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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.

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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

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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

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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

The more terms we add, the better our approximation.

On the TI-Nspire, the factorial symbol is b6

Probability

Hint:

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 “Stuff You Must Know Cold”.

QUIZ SOON!

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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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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