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Unformatted text preview: Taylors Theorem If you look up Taylors theorem in any decent calculus book, it will tell you that you can take any function and express it as a sum of polynomial terms. You can use Taylors theorem to find alternative ways of calculating transcendental functions such as sine, cosine and exponentiation. . . cos = 1- 2 2! + 4 4!- 6 6! . . . sin = - 3 3! + 5 5!- 7 7! . . . e x = 1 + x + x 2 2! + x 3 3! + x 4 4! . . . For fun, take the derivative of each of these polynomial expressions. If everything was done correctly, the derivative of the function for sine should give you the function for cosine, the derivative of cosine should give you negative sine, and the derivative of e x should give you e x . Pretty cool, huh? You should also try plotting these polynomials on your calculator (do the trig functions in radians ). Imaginary Numbers In short, the numbers youre used to playing with are members of the set of real numbers. Theres another set that most people dont have occasion to use very often - the set of imaginary numbers. Take the square root of a positive real number - youll get a real number for an answer. Take the square root of a negative number, and youll get an imaginary number. - 1 i Imaginary numbers are multiples of i ....
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This note was uploaded on 11/12/2010 for the course PHYSICS 1B 318007241 taught by Professor Corbin during the Spring '10 term at UCLA.
- Spring '10