advertisement

Taylor & MacClaurin Series Certain functions can be expressed as power series: f (x) an x an x n 0 n n0 n a0 a1x a2 x a3 x a4 x ... 2 How can we find the values of a0, a0 f (0) a1 ? 3 4 a1, a2, a3…? Plug in 0 for x and watch all the other terms go away a1 f (0) Plug in 0 for x again 2 3 f ( x) a1 2a2 x 3a3 x 4a4 x ... a0 f (0) a1 ? Plug in 0 for x a1 f (0) f ( x) a1 2a2 x 3a3 x 4a4 x ... 2 3 f (0) a2 a2 ? 2 2 f ( x) 2a2 6a3 x 12a4 x ... f (0) Can you find the pattern for finding an? a3 ? a3 6 f ( x) 6a3 24a4 x ... So the power series that converges to f (x) can be written as… n f (0) an n! f (x) an x n n 0 n 0 n f (0) n x n! f (0) 2 f (0) 3 f IV (0) 4 f ( x) f (0) f (0) x x x x ... 2 6 24 This is called the Taylor Series for f (x) centered at x = 0 because we used 0 to find all the terms. A series centered at x = 0 is also called a MacClaurin Series. In class, we will discuss how to generate a series centered at a point other than x = 0 Find the Taylor Series centered at x = 0 for the function f ( x) e x We would start by finding the first few derivatives and then looking for a pattern. But in this case the derivatives are easy… n 0 n f (0) n x n! f (0) f (0) f (0) f (0)... 1 1 2 1 3 1 4 e 1 x x x x ... 2 3! 4! x f (0) 2 f (0) 3 f IV (0) 4 f ( x) f (0) f (0) x x x x ... 2 6 24 Find the Taylor Series centered at x = 0 for the function f ( x) e x We would start by finding the first few derivatives and then looking for a pattern. But in this case the derivatives are easy… f (0) f (0) f (0) f (0)... 1 1 2 1 3 1 4 e 1 x x x x ... 2 3! 4! x …and the answer is… n x e n 0 n! x We can approximate the graph of ex by generating terms of the series… 2 x e 1 x 2 x f (x) = ex We can approximate the graph of ex by generating terms of the series… 2 x e 1 x 2 x f (x) = ex We can approximate the graph of ex by generating terms of the series… 2 3 x x e 1 x 2 6 x f (x) = ex We can approximate the graph of ex by generating terms of the series… 2 3 4 x x x e 1 x 2 6 24 x f (x) = ex What happens every time we add a term? The approximation gets better Where does the overlap appear to be centered? At x=0 In class you will need to be prepared to generate other series centered at any value of x A number of MacClaurin Series for some familiar functions can be found on page 477