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## Brook Taylor 1685 - 1731

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**12.4: Taylor Series**Brook Taylor was an accomplished musician and painter. He did research in a variety of areas, but is most famous for his development of ideas regarding infinite series. He added to mathematics a new branch now called the "calculus of finite differences", invented integration by parts, and discovered the celebrated series known as Taylor's expansion. Brook Taylor 1685 - 1731 http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Taylor.html Greg Kelly, Hanford High School, Richland, Washington**If we make , and the first, second, third**and fourth derivatives the same, then we would have a pretty good approximation. Suppose we wanted to find a fourth degree polynomial of the form: that approximates the behavior of at**If we plot both functions, we see that near zero the**functions match very well!**This pattern occurs no matter what the original function**was! Our polynomial: has the form: or:**Maclaurin Series:**(generated by f at ) Taylor Series: (generated by f at ) If we want to center the series (and it’s graph) at some point other than zero, we get the Taylor Series:**The more terms we add, the better our approximation.**On the TI-Nspire, the factorial symbol is b6 Probability Hint:**example:**Rather than start from scratch, we can use the function that we already know:**There are some Maclaurin series that occur often enough that**they should be memorized. They are on your “Stuff You Must Know Cold”. QUIZ SOON!**The 3rd order polynomial for is , but**it is degree 2. When referring to Taylor polynomials, we can talk about number of terms, order or degree. This is a polynomial in 3 terms. It is a 4th order Taylor polynomial, because it was found using the 4th derivative. It is also a 4th degree polynomial, because x is raised to the 4th power. The x3 term drops out when using the third derivative. A recent AP exam required the student to know the difference between order and degree. This is also the 2nd order polynomial.**Brackets means this is optional. If you don’t put anything**the default is 0, i.e. a Maclaurin series. taylor taylor taylor The TI-Nspire finds Taylor Polynomials: taylor (expression, variable, order [,point]) To be reminded of this notation look in the catalog. To jump to the T’s press b59 p