Provides a way to approximate the value of a function at a point using its derivatives at another point.

Approximating function values, estimating errors in approximation, and defining functions.

Lagrange Error Bound is a direct result of Taylor's Theorem, providing an upper bound for the error in Taylor approximations.

The error in approximating f(x) by its nth-degree Taylor polynomial is bounded by the (n+1)th term of the Taylor series, with the (n+1)th derivative evaluated at some point between the center and x.

Taylor's Theorem provides the theoretical basis for approximating functions with polynomials.

The existence of a point 'c' in the interval (a, x) such that the remainder term can be expressed in terms of the (n+1)th derivative at 'c'.

It is used to develop numerical methods for approximating solutions to equations and integrals.

The function must be n+1 times differentiable on an interval containing the point of approximation and the center.

By providing a formula for the remainder term, which represents the difference between the actual function value and the polynomial approximation.

The function value at the center, derivatives at the center, the point of approximation, and the remainder term.

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.

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