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Bekki George Lecture notes 13

Bekki George Lecture notes 13 - Math 1432 Notes Week 13...

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Math 1432 Notes – Week 13 Recap of series:
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LecPop13_2 1. Determine whether the series converges or diverges. a) diverges b) converges c) cannot be determined
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11.5 Taylor Polynomials General procedure for finding an infinite polynomial series that converges to a "non7 polynomial" function. Taylor Polynomial at 0 The n th degree Taylor polynomial for f at x = 0 is P n ( x ) = f (0) + f '(0) x + f ''(0) 2! x 2 + f '''(0) 3! x 3 + + f n (0) n ! x n = f k (0) k ! x k k = 0 n provided f has n derivatives at 0. Example 1: Find ( ) 4 P x for ( ) x f x e = k ( ) k f x (0) k f (0) ! k f k COMPARE the graph of e x and the Taylor approximation to increasing values of n on the calculator & discuss "interval of convergence"
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Example 2: Find ( ) 5 P x for ( ) sin( ) f x x = k ( ) k f x (0) k f (0) ! k f k Example 3 a) Find ( ) 6 P x for 2 ( ) sin( ) f x x = b) Find ( ) 7 P x for ( ) 2 2 ( ) x f x xe =
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Example 4: Use the values in the table below and the formula for Taylor polynomials to give the 4 th degree Taylor polynomial for f centered at x = 0. f(0) f '(0) f ''(0) f '''(0) f (4) (0) -5 -2 -2 4 -5 Example 5: Find the Taylor polynomial P 4 ( x ) for the given function.
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Taylor’s Theorem: If f has n+1
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