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In this four-lesson series, we'll define, derive and evaluate Taylor Polynomials and Maclaurin polynomials. We'll also learn about the remainder of a Taylor polynomial and calculate the degree of error in your approximation.
Higher-degree polynomial approximations can be represented using sigma notation. There is a mathematical convention that zero factorial equals one. These polynomial approximations are called Taylor polynomials. To find a Taylor polynomial, find all the necessary derivatives and evaluate them at x = c, then insert the values into the Taylor polynomial function. If you center the Taylor polynomial approximation of a function about the point x = 0, you get the Maclaurin polynomial approximation of the function.
Since the Taylor approximation is not exactly equal to the original function, there is an error term, called the remainder. If you add the remainder to the Taylor approximation, then the result does equal the original function. You can use the remainder expression to determine what the largest possible error is for your approximation.
Taught by Professor Edward Burger, this series comes from a comprehensive Calculus course. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/calculus. The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hopital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.
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