x−a)^k, center a=4, evaluation point=3. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([17, 34, 51], [17, 34, 102], [17, 34, 51], 3, 34, 34, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([17, 34, 51], [17, 34, 102], [17, 34, 51], 3, 34, 34, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0067/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 68. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[18, 36, 54, 72] of the explicit polynomial sum c_k*(x−a)^k, center a=5, evaluation point=5. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([18, 36, 54, 72], [18, 36, 108, 432], [18, 36, 54, 72], 5, 18, 18, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([18, 36, 54, 72], [18, 36, 108, 432], [18, 36, 54, 72], 5, 18, 18, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0068/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 69. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[19, 38, 57, 76, 95] of the explicit polynomial sum c_k*(x−a)^k, center a=6, evaluation point=7. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([19, 38, 57, 76, 95], [19, 38, 114, 456, 2280], [19, 38, 57, 76, 95], 7, 285, 285, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([19, 38, 57, 76, 95], [19, 38, 114, 456, 2280], [19, 38, 57, 76, 95], 7, 285, 285, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0069/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 70. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[20, 40, 60, 80, 100, 120] of the explicit polynomial sum c_k*(x−a)^k, center a=7, evaluation point=9. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([20, 40, 60, 80, 100, 120], [20, 40, 120, 480, 2400, 14400], [20, 40, 60, 80, 100, 120], 9, 6420, 6420, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([20, 40, 60, 80, 100, 120], [20, 40, 120, 480, 2400, 14400], [20, 40, 60, 80, 100, 120], 9, 6420, 6420, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0070/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 71. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[21, 42, 63] of the explicit polynomial sum c_k*(x−a)^k, center a=8, evaluation point=6. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([21, 42, 63], [21, 42, 126], [21, 42, 63], 6, 189, 189, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([21, 42, 63], [21, 42, 126], [21, 42, 63], 6, 189, 189, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0071/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 72. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[22, 44, 66, 88] of the explicit polynomial sum c_k*(x−a)^k, center a=9, evaluation point=8. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([22, 44, 66, 88], [22, 44, 132, 528], [22, 44, 66, 88], 8, -44, -44, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([22, 44, 66, 88], [22, 44, 132, 528], [22, 44, 66, 88], 8, -44, -44, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0072/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 73. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[23, 46, 69, 92, 115] of the explicit polynomial sum c_k*(x−a)^k, center a=10, evaluation point=10. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([23, 46, 69, 92, 115], [23, 46, 138, 552, 2760], [23, 46, 69, 92, 115], 10, 23, 23, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([23, 46, 69, 92, 115], [23, 46, 138, 552, 2760], [23, 46, 69, 92, 115], 10, 23, 23, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0073/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 74. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[24, 48, 72, 96, 120, 144] of the explicit polynomial sum c_k*(x−a)^k, center a=11, evaluation point=12. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([24, 48, 72, 96, 120, 144], [24, 48, 144, 576, 2880, 17280], [24, 48, 72, 96, 120, 144], 12, 504, 504, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([24, 48, 72, 96, 120, 144], [24, 48, 144, 576, 2880, 17280], [24, 48, 72, 96, 120, 144], 12, 504, 504, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0074/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 75. A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Use shifted_coefficients=[25, 50, 75] of the explicit polynomial sum c_k*(x−a)^k, center a=12, evaluation point=14. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

View answer and explanation([25, 50, 75], [25, 50, 150], [25, 50, 75], 14, 425, 425, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([25, 50, 75], [25, 50, 150], [25, 50, 75], 14, 425, 425, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0075/)

Taylor polynomial · qf-mod43-taylor-exact-polynomial-reconstruction · candidate





## 76. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=missing_derivative, supplied derivatives=[1, 2], center=-12, order=2, alternate_center=-11. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(missing_derivative, 2, [1, 2], -12, -11, No, all_derivatives_zero_through_n_are_required)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_derivative, 2, [1, 2], -12, -11, No, all_derivatives_zero_through_n_are_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0076/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 77. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=wrong_center, supplied derivatives=[2, 4, 6], center=-11, order=2, alternate_center=-10. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(wrong_center, 2, [2, 4, 6], -11, -10, No, a_jet_is_bound_to_its_declared_center)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (wrong_center, 2, [2, 4, 6], -11, -10, No, a_jet_is_bound_to_its_declared_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0077/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 78. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=missing_factorial, supplied derivatives=[3, 6, 9], center=-10, order=2, alternate_center=-9. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(missing_factorial, 2, [3, 6, 9], -10, -9, No, coefficients_require_division_by_k_factorial)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_factorial, 2, [3, 6, 9], -10, -9, No, coefficients_require_division_by_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0078/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 79. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=remainder_claim, supplied derivatives=[4, 8, 12], center=-9, order=2, alternate_center=-8. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(remainder_claim, 2, [4, 8, 12], -9, -8, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (remainder_claim, 2, [4, 8, 12], -9, -8, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0079/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 80. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=missing_derivative, supplied derivatives=[5, 10], center=-8, order=2, alternate_center=-7. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(missing_derivative, 2, [5, 10], -8, -7, No, all_derivatives_zero_through_n_are_required)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_derivative, 2, [5, 10], -8, -7, No, all_derivatives_zero_through_n_are_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0080/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 81. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=wrong_center, supplied derivatives=[6, 12, 18], center=-7, order=2, alternate_center=-6. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(wrong_center, 2, [6, 12, 18], -7, -6, No, a_jet_is_bound_to_its_declared_center)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (wrong_center, 2, [6, 12, 18], -7, -6, No, a_jet_is_bound_to_its_declared_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0081/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 82. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=missing_factorial, supplied derivatives=[7, 14, 21], center=-6, order=2, alternate_center=-5. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(missing_factorial, 2, [7, 14, 21], -6, -5, No, coefficients_require_division_by_k_factorial)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_factorial, 2, [7, 14, 21], -6, -5, No, coefficients_require_division_by_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0082/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 83. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=remainder_claim, supplied derivatives=[8, 16, 24], center=-5, order=2, alternate_center=-4. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(remainder_claim, 2, [8, 16, 24], -5, -4, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (remainder_claim, 2, [8, 16, 24], -5, -4, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0083/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 84. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=missing_derivative, supplied derivatives=[9, 18], center=-4, order=2, alternate_center=-3. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(missing_derivative, 2, [9, 18], -4, -3, No, all_derivatives_zero_through_n_are_required)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_derivative, 2, [9, 18], -4, -3, No, all_derivatives_zero_through_n_are_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0084/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 85. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=wrong_center, supplied derivatives=[10, 20, 30], center=-3, order=2, alternate_center=-2. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(wrong_center, 2, [10, 20, 30], -3, -2, No, a_jet_is_bound_to_its_declared_center)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (wrong_center, 2, [10, 20, 30], -3, -2, No, a_jet_is_bound_to_its_declared_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0085/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 86. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=missing_factorial, supplied derivatives=[11, 22, 33], center=-2, order=2, alternate_center=-1. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(missing_factorial, 2, [11, 22, 33], -2, -1, No, coefficients_require_division_by_k_factorial)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_factorial, 2, [11, 22, 33], -2, -1, No, coefficients_require_division_by_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0086/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 87. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=remainder_claim, supplied derivatives=[12, 24, 36], center=-1, order=2, alternate_center=0. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(remainder_claim, 2, [12, 24, 36], -1, 0, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (remainder_claim, 2, [12, 24, 36], -1, 0, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0087/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 88. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=missing_derivative, supplied derivatives=[13, 26], center=0, order=2, alternate_center=1. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(missing_derivative, 2, [13, 26], 0, 1, No, all_derivatives_zero_through_n_are_required)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_derivative, 2, [13, 26], 0, 1, No, all_derivatives_zero_through_n_are_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0088/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 89. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=wrong_center, supplied derivatives=[14, 28, 42], center=1, order=2, alternate_center=2. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(wrong_center, 2, [14, 28, 42], 1, 2, No, a_jet_is_bound_to_its_declared_center)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (wrong_center, 2, [14, 28, 42], 1, 2, No, a_jet_is_bound_to_its_declared_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0089/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 90. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=missing_factorial, supplied derivatives=[15, 30, 45], center=2, order=2, alternate_center=3. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(missing_factorial, 2, [15, 30, 45], 2, 3, No, coefficients_require_division_by_k_factorial)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_factorial, 2, [15, 30, 45], 2, 3, No, coefficients_require_division_by_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0090/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 91. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=remainder_claim, supplied derivatives=[16, 32, 48], center=3, order=2, alternate_center=4. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(remainder_claim, 2, [16, 32, 48], 3, 4, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (remainder_claim, 2, [16, 32, 48], 3, 4, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0091/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 92. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=missing_derivative, supplied derivatives=[17, 34], center=4, order=2, alternate_center=5. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(missing_derivative, 2, [17, 34], 4, 5, No, all_derivatives_zero_through_n_are_required)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_derivative, 2, [17, 34], 4, 5, No, all_derivatives_zero_through_n_are_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0092/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 93. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=wrong_center, supplied derivatives=[18, 36, 54], center=5, order=2, alternate_center=6. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(wrong_center, 2, [18, 36, 54], 5, 6, No, a_jet_is_bound_to_its_declared_center)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (wrong_center, 2, [18, 36, 54], 5, 6, No, a_jet_is_bound_to_its_declared_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0093/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 94. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=missing_factorial, supplied derivatives=[19, 38, 57], center=6, order=2, alternate_center=7. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(missing_factorial, 2, [19, 38, 57], 6, 7, No, coefficients_require_division_by_k_factorial)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_factorial, 2, [19, 38, 57], 6, 7, No, coefficients_require_division_by_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0094/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 95. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=remainder_claim, supplied derivatives=[20, 40, 60], center=7, order=2, alternate_center=8. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(remainder_claim, 2, [20, 40, 60], 7, 8, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (remainder_claim, 2, [20, 40, 60], 7, 8, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0095/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 96. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=missing_derivative, supplied derivatives=[21, 42], center=8, order=2, alternate_center=9. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(missing_derivative, 2, [21, 42], 8, 9, No, all_derivatives_zero_through_n_are_required)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_derivative, 2, [21, 42], 8, 9, No, all_derivatives_zero_through_n_are_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0096/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 97. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=wrong_center, supplied derivatives=[22, 44, 66], center=9, order=2, alternate_center=10. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(wrong_center, 2, [22, 44, 66], 9, 10, No, a_jet_is_bound_to_its_declared_center)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (wrong_center, 2, [22, 44, 66], 9, 10, No, a_jet_is_bound_to_its_declared_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0097/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 98. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=missing_factorial, supplied derivatives=[23, 46, 69], center=10, order=2, alternate_center=11. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(missing_factorial, 2, [23, 46, 69], 10, 11, No, coefficients_require_division_by_k_factorial)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_factorial, 2, [23, 46, 69], 10, 11, No, coefficients_require_division_by_k_factorial).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0098/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 99. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=remainder_claim, supplied derivatives=[24, 48, 72], center=11, order=2, alternate_center=12. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(remainder_claim, 2, [24, 48, 72], 11, 12, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (remainder_claim, 2, [24, 48, 72], 11, 12, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0099/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate





## 100. Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries

Use mode=missing_derivative, supplied derivatives=[25, 50], center=12, order=2, alternate_center=13. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).

zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims.

View answer and explanation(missing_derivative, 2, [25, 50], 12, 13, No, all_derivatives_zero_through_n_are_required)

zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_derivative, 2, [25, 50], 12, 13, No, all_derivatives_zero_through_n_are_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0100/)

Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261112-r02-taylor-polynomial-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261112-r02-taylor-polynomial-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261112-r02-taylor-polynomial-foundations/validation-report.json)