Taylor+Polynomials+(BC)

__About the Taylor Series__: The "Taylor Series" is a special type of Power Series that allows us as mathematicians to express any given mathematical function (either complex or real) in terms of its n derivatives. Finding a Taylor Series is made possible by finding the Taylor Polynomials (of differing degrees) of a given function. Each term in the function is a power of //x// and multiplied by a different constant. In a sense, it's the power series around a given point.

Ex:

are determined based on the function's derivatives. For example, is based off of the function's second derivative, and so on. (The term's subscript indicates the value of the derivative used.)

 The Taylor Series is proved by a combination of the Fundamental Theorem of Calculus Part 1 and Part 2, as well as the Mean Value Theorem.

A Taylor Series that is centered at n=0 is also called a Maclaurin Series.

As more n values are taken into account (or as the degree of the polynomial increases), the function is able to "match" the original function better, as displayed by the graph above. This will more accurately express f(x) and will result in a smaller remainder. This fact is proven true through The Remainder Theorem.

__Taylor Polynomials in Context__: Take the function f(x) = sin(x) for example.

With the expansion point n=0, the series expansion would look like this:

However, as the expansion point is increased, the series changes. With the expansion point n=3, the series expansion would look like this:



__Taylor Polynomial Practice__: **Non-Calculator** Question: Find the third degree Taylor polynomial of  with center x <span style="font-family: Georgia,serif; font-size: 120%; vertical-align: sub;">0 <span style="font-family: Georgia,serif; font-size: 120%; vertical-align: super;">=3. <span style="font-family: Georgia,serif; font-size: 120%;">Answer: We must start by finding the first three derivatives of //f//(//x//) at x=3. <span style="font-family: Georgia,serif; line-height: 23px;"> <span style="font-family: Georgia,serif; line-height: 23px; vertical-align: super;">OR <span style="font-family: Georgia,serif; vertical-align: super;">OR

<span style="font-family: Georgia,serif; font-size: 120%;">So... <span style="font-family: Georgia,serif; font-size: 120%;"> <span style="font-family: Georgia,serif; font-size: 120%;"> <span style="font-family: Georgia,serif; font-size: 120%;">

<span style="font-family: Georgia,serif; font-size: 120%;">Therefore, the third degree Taylor polynomial would be as follows: <span style="font-family: Georgia,serif; font-size: 120%;"> <span style="font-family: Georgia,serif; font-size: 120%;">or with the series representation, <span style="font-family: Georgia,serif; font-size: 120%;">

<span style="font-family: Georgia,serif;">Question: Compute the Taylor series for the following: <span style="font-family: Georgia,serif; font-size: 120%;">Answer: We first must find the first 3 derivatives of the function f(x). <span style="font-family: Georgia,serif; font-size: 120%;">Using these derivatives, we can reason that f(0)=1, f'(0)=-1, f(0)=2, and f'(0)=-6. Therefore, the series would be:
 * // p // ∞(// x //) || = || [[image:http://img.sparknotes.com/figures/1/14f90917813a241b1325542a6f6f9d56/latex_img6.gif]][[image:http://img.sparknotes.com/figures/1/14f90917813a241b1325542a6f6f9d56/latex_img85.gif]]// x // n ||  ||
 * || = || [[image:http://img.sparknotes.com/figures/1/14f90917813a241b1325542a6f6f9d56/latex_img6.gif]](- 1)n // x // n ||  ||
 * || = || 1 - // x // + // x // 2 - // x // 3 + ... ||

sources? [] [] [] [] [] [] [|http://www.sparknotes.com/math/calcbc2/taylorseries/problems.html#explanation1]