# A Taylor polynomial retaining the center and factorials

[← Guide index](https://ocme.evemisslab.com/en/derivations/) · [Original MKO](https://ocme.evemisslab.com/en/mko/mko-taylor-polynomial/)

This is an owner-adopted natural-language teaching guide. Finite/arithmetic checks are not universal proofs; no Lean was run, and review scope is recorded per revision. Original source status labels are retained as historical data. Formulas retain exact TeX; no new semantic AST is invented.

derivation-mko-taylor-polynomial-01 · ocme-new142-reader-adoption-20261021



## Goal and scope

Construct T_2 at center a=1 for the specified cubic polynomial.

Construct a finite Taylor polynomial only; a finite jet is no general remainder or analyticity certificate.





## Givens and symbols


- algebra
- polynomial
- definition_here


- G1 — f(x)=1+2(x-1)+3(x-1)^2-(x-1)^3, a=1, n=2.


- h=x-1 — center shift
- T_n — finite Taylor polynomial





## Rules and sources


- mko_definition · cited — Fix center a and nonnegative order n and require every f^(k)(a) for0≤k≤n to exist. T_n(x)=sum over k=0..n of f^(k)(a)*(x−a)^k/k!; at n=0 this is constant f(a). The polynomial itself gives no remainder bound or guarantee that an infinite Taylor series converges or equals f. A polynomial f of degree at most n reconstructs exactly; this does not make an arbitrary smooth function analytic. · mko-taylor-polynomial
- algebra · prerequisite — Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities.
- polynomial · prerequisite — Real polynomials are continuous and differentiate termwise: the derivative of x^k is k*x^(k-1), and constants differentiate to zero.
- definition_here · prerequisite — The Taylor polynomial of order n at a sums f^(k)(a)*(x-a)^k/k! from k=0 through n and requires the complete center jet; at n=0 it is f(a).





## Main derivation

S01 · definitionBefore

h=x-1

After

f(x)=1+2h+3h^2-h^3

Conditions: none

Rule: definition_here · Uses: G1

Leaf basis: definition_here

Fix the shift h so every jet and power basis uses the same center a.



S02 · impliesBefore

f'(x)=2+6h-3h^2

After

f(1)=1,\quad f'(1)=2

Conditions: none

Rule: polynomial · Uses: S01

Leaf basis: polynomial

The derivative of h with respect to x is 1; setting h=0 gives the first two jet values.



S03 · impliesBefore

f''(x)=6-6h

After

f''(1)=6

Conditions: none

Rule: polynomial · Uses: S02

Leaf basis: polynomial

Differentiate again and evaluate at the center, completing the entire jet for orders zero through two.



S04 · definitionBefore

T_2(x)=\sum_{k=0}^{2}\frac{f^{(k)}(1)}{k!}(x-1)^k

After

T_2(x)=1+2h+\frac6{2!}h^2

Conditions: none

Rule: definition_here · Uses: S02, S03

Leaf basis: definition_here

Scale each jet value by its factorial in the centered power basis.



S05 · equivalentBefore

2!=2

After

T_2(x)=1+2(x-1)+3(x-1)^2

Conditions: none

Rule: algebra · Uses: S04

Leaf basis: algebra

Exact factorial division gives the required finite polynomial.



S06 · equivalentBefore

f(x)-T_2(x)

After

f(x)-T_2(x)=-(x-1)^3

Conditions: none

Rule: algebra · Uses: G1, S05

Leaf basis: algebra

The full function permits exact subtraction here; a general finite jet would not provide this difference formula.



S07 · impliesBefore

f^{(3)}(1)=-6

After

T_3(x)=T_2(x)-h^3=f(x)

Conditions: none

Rule: polynomial · Uses: G1

Leaf basis: polynomial

Since this f has degree three, adding the cubic coefficient -6/3!=-1 reconstructs it exactly.



S08 · definitionBefore

n=0\text{ as a separate order boundary}

After

T_0(x)=f(1)=1

Conditions: none

Rule: definition_here · Uses: S02, S05

Leaf basis: definition_here

At order zero there are no positive-power terms; the constant polynomial remains well-defined.







## Substeps and lemmas

No substeps in this guide.





## Reverse audits


- S08 · premise_audit · S02, S05 — Order zero does not mean there are no terms.
- S07 · premise_audit · G1 — Exact reconstruction for degree≤n is not extrapolated to arbitrary smooth functions.
- S06 · inverse_equivalence · G1, S05 — This identity does not claim a general Taylor remainder theorem.
- S05 · inverse_equivalence · S04 — Multiplying coefficient 3 by 2! recovers the second derivative 6.
- S04 · premise_audit · S02, S03 — The quadratic coefficient is not the unscaled derivative 6.
- S03 · premise_audit · S02 — No lower order is omitted in favor of only the highest one.
- S02 · premise_audit · S01 — Every derivative is evaluated at the same center 1.
- S01 · premise_audit · G1 — h is not x; the shift -1 cannot be omitted.





## Error controls and negative controls

FactorialsDivide each k-th derivative by k!, including 0!=1.

No implicit remainder promiseThe finite polynomial alone promises no interval error bound.


- The quadratic coefficient is 6. → reject — Division by 2! gives 3.
- The same jet can be moved directly to center zero. → reject — The jet is bound to center 1; a new center requires reevaluation.
- T_2 necessarily equals the full f. → reject — The difference here is a nonzero cubic term.





## Result and limitations

T_2=1+2(x-1)+3(x-1)^2, with difference -(x-1)^3; T_3 equals this f.







## Verification and source binding

1 declared checks for this guide; only the listed expressions are covered.


- source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
- adopted source SHA-256: 933a8e53367e588e2e0bd950532d7fde4ed554f553b2e996d3de73948108fed0
- original source SHA-256: 933a8e53367e588e2e0bd950532d7fde4ed554f553b2e996d3de73948108fed0
- Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

[Same-source full JSON (formulas, ASTs, controls and revision log)](https://ocme.evemisslab.com/data/derivations/coverage-20261021/guides/derivation-mko-taylor-polynomial-01.json)