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Brook Taylor 1685 - 1731 9.2: 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.

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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.

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

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

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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 )

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example:

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The more terms we add, the better our approximation. Hint:On the TI-89, the factorial symbol is:

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

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example:

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There are some Maclaurin series that occur often enough that they should be memorized. They are on your formula sheet.

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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.

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The TI-89 finds Taylor Polynomials: taylor (expression, variable, order, [point]) taylor 9F3

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Warm up Construct the Taylor polynomial of degree 5 about x = 0 for the function f(x)=ex. Graph f and your approximation function for a graphical comparison.

Warm up Construct the Taylor polynomial of degree 5 about x = 0 for the function f(x)=ex. Graph f and your approximation function for a graphical comparison.

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