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Taylor & MacClaurin Series

Taylor & MacClaurin Series. Certain functions can be expressed as power series:. How can we find the values of a 0 , a 1 , a 2 , a 3 …?. Plug in 0 for x and watch all the other terms go away. Plug in 0 for x again. Plug in 0 for x. Can you find the pattern for finding a n ?.

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Taylor & MacClaurin Series

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  1. Taylor & MacClaurin Series

  2. Certain functions can be expressed as power series: How can we find the values of a0, a1, a2, a3…? Plug in 0 for x and watch all the other terms go away Plug in 0 for x again

  3. Plug in 0 for x Can you find the pattern for finding an?

  4. So the power series that converges to f (x) can be written as… This is called the Taylor Series for f (x)centered at x = 0 because we used 0 to find all the terms. A series centered at x = 0 is also called a MacClaurin Series. In class, we will discuss how to generate a series centered at a point other than x = 0

  5. Find the Taylor Series centered at x = 0 for the function We would start by finding the first few derivatives and then looking for a pattern. But in this case the derivatives are easy…

  6. Find the Taylor Series centered at x = 0 for the function We would start by finding the first few derivatives and then looking for a pattern. But in this case the derivatives are easy… …and the answer is…

  7. We can approximate the graph of ex by generating terms of the series… f (x) = ex

  8. We can approximate the graph of ex by generating terms of the series… f (x) = ex

  9. We can approximate the graph of ex by generating terms of the series… f (x) = ex

  10. We can approximate the graph of ex by generating terms of the series… f (x) = ex What happens every time we add a term? The approximation gets better Where does the overlap appear to be centered? At x = 0

  11. In class you will need to be prepared to generate other series centered at any value of x A number of MacClaurin Series for some familiar functions can be found on page 477

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