## taylor polynomials

A Taylor Polynomial has the general equation:

## taylor polynomials and maclaurin series

Taylor polynomials are very similar in form to the Maclaurin series; however, there is one major difference: Maclaurin series are always centered at 0; the "a" value for Maclaurin series is 0. Always. The "a" value for Taylor polynomials can be anything - including 0. The important thing to remember: a Maclaurin series is always a Taylor polynomial, but a Taylor polynomial is most often not a Maclaurin series. (See "Maclaurin series")

## terms to know

**ORDER:**

An

**order**is how many derivatives it takes to get to x amount of terms.

Example: cos(x) = 1 - x^2/2 + x^4/(4!) could be a fourth or fifth order Taylor polynomial, for the fifth derivative of

cos(x) = 0.

**DEGREE:**

A

**degree**is the power of the highest derivative term.

Example: cos(x) = 1 - x^2/2 + x^4(4!) is a fourth degree Taylor polynomial, for the highest power in a term is 4.

Note: Order and degree are the same if no derivatives of the original equation equal 0; otherwise, the order can be one more than the degree.

## uses and relation to other topics

Taylor Polynomials are generally used to find the graph of functions that are not common, such as sin(7 degrees). The same basic form (without the sigma) is used for Maclaurin series (where a = 0) and for finding the error of a truncated series.