MATLAB Laboratory 10/14/10 Lecture. Taylor Polynomials
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1 MATLAB Laboratory 10/14/10 Lecture Taylor Polynomials Lisa A. Oberbroeckling Loyola University Maryland L. Oberbroeckling (Loyola University) MATLAB 10/14/10 Lecture 1 / 7
2 Taylor/Maclaurin Series Definition The general formula for the Taylor Series of a function f(x) centered at a is: L. Oberbroeckling (Loyola University) MATLAB 10/14/10 Lecture 2 / 7
3 Taylor/Maclaurin Series Definition The general formula for the Taylor Series of a function f(x) centered at a is: f(x) = f (k) (a) k! (x a) k L. Oberbroeckling (Loyola University) MATLAB 10/14/10 Lecture 2 / 7
4 Taylor/Maclaurin Series Definition The general formula for the Taylor Series of a function f(x) centered at a is: f(x) = f (k) (a) k! (x a) k using the conventions that f (0) = f and 0! = 1. L. Oberbroeckling (Loyola University) MATLAB 10/14/10 Lecture 2 / 7
5 Taylor/Maclaurin Series Definition The general formula for the Taylor Series of a function f(x) centered at a is: f(x) = f (k) (a) k! (x a) k using the conventions that f (0) = f and 0! = 1. The Maclaurin Series is a Taylor series when a = 0. L. Oberbroeckling (Loyola University) MATLAB 10/14/10 Lecture 2 / 7
6 Taylor/Maclaurin Series Definition The general formula for the Taylor Series of a function f(x) centered at a is: f(x) = f (k) (a) k! (x a) k using the conventions that f (0) = f and 0! = 1. The Maclaurin Series is a Taylor series when a = 0. This series converges for any x within the radius of convergence R. L. Oberbroeckling (Loyola University) MATLAB 10/14/10 Lecture 2 / 7
7 Taylor/Maclaurin Series Definition The general formula for the Taylor Series of a function f(x) centered at a is: f(x) = f (k) (a) k! (x a) k using the conventions that f (0) = f and 0! = 1. The Maclaurin Series is a Taylor series when a = 0. This series converges for any x within the radius of convergence R. Many, many uses L. Oberbroeckling (Loyola University) MATLAB 10/14/10 Lecture 2 / 7
8 Some Maclaurin Series Examples cos x cos x = ( 1) k x2k (2k)!, R = L. Oberbroeckling (Loyola University) MATLAB 10/14/10 Lecture 3 / 7
9 Some Maclaurin Series Examples cos x cos x = ( 1) k x2k (2k)!, R = sin x sin x = ( 1) k x 2k+1 (2k + 1)!, R = L. Oberbroeckling (Loyola University) MATLAB 10/14/10 Lecture 3 / 7
10 Some Maclaurin Series Examples cos x cos x = ( 1) k x2k (2k)!, R = sin x sin x = ( 1) k x 2k+1 (2k + 1)!, R = 1 1 x 1 1 x = x k, R = 1 L. Oberbroeckling (Loyola University) MATLAB 10/14/10 Lecture 3 / 7
11 Taylor Polynomials Definition The general formula for the n-th degree Taylor Polynomial P n of f(x) at a is: P n (x) = n f (k) (a) k! (x a) k L. Oberbroeckling (Loyola University) MATLAB 10/14/10 Lecture 4 / 7
12 Taylor Polynomials Definition The general formula for the n-th degree Taylor Polynomial P n of f(x) at a is: P n (x) = n f (k) (a) k! Main use: for approximating functions (x a) k L. Oberbroeckling (Loyola University) MATLAB 10/14/10 Lecture 4 / 7
13 Cosine Example Recall that cos x = ( 1) k x2k (2k)!, R = L. Oberbroeckling (Loyola University) MATLAB 10/14/10 Lecture 5 / 7
14 Cosine Example Recall that cos x = ( 1) k x2k (2k)!, R = Notice that this is obtained by using the general formula, then simplifying it to the above summation. L. Oberbroeckling (Loyola University) MATLAB 10/14/10 Lecture 5 / 7
15 Cosine Example Recall that cos x = ( 1) k x2k (2k)!, R = Notice that this is obtained by using the general formula, then simplifying it to the above summation. Then P 6 (x) for cos x would be P 6 (x) = 1 x2 2 + x4 4! x6 6! L. Oberbroeckling (Loyola University) MATLAB 10/14/10 Lecture 5 / 7
16 Cosine Example Recall that cos x = ( 1) k x2k (2k)!, R = Notice that this is obtained by using the general formula, then simplifying it to the above summation. Then P 6 (x) for cos x would be P 6 (x) = 1 x2 2 + x4 4! x6 6! Notice that this actually corresponds to P 6 (x) = 3 ( 1) k x2k (2k)!, R = L. Oberbroeckling (Loyola University) MATLAB 10/14/10 Lecture 5 / 7
17 Exponential Example Another common example is e x = x k k!, R = L. Oberbroeckling (Loyola University) MATLAB 10/14/10 Lecture 6 / 7
18 Exponential Example Another common example is e x = x k k!, R = Taylor polynomials for e x can be used to estimate e, for example. L. Oberbroeckling (Loyola University) MATLAB 10/14/10 Lecture 6 / 7
19 Inverse Tangent Example tan 1 x = ( 1) k 2k + 1 x2k+1, R = 1 L. Oberbroeckling (Loyola University) MATLAB 10/14/10 Lecture 7 / 7
20 Inverse Tangent Example tan 1 x = ( 1) k 2k + 1 x2k+1, R = y = tan 1 x y = P 3 y = P 7 Taylor Polynomial Example L. Oberbroeckling (Loyola University) MATLAB 10/14/10 Lecture 7 / 7
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