1. Brook Taylor 1685 - 1731 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. Greg Kelly, Hanford High School, Richland, Washington 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. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Taylor.html

2. Suppose we wanted to find a fourth degree polynomial of the form: atthat approximates the behavior of If we make, and the first, second, third and fourth derivatives the same, then we would have a pretty good approximation.

5. If we plot both functions, we see that near zero the functions match very well!

6. This pattern occurs no matter what the original function was! Our polynomial: has the form: or:

7. Maclaurin 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: Taylor Series: (generated by f at )

8. example:

9. The more terms we add, the better our approximation. Hint: On the TI-Nspire, the factorial symbol is b6 Probability

10. example: Rather than start from scratch, we can use the function that we already know:

11. example:

12. 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!

13. 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 3rd order polynomial for is, but it is degree 2. The x 3 term drops out when using the third derivative. This is also the 2nd order polynomial. A recent AP exam required the student to know the difference between order and degree.

14. taylor (expression, variable, order [,point]) taylor  b59 The TI-Nspire finds Taylor Polynomials: Brackets means this is optional. If you don’t put anything the default is 0, i.e. a Maclaurin series. To be reminded of this notation look in the catalog. To jump to the T’s press