Taylor polynomials

Definition

Taylor polynomials are approximations of functions as polynomials derived from the function's derivatives at a single point. They provide a means to estimate the value of functions near that point.

5 Must Know Facts For Your Next Test

1. The $n$th-degree Taylor polynomial for a function $f(x)$ centered at $a$ is given by $$P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n$$.
2. Taylor polynomials are used to approximate functions locally around the center point $a$.
3. The error in approximating a function using its Taylor polynomial is given by the remainder term, often denoted as $R_n(x)$.
4. When the center point $a = 0$, the Taylor polynomial is specifically called a Maclaurin polynomial.
5. Taylor series can converge to the function they approximate if certain conditions are met regarding the limit of their remainder term.

Review Questions

• What is the general form of an $n$th-degree Taylor polynomial for a function $f(x)$ centered at a point $a$?
• How does a Maclaurin polynomial differ from a Taylor polynomial?
• What role does the remainder term play in the accuracy of Taylor polynomial approximations?

Related terms

Power Series: An infinite series of the form $$\sum_{n=0}^{\infty} c_n (x - a)^n$$ where $c_n$ are coefficients and $a$ is the center.

Maclaurin Series: A special case of Taylor series centered at zero, i.e., $$\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n$$.

Radius of Convergence: $R$, such that within this radius, a power series converges to its corresponding function.