Calculus 9.2 day 2

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9.2 day 2
Finding Common Maclaurin Series
Liberty Bell, Philadelphia, PA
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There are some Maclaurin series that occur often
enough that they should be memorized. They are
on your formula sheet, but today we are going to
look at where they come from.
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This is a geometric series with a 1 and r x.
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We could generate this same series for
with polynomial long division
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This is a geometric series with a 1 and r -x.
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We wouldnt expect to use the previous two series
to evaluate the functions, since we can evaluate
the functions directly.
They do help to explain where the formula for the
sum of an infinite geometric comes from.
We will find other uses for these series, as well.
A more impressive use of Taylor series is to
evaluate transcendental functions.
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Both sides are even functions.
Cos (0) 1 for both sides.
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Both sides are odd functions.
Sin (0) 0 for both sides.
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If we start with this function
This is a geometric series with a 1 and r -x2.
If we integrate both sides
This looks the same as the series for sin (x),
but without the factorials.
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We have saved the best for last!
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