Taylor Series (11/12/08) Given a nice smooth function f (x): What is the best constant function to approximate it near 0? Best linear function to approximate it near 0? Best quadratic function to approximate it near 0? Best cubic function to approximate it near 0? Etc. etc.

Taylor Series continued If f(x) is a function for which we can compute all of its derivatives (i.e., first derivative f '(x), second derivative f (2) (x), third derivative f (3) (x), and so on), then we can write down a representation of f as a power series in terms of the values of those derivatives at the center of the interval of convergence. This is called the Taylor Series for f(x).

Taylor Series centered at 0 (also known as Maclaurin Series) We get that, on the interval of convergence, f(x) = f(0) + f '(0) x + f (2) (0) x 2 / 2! +… + f (n) (0) x n / n! +… = To estimate f, we can use the first so many terms (as a calculator does, and as you did in lab Monday).

Examples of series centered at 0 Since if f (x) = e x, then for every n, f (n) (x) = e x also, so f (n) (0) = 1. Hence the Taylor Series for e x is very simple: e x = 1 + x + x 2 /2! + … + x n /n! + …, which converges everywhere. Show that the Taylor series for sin(x) is sin(x) = x – x 3 /3! + x 5 /5! – x 7 /7! + …. By computing the derivatives of sin at 0.

Taylor Series centered at a If it is easier (or more useful) to evaluate f ‘s derivatives at a point a rather than zero, then we simply center the power series at a and evaluate the derivative at a: f(x) = f(a) + f '(a) (x-a) + f (2) (a) (x-a) 2 / 2! + …+ f (n) (a) (x-a) n / n! +… For example, what is the Taylor Series for ln(x) centered at x = 1? What is its interval of convergence?

Assignment For Friday, please read the following parts of Section 11.10: Page 734 through Example 1 on page 736, Examples 3 - 7, and Table 1 on page 743. On page 746 do Exercises 5, 7, 9, 13, 15, & 17. Monday will be an optional Q & A session. Wednesday (11/19) is Test #2.