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Unformatted text preview: MTH 122 — Calculus II Essex County College — Division of Mathematics and Physics 1 Lecture Notes #11 — Sakai Web Project Material 1 Approximating Functions with Polynomials This particular sheet is based on an extra credit problem that I gave my MTH-121 students. It is based on taking derivatives and then fitting points to a particular polynomial, even though the differentiated function was not a polynomial. The basis of the project was to use increasing degree polynomials to approximate functions that have no obvious relationship to polynomials. For example the sine function near the origin can be approximated with an ever increasing degree polynomial of the form: sin x = x- x 3 3! + x 5 5!- x 7 7! + x 9 9!- x 11 11! + ··· Let’s take a look, and I’d like you to label the curves directly on the graph. Yes, they’re related to the sine function and the expansion above. One curve is the sine function, then a sequence of polynomials of degree 1, 3, 5, 7, 9 and 11. Take a look, it’s really amazing to see how this sequence proceeds. Figure 1: Sine and a sequence of polynomial functions. Seven graphs in all....
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