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The first Taylor polynomial approximation to $\cos (x)$ centered at $a=0$ is the constant line $y=1$. The second Taylor polynomial approximation is the quadratic curve $y=1-x^2/2$.

Conditions: Function is $\cos (x)$; Center is $a=0$

The first Taylor polynomial approximation to $\cos (x)$ centered at $a=0$ is the constant line $y=1$. The second Taylor polynomial approximation is the quadratic curve $y=1-x^2/2$.

Conditions: Function is $\cos (x)$; Center is $a=0$

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.