8.8 Taylor &amp; Maclauren Series

# 8.8 Taylor &amp; Maclauren Series - 8.8   T a y l o...

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Unformatted text preview: 8 .8   T a y l o r  a n d  M a c l a u r e n  S e r ie s   Just as it’s easier to approximate some numbers  (π ≈ 3.14, 2 ≈ 1.414 ) , sometimes we want to  approximate a function close to a point with simpler terms –  for example we might want to use a polynomial.  Ex.  f ( x ) = e x  at x = 1  y x                                     In general the formula for the Taylor series of a function f at   x = a is  ∞ ∑ k =0 f ( k ) ( a) f ′′ ( a) f ( n ) ( a) k 2 ( x − a) = f ( a) + f ′ (a)( x − a) + ( x − a) + ... + ( x − a) n + ....  k! 2! n!   If a = 0, then the series is a Maclaurin series.  ∞ ∑ k =0 f ( k ) (0) k f ′′ (0) 2 f ( n ) (0) n ( x ) = f (0) + f ′ (0)( x ) + ( x ) + ... + ( x ) + ....  k! 2! n!   Ex.  The Maclaurin series for                f ( x ) = e x  is  Ex.  Find the Maclaurin series for  y = sin x .                            y = sin x  at  a = π .  Ex.  Find the Taylor series for  2           x Ex.  Find the Maclaurin series for  f ( x ) = 2 .                            x Ex.  Find the Taylor series for  f ( x ) = 2  at a = 1.              Ex.  Suppose we want to find the Maclaurin series for  f ( x ) = 2( −7 x) .                1 f ( x) = Do: 1.  Find the Maclaurin series for  2 − x .    1 f ( x) =         2.  Find the Taylor series for  2 − x  at a = 1.            ...
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## This note was uploaded on 10/02/2011 for the course AERO 1234 at Virginia Tech.

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