Complete Centered Jets, Exact Factorial Coefficients and Finite Taylor Boundaries
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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.
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Taylor polynomial · qf-mod43-taylor-definition-boundary · candidate
Canonical question JSONL · English JSONL · Validation report