Question 11590: A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Original question and givens

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.

• G001 — 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.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-partial-derivative

Step-by-step derivation

original coefficients=[2, 2, -3, 1, 1, -1];basis=[1,x,y,x^2,xy,y^2]

u(x,y)=(2)+(2)*x+(-3)*y+(1)*x^2+(1)*x*y+(-1)*y^2

fixed-y slice fixed coordinate squared := mul(2, 2)

fixed-y slice fixed coordinate squared = 4

fixed-y slice fixed linear term := mul(-3, 2)

fixed-y slice fixed linear term = -6

fixed-y slice fixed square term := mul(-1, 4)

fixed-y slice fixed square term = -4

fixed-y slice constant coefficient := sum(2, -6, -4)

fixed-y slice constant coefficient = -8

fixed-y slice cross contribution := mul(1, 2)

fixed-y slice cross contribution = 2

fixed-y slice linear coefficient := sum(2, 2)

fixed-y slice linear coefficient = 4

fixed-y slice:vary x;fix y=2;original coefficients=[2, 2, -3, 1, 1, -1]

ascending x slice coefficients=[-8, 4, 1]

x slice derivative:g(t)=(-8)+(4)*t+(1)*t^2;point=-3;h!=0

g(-3+h)-g(-3)=(4)*h+(1)*(2*(-3)*h+h^2)

x slice derivative twice quadratic coefficient := mul(2, 1)

x slice derivative twice quadratic coefficient = 2

x slice derivative derivative point term := mul(2, -3)

x slice derivative derivative point term = -6

x slice derivative derivative value := sum(4, -6)

x slice derivative derivative value = -2

x slice derivative:h!=0;increment=h*((4)+2*(1)*(-3)+(1)*h)

difference quotient=-2+(1)*h

x slice derivative:quotient=-2+(1)*h;epsilon>0;delta=epsilon/1

0<abs(h)<delta gives abs(quotient-(-2))=abs(1)*abs(h)<epsilon;limit=-2

Reverse checks

• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Recompute x slice derivative derivative value with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute x slice derivative derivative point term with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute x slice derivative twice quadratic coefficient with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Recompute fixed-y slice linear coefficient with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute fixed-y slice cross contribution with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute fixed-y slice constant coefficient with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute fixed-y slice fixed square term with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute fixed-y slice fixed linear term with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute fixed-y slice fixed coordinate squared with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([2, 2, -3, 1, 1, -1], 2, [-8, 4, 1], [4, 2], -3, -2)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.