Notes_on_Taylor_Polynomials_and_Approximations.doc - Taylor Polynomials and Approximations Polynomial functions can be used to approximate other

# Notes_on_Taylor_Polynomials_and_Approximations.doc - Taylor...

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Taylor Polynomials and Approximations Polynomial functions can be used to approximate other elementary functions such as sin x , x e , and ln x . On your calculator, graph: 1 sin y x 3 5 2 3! 5! x x y x and compare. _______________________________________________________________________________ Definition of an nth-degree Taylor polynomial : If f has n derivatives at x = c , then the polynomial   2 ... 2! ! n n n f c f c P x f c f c x c x c x c n  is called the n th-degree Taylor polynomial for f at c , named after Brook Taylor, an English mathematician. ________________________________________________________________________________ If c = 0, then 2 0 0 0 0 ... 2! ! n n n f f P x f f x x x n  is called the n th-degree Maclaurin polynomial for f , named after another English mathematician, Colin Maclaurin. ________________________________________________________________________________ Ex . Find the Maclaurin polynomial of degree n = 4 for cos f x x , and then use 4 P x to approximate the value of cos 0.8, cos 1.1, and cos 4. Check your answers. What do you notice about your approximations?