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
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 =
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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