Fixed-Coordinate Polynomial Partials and Ordered Gradients
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261013-advance-20261024-r04-partial-derivative-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[-12, -2, -3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-2, y=-3. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([-12, -2, -3, 1, -1, -1], -2, -3, -3)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -1], -2, -3, -3).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
2. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[-11, -1, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-1, y=-2. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([-11, -1, -2, 1, 0, -1], -1, -2, -3)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 1, 0, -1], -1, -2, -3).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
3. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[-10, 0, -1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=0, y=-1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([-10, 0, -1, 1, 1, -1], 0, -1, -1)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 1, 1, -1], 0, -1, -1).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
4. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[-9, 1, 0, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=1, y=0. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([-9, 1, 0, 1, -1, -1], 1, 0, 3)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 1, -1, -1], 1, 0, 3).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
5. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[-8, 2, 1, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=2, y=1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([-8, 2, 1, 1, 0, -1], 2, 1, 6)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, -1], 2, 1, 6).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
6. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[-7, -2, 2, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-2, y=2. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([-7, -2, 2, 1, 1, -1], -2, 2, -4)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 1, 1, -1], -2, 2, -4).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
7. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[-6, -1, 3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-1, y=3. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([-6, -1, 3, 1, -1, -1], -1, 3, -6)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 1, -1, -1], -1, 3, -6).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
8. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[-5, 0, -3, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=0, y=-3. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([-5, 0, -3, 1, 0, -1], 0, -3, 0)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 1, 0, -1], 0, -3, 0).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
9. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[-4, 1, -2, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=1, y=-2. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([-4, 1, -2, 1, 1, -1], 1, -2, 1)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, -1], 1, -2, 1).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
10. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[-3, 2, -1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=2, y=-1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([-3, 2, -1, 1, -1, -1], 2, -1, 7)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 1, -1, -1], 2, -1, 7).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
11. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[-2, -2, 0, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-2, y=0. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([-2, -2, 0, 1, 0, -1], -2, 0, -6)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 1, 0, -1], -2, 0, -6).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
12. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[-1, -1, 1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-1, y=1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([-1, -1, 1, 1, 1, -1], -1, 1, -2)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 1, 1, -1], -1, 1, -2).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
13. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[0, 0, 2, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=0, y=2. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([0, 0, 2, 1, -1, -1], 0, 2, -2)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, -1], 0, 2, -2).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
14. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[1, 1, 3, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=1, y=3. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([1, 1, 3, 1, 0, -1], 1, 3, 3)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 1, 0, -1], 1, 3, 3).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
15. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[2, 2, -3, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=2, y=-3. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([2, 2, -3, 1, 1, -1], 2, -3, 3)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 1, 1, -1], 2, -3, 3).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
16. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[3, -2, -2, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-2, y=-2. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([3, -2, -2, 1, -1, -1], -2, -2, -4)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 1, -1, -1], -2, -2, -4).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
17. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[4, -1, -1, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-1, y=-1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([4, -1, -1, 1, 0, -1], -1, -1, -3)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], -1, -1, -3).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
18. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[5, 0, 0, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=0, y=0. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([5, 0, 0, 1, 1, -1], 0, 0, 0)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 1, 1, -1], 0, 0, 0).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
19. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[6, 1, 1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=1, y=1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([6, 1, 1, 1, -1, -1], 1, 1, 2)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 1, -1, -1], 1, 1, 2).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
20. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[7, 2, 2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=2, y=2. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([7, 2, 2, 1, 0, -1], 2, 2, 6)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 1, 0, -1], 2, 2, 6).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
21. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[8, -2, 3, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-2, y=3. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([8, -2, 3, 1, 1, -1], -2, 3, -3)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -1], -2, 3, -3).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
22. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[9, -1, -3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-1, y=-3. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([9, -1, -3, 1, -1, -1], -1, -3, 0)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 1, -1, -1], -1, -3, 0).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
23. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[10, 0, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=0, y=-2. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([10, 0, -2, 1, 0, -1], 0, -2, 0)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 1, 0, -1], 0, -2, 0).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
24. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[11, 1, -1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=1, y=-1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([11, 1, -1, 1, 1, -1], 1, -1, 2)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 1, 1, -1], 1, -1, 2).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
25. Differentiate in the First Coordinate while Holding the Second Fixed
Use coefficients=[12, 2, 0, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=2, y=0. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).
View answer and explanation
([12, 2, 0, 1, -1, -1], 2, 0, 6)
Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, -1], 2, 0, 6).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-x · candidate
26. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[-12, -2, -3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-2, y=-3. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([-12, -2, -3, 1, -1, -1], -2, -3, 5)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -1], -2, -3, 5).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
27. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[-11, -1, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-1, y=-2. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([-11, -1, -2, 1, 0, -1], -1, -2, 2)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 1, 0, -1], -1, -2, 2).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
28. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[-10, 0, -1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=0, y=-1. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([-10, 0, -1, 1, 1, -1], 0, -1, 1)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 1, 1, -1], 0, -1, 1).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
29. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[-9, 1, 0, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=1, y=0. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([-9, 1, 0, 1, -1, -1], 1, 0, -1)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 1, -1, -1], 1, 0, -1).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
30. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[-8, 2, 1, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=2, y=1. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([-8, 2, 1, 1, 0, -1], 2, 1, -1)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, -1], 2, 1, -1).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
31. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[-7, -2, 2, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-2, y=2. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([-7, -2, 2, 1, 1, -1], -2, 2, -4)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 1, 1, -1], -2, 2, -4).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
32. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[-6, -1, 3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-1, y=3. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([-6, -1, 3, 1, -1, -1], -1, 3, -2)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 1, -1, -1], -1, 3, -2).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
33. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[-5, 0, -3, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=0, y=-3. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([-5, 0, -3, 1, 0, -1], 0, -3, 3)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 1, 0, -1], 0, -3, 3).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
34. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[-4, 1, -2, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=1, y=-2. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([-4, 1, -2, 1, 1, -1], 1, -2, 3)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, -1], 1, -2, 3).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
35. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[-3, 2, -1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=2, y=-1. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([-3, 2, -1, 1, -1, -1], 2, -1, -1)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 1, -1, -1], 2, -1, -1).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
36. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[-2, -2, 0, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-2, y=0. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([-2, -2, 0, 1, 0, -1], -2, 0, 0)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 1, 0, -1], -2, 0, 0).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
37. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[-1, -1, 1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-1, y=1. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([-1, -1, 1, 1, 1, -1], -1, 1, -2)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 1, 1, -1], -1, 1, -2).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
38. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[0, 0, 2, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=0, y=2. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([0, 0, 2, 1, -1, -1], 0, 2, -2)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, -1], 0, 2, -2).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
39. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[1, 1, 3, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=1, y=3. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([1, 1, 3, 1, 0, -1], 1, 3, -3)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 1, 0, -1], 1, 3, -3).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
40. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[2, 2, -3, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=2, y=-3. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([2, 2, -3, 1, 1, -1], 2, -3, 5)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 1, 1, -1], 2, -3, 5).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
41. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[3, -2, -2, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-2, y=-2. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([3, -2, -2, 1, -1, -1], -2, -2, 4)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 1, -1, -1], -2, -2, 4).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
42. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[4, -1, -1, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-1, y=-1. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([4, -1, -1, 1, 0, -1], -1, -1, 1)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], -1, -1, 1).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
43. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[5, 0, 0, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=0, y=0. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([5, 0, 0, 1, 1, -1], 0, 0, 0)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 1, 1, -1], 0, 0, 0).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
44. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[6, 1, 1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=1, y=1. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([6, 1, 1, 1, -1, -1], 1, 1, -2)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 1, -1, -1], 1, 1, -2).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
45. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[7, 2, 2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=2, y=2. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([7, 2, 2, 1, 0, -1], 2, 2, -2)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 1, 0, -1], 2, 2, -2).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
46. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[8, -2, 3, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-2, y=3. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([8, -2, 3, 1, 1, -1], -2, 3, -5)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -1], -2, 3, -5).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
47. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[9, -1, -3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-1, y=-3. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([9, -1, -3, 1, -1, -1], -1, -3, 4)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 1, -1, -1], -1, -3, 4).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
48. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[10, 0, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=0, y=-2. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([10, 0, -2, 1, 0, -1], 0, -2, 2)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 1, 0, -1], 0, -2, 2).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
49. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[11, 1, -1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=1, y=-1. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([11, 1, -1, 1, 1, -1], 1, -1, 2)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 1, 1, -1], 1, -1, 2).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
50. Differentiate in the Second Coordinate while Holding the First Fixed
Use coefficients=[12, 2, 0, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=2, y=0. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).
View answer and explanation
([12, 2, 0, 1, -1, -1], 2, 0, -2)
Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, -1], 2, 0, -2).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-partial-y · candidate
51. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[-12, -2, -3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=-2, y=-3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([-12, -2, -3, 1, -1, -1], -2, -3, [-3, 5])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -1], -2, -3, [-3, 5]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
52. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[-11, -1, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=-1, y=-2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([-11, -1, -2, 1, 0, -1], -1, -2, [-3, 2])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 1, 0, -1], -1, -2, [-3, 2]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
53. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[-10, 0, -1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=0, y=-1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([-10, 0, -1, 1, 1, -1], 0, -1, [-1, 1])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 1, 1, -1], 0, -1, [-1, 1]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
54. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[-9, 1, 0, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=1, y=0. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([-9, 1, 0, 1, -1, -1], 1, 0, [3, -1])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 1, -1, -1], 1, 0, [3, -1]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
55. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[-8, 2, 1, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=2, y=1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([-8, 2, 1, 1, 0, -1], 2, 1, [6, -1])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, -1], 2, 1, [6, -1]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
56. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[-7, -2, 2, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=-2, y=2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([-7, -2, 2, 1, 1, -1], -2, 2, [-4, -4])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 1, 1, -1], -2, 2, [-4, -4]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
57. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[-6, -1, 3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=-1, y=3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([-6, -1, 3, 1, -1, -1], -1, 3, [-6, -2])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 1, -1, -1], -1, 3, [-6, -2]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
58. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[-5, 0, -3, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=0, y=-3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([-5, 0, -3, 1, 0, -1], 0, -3, [0, 3])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 1, 0, -1], 0, -3, [0, 3]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
59. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[-4, 1, -2, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=1, y=-2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([-4, 1, -2, 1, 1, -1], 1, -2, [1, 3])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, -1], 1, -2, [1, 3]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
60. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[-3, 2, -1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=2, y=-1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([-3, 2, -1, 1, -1, -1], 2, -1, [7, -1])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 1, -1, -1], 2, -1, [7, -1]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
61. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[-2, -2, 0, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=-2, y=0. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([-2, -2, 0, 1, 0, -1], -2, 0, [-6, 0])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 1, 0, -1], -2, 0, [-6, 0]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
62. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[-1, -1, 1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=-1, y=1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([-1, -1, 1, 1, 1, -1], -1, 1, [-2, -2])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 1, 1, -1], -1, 1, [-2, -2]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
63. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[0, 0, 2, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=0, y=2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([0, 0, 2, 1, -1, -1], 0, 2, [-2, -2])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, -1], 0, 2, [-2, -2]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
64. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[1, 1, 3, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=1, y=3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([1, 1, 3, 1, 0, -1], 1, 3, [3, -3])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 1, 0, -1], 1, 3, [3, -3]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
65. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[2, 2, -3, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=2, y=-3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([2, 2, -3, 1, 1, -1], 2, -3, [3, 5])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 1, 1, -1], 2, -3, [3, 5]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
66. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[3, -2, -2, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=-2, y=-2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([3, -2, -2, 1, -1, -1], -2, -2, [-4, 4])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 1, -1, -1], -2, -2, [-4, 4]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
67. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[4, -1, -1, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=-1, y=-1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([4, -1, -1, 1, 0, -1], -1, -1, [-3, 1])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], -1, -1, [-3, 1]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
68. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[5, 0, 0, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=0, y=0. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([5, 0, 0, 1, 1, -1], 0, 0, [0, 0])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 1, 1, -1], 0, 0, [0, 0]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
69. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[6, 1, 1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=1, y=1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([6, 1, 1, 1, -1, -1], 1, 1, [2, -2])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 1, -1, -1], 1, 1, [2, -2]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
70. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[7, 2, 2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=2, y=2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([7, 2, 2, 1, 0, -1], 2, 2, [6, -2])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 1, 0, -1], 2, 2, [6, -2]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
71. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[8, -2, 3, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=-2, y=3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([8, -2, 3, 1, 1, -1], -2, 3, [-3, -5])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -1], -2, 3, [-3, -5]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
72. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[9, -1, -3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=-1, y=-3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([9, -1, -3, 1, -1, -1], -1, -3, [0, 4])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 1, -1, -1], -1, -3, [0, 4]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
73. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[10, 0, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=0, y=-2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([10, 0, -2, 1, 0, -1], 0, -2, [0, 2])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 1, 0, -1], 0, -2, [0, 2]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
74. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[11, 1, -1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=1, y=-1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([11, 1, -1, 1, 1, -1], 1, -1, [2, 2])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 1, 1, -1], 1, -1, [2, 2]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
75. Collect the Two Coordinate Partials in Their Declared Order
Use coefficients=[12, 2, 0, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=2, y=0. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.
View answer and explanation
([12, 2, 0, 1, -1, -1], 2, 0, [6, -2])
Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, -1], 2, 0, [6, -2]).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-gradient · candidate
76. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[-12, -2, -3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=-2, evaluation x=-3. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([-12, -2, -3, 1, -1, -1], -2, [-10, 0, 1], [0, 2], -3, -6)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -1], -2, [-10, 0, 1], [0, 2], -3, -6).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
77. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[-11, -1, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=-1, evaluation x=-2. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([-11, -1, -2, 1, 0, -1], -1, [-10, -1, 1], [-1, 2], -2, -5)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 1, 0, -1], -1, [-10, -1, 1], [-1, 2], -2, -5).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
78. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[-10, 0, -1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=0, evaluation x=-1. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([-10, 0, -1, 1, 1, -1], 0, [-10, 0, 1], [0, 2], -1, -2)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 1, 1, -1], 0, [-10, 0, 1], [0, 2], -1, -2).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
79. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[-9, 1, 0, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=1, evaluation x=0. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([-9, 1, 0, 1, -1, -1], 1, [-10, 0, 1], [0, 2], 0, 0)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 1, -1, -1], 1, [-10, 0, 1], [0, 2], 0, 0).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
80. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[-8, 2, 1, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=2, evaluation x=1. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([-8, 2, 1, 1, 0, -1], 2, [-10, 2, 1], [2, 2], 1, 4)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, -1], 2, [-10, 2, 1], [2, 2], 1, 4).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
81. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[-7, -2, 2, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=-2, evaluation x=2. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([-7, -2, 2, 1, 1, -1], -2, [-15, -4, 1], [-4, 2], 2, 0)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 1, 1, -1], -2, [-15, -4, 1], [-4, 2], 2, 0).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
82. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[-6, -1, 3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=-1, evaluation x=3. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([-6, -1, 3, 1, -1, -1], -1, [-10, 0, 1], [0, 2], 3, 6)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 1, -1, -1], -1, [-10, 0, 1], [0, 2], 3, 6).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
83. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[-5, 0, -3, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=0, evaluation x=-3. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([-5, 0, -3, 1, 0, -1], 0, [-5, 0, 1], [0, 2], -3, -6)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 1, 0, -1], 0, [-5, 0, 1], [0, 2], -3, -6).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
84. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[-4, 1, -2, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=1, evaluation x=-2. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([-4, 1, -2, 1, 1, -1], 1, [-7, 2, 1], [2, 2], -2, -2)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, -1], 1, [-7, 2, 1], [2, 2], -2, -2).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
85. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[-3, 2, -1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=2, evaluation x=-1. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([-3, 2, -1, 1, -1, -1], 2, [-9, 0, 1], [0, 2], -1, -2)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 1, -1, -1], 2, [-9, 0, 1], [0, 2], -1, -2).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
86. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[-2, -2, 0, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=-2, evaluation x=0. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([-2, -2, 0, 1, 0, -1], -2, [-6, -2, 1], [-2, 2], 0, -2)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 1, 0, -1], -2, [-6, -2, 1], [-2, 2], 0, -2).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
87. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[-1, -1, 1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=-1, evaluation x=1. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([-1, -1, 1, 1, 1, -1], -1, [-3, -2, 1], [-2, 2], 1, 0)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 1, 1, -1], -1, [-3, -2, 1], [-2, 2], 1, 0).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
88. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[0, 0, 2, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=0, evaluation x=2. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([0, 0, 2, 1, -1, -1], 0, [0, 0, 1], [0, 2], 2, 4)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, -1], 0, [0, 0, 1], [0, 2], 2, 4).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
89. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[1, 1, 3, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=1, evaluation x=3. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([1, 1, 3, 1, 0, -1], 1, [3, 1, 1], [1, 2], 3, 7)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 1, 0, -1], 1, [3, 1, 1], [1, 2], 3, 7).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
90. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[2, 2, -3, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=2, evaluation x=-3. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([2, 2, -3, 1, 1, -1], 2, [-8, 4, 1], [4, 2], -3, -2)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 1, 1, -1], 2, [-8, 4, 1], [4, 2], -3, -2).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
91. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[3, -2, -2, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=-2, evaluation x=-2. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([3, -2, -2, 1, -1, -1], -2, [3, 0, 1], [0, 2], -2, -4)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 1, -1, -1], -2, [3, 0, 1], [0, 2], -2, -4).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
92. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[4, -1, -1, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=-1, evaluation x=-1. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([4, -1, -1, 1, 0, -1], -1, [4, -1, 1], [-1, 2], -1, -3)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], -1, [4, -1, 1], [-1, 2], -1, -3).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
93. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[5, 0, 0, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=0, evaluation x=0. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([5, 0, 0, 1, 1, -1], 0, [5, 0, 1], [0, 2], 0, 0)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 1, 1, -1], 0, [5, 0, 1], [0, 2], 0, 0).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
94. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[6, 1, 1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=1, evaluation x=1. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([6, 1, 1, 1, -1, -1], 1, [6, 0, 1], [0, 2], 1, 2)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 1, -1, -1], 1, [6, 0, 1], [0, 2], 1, 2).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
95. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[7, 2, 2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=2, evaluation x=2. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([7, 2, 2, 1, 0, -1], 2, [7, 2, 1], [2, 2], 2, 6)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 1, 0, -1], 2, [7, 2, 1], [2, 2], 2, 6).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
96. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[8, -2, 3, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=-2, evaluation x=3. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([8, -2, 3, 1, 1, -1], -2, [-2, -4, 1], [-4, 2], 3, 2)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -1], -2, [-2, -4, 1], [-4, 2], 3, 2).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
97. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[9, -1, -3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=-1, evaluation x=-3. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([9, -1, -3, 1, -1, -1], -1, [11, 0, 1], [0, 2], -3, -6)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 1, -1, -1], -1, [11, 0, 1], [0, 2], -3, -6).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
98. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[10, 0, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=0, evaluation x=-2. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([10, 0, -2, 1, 0, -1], 0, [10, 0, 1], [0, 2], -2, -4)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 1, 0, -1], 0, [10, 0, 1], [0, 2], -2, -4).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
99. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[11, 1, -1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=1, evaluation x=-1. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([11, 1, -1, 1, 1, -1], 1, [9, 2, 1], [2, 2], -1, 0)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 1, 1, -1], 1, [9, 2, 1], [2, 2], -1, 0).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
100. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative
Use coefficients=[12, 2, 0, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=2, evaluation x=0. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.
View answer and explanation
([12, 2, 0, 1, -1, -1], 2, [8, 0, 1], [0, 2], 0, 0)
Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, -1], 2, [8, 0, 1], [0, 2], 0, 0).
Reviewed step-by-step derivation
Partial derivative · qf-mod24-partial-x-slice-derivative · candidate
Canonical question JSONL · English JSONL · Validation report