In this section, we will investigate how we can approximate any function with a polynomial.

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Presentation transcript:

In this section, we will investigate how we can approximate any function with a polynomial.

Polynomials are “nice” functions. We want to construct, for any function, a degree n polynomial that matches a given function “perfectly” at some base point, x 0. But what does it mean to say “matches perfectly”?

Suppose f is the function of interest and p is the degree n polynomial to be constructed. We need: How does this happen?

Let f be any function such that its first n derivatives all exist at x = x 0. The Taylor Polynomial of Order n, based at x 0, is defined by: where

The Taylor polynomial of order 1 (i.e. n = 1) is called the linear approximation of f. The Taylor polynomial of order 2 (i.e. n = 2) is called the quadratic approximation of f. A Taylor polynomial with x 0 = 0 is often called a Maclaurin Polynomial.

Find the 2 nd, 4 th, and 6 th order Maclaurin polynomial for the function.

Find the 3 rd and 4 th order Taylor polynomial for the functioncentered at x = 7.

Find the quadratic approximation for the function centered at x = 2.