Taylor polynomials use derivatives at a single point to create a polynomial that mimics the function's behavior near that point.

Higher degree polynomials generally provide better approximations, but require more calculations.

It provides the worst-case scenario for the error, ensuring the error is less than the calculated bound.

The closer x is to 'a', the more accurate the Taylor series approximation is likely to be.

It quantifies the maximum possible error when using a Taylor polynomial to approximate a function.

A Maclaurin series is a special case of a Taylor series where the series is centered at x=0.

The factorials compensate for the repeated differentiation, ensuring the polynomial matches the function's derivatives at the center.

The remainder represents the error between the actual function value and the Taylor polynomial approximation.

It gives an upper limit on the absolute value of the error, ensuring the approximation is within a certain range of accuracy.

It allows us to determine the degree of the Taylor polynomial needed to achieve a desired level of accuracy.

Approximation of functions using polynomial expressions by finding derivatives.

A Taylor polynomial centered at 0.

The maximum possible error when approximating a function using a Taylor polynomial.

The difference between the actual function value and the Taylor polynomial approximation: $f(x) - P_n(x)$.

The point 'a' around which the Taylor series approximates the function's behavior.

A Taylor polynomial of degree n, approximating a function f(x).

The nth derivative of the function f(x) evaluated at x=a.

The maximum possible error in a Taylor polynomial approximation.

The function f(x) equals the Taylor polynomial $P_n(x)$ plus the remainder $R_n(x)$.

The interval between the center of the Taylor series, 'a', and the point of approximation, 'x'.

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

$R_n(x) = \dfrac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$

$f(x) = P_n(x) + R_n(x)$

$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$

$sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ...$

$cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ...$

$\frac{1}{1-x} = 1 + x + x^2 + x^3 + ...$

$ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ...$

$|R_n(x)| \le \frac{M}{(n+1)!}|x-a|^{n+1}$, where M is the max of $|f^{(n+1)}(z)|$ on the interval between a and x.

$P_3(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3$