# M329L3 - f x = P x R x = f x |{z P x f ξ x x − x |{z R o...

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2. REVIEW OF CALCULUS 3 Theorem (Taylor’s Theorem) . Let f C n [ a,b ] with f ( n +1) existing on [ a,b ] and x 0 [ a,b ] . For every x [ a,b ] , there exists a number ξ ( x ) between x 0 and x with f ( x ) = P n ( x ) + R n ( x ) where P n ( x ) = f ( x 0 ) + f 0 ( x 0 )( x x 0 ) + f 00 ( x 0 ) 2! ( x x 0 ) 2 + · · · + f ( n ) ( x 0 ) n ! ( x x 0 ) n and R n ( x ) = f ( n +1) ( ξ ( x )) ( n + 1)! ( x x 0 ) n +1 . Note. (1) f is the sum of a Taylor polynomial of order n (also called a Maclaurin polynomial if x 0 = 0) and a remainder term (or truncation error). (2) The case n = 0 is just the Mean Value Theorem (MVT).
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Unformatted text preview: f ( x ) = P ( x ) + R ( x ) = f ( x ) | {z } P ( x ) + f ( ξ ( x ))( x − x ) | {z } R o ( x ) (3) If f is a polynomial of degree r , then for n ≥ r , f ( x ) = P n ( x ). Maple. See taylor polynomials.mw and/or taylor polynomials.pdf...
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