What would you like to understand?

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.

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 $a=0$

When the center changes from 0 to pi, the Taylor approximations shift their region of accuracy to the neighborhood of x=pi. The quadratic approximation changes from $1-x^2/2$ to -1+(x-pi)^$2/2$.

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

When the center changes from 0 to pi, the Taylor approximations shift their region of accuracy to the neighborhood of x=pi. The quadratic approximation changes from $1-x^2/2$ to -1+(x-pi)^$2/2$.

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