To approximate the value of a function using a polynomial, especially useful when the function is difficult to compute directly.

Because they are centered at $x=0$, which often simplifies calculations.

The function $f(x)$ evaluated at $x=a$, i.e., $f(a)$.

Generally, a higher degree polynomial provides a more accurate approximation, especially closer to the center $x=a$.

A Taylor polynomial is a truncated Taylor series, taking only the first few terms of the infinite series.

Derivatives determine the coefficients of the polynomial terms, reflecting the function's rate of change at the center.

An infinite sum of terms expressed in terms of the function's derivatives at a single point.

A Taylor series centered at $x=0$.

The $n^{\text{th}}$ derivative of the function $f(x)$ evaluated at $x=a$.

The $n^{\text{th}}$ partial sum of the Taylor series.

Using a Taylor polynomial to estimate the value of a function at a specific point.

$\sum_{n=0}^infty \frac{f^{(n)}(a)}{n!}(x-a)^n$

$f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+...+\frac{f^{(n)}(a)}{n!}(x-a)^n$

$\frac{f^{(n)}(a)}{n!}(x-a)^n$

$\sum_{n=0}^infty \frac{f^{(n)}(0)}{n!}x^n$

$1+5x+\frac{25}{2}x^2+\frac{125}{6}x^3$