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Understanding Maclaurin Series: Foundations and Applications in Calculus

This resource provides an in-depth exploration of Maclaurin series, their derivation, and applications. It highlights commonly memorized series while explaining their origins through function derivatives. You'll learn about geometric series and how they relate to the evaluation of transcendental functions. Key concepts such as the infinite geometric series and Taylor series are illustrated, along with the unique interrelations of fundamental mathematical constants. Perfect for students and educators seeking clarity in calculus concepts.

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Understanding Maclaurin Series: Foundations and Applications in Calculus

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  1. 9.2 day 2 Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington Maclaurin Series Liberty Bell, Philadelphia, PA

  2. Maclaurin Series: (generated by f at ) 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.

  3. List the function and its derivatives.

  4. List the function and its derivatives. Evaluate column one for x = 0. This is a geometric series with a = 1 and r = x.

  5. We could generate this same series for with polynomial long division:

  6. This is a geometric series with a = 1 and r = -x.

  7. We wouldn’t 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.

  8. Both sides are even functions. Cos (0) = 1 for both sides.

  9. Both sides are odd functions. Sin (0) = 0 for both sides.

  10. and substitute for , we get: 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.

  11. We have saved the best for last!

  12. An amazing use for infinite series: Substitute xi for x. Factor out the i terms.

  13. Let This is the series for cosine. This is the series for sine. This amazing identity contains the five most famous numbers in mathematics, and shows that they are interrelated. p

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