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Brook Taylor 1685 - 1731

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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. Brook Taylor 1685 - 1731. Greg Kelly, Hanford High School, Richland, Washington.

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slide1
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.

Brook Taylor

1685 - 1731

Greg Kelly, Hanford High School, Richland, Washington

slide2
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

slide7
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:

slide9
Hint:

On the TI-89, the factorial symbol is:

The more terms we add, the better our approximation.

slide10
example:

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

slide12
There are some Maclaurin series that occur often enough that they should be memorized. They are on your formula sheet.
slide13
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.

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taylor

taylor

taylor

The TI-89 finds Taylor Polynomials:

taylor (expression, variable, order, [point])

F3

9

p

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