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Taylor polynomial approximations are considered local because their coefficients are computed using derivative information only at a single chosen center point a. While this guarantees a very close match at $x=a$ and in its immediate neighborhood, the polynomial can diverge drastically from the original function as x moves farther away from the center.

Conditions: A center point a must be chosen.; Approximation quality is strongest near a.

Taylor polynomial approximations are considered local because their coefficients are computed using derivative information only at a single chosen center point a. While this guarantees a very close match at $x=a$ and in its immediate neighborhood, the polynomial can diverge drastically from the original function as x moves farther away from the center.

Conditions: A center point a must be chosen.; Approximation quality is strongest near a.