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Unformatted text preview: x, y ) within 1 / 4 of (0 , π/ 4). Find an upper bound for the error in the linear approximation L (0 ,π/ 4) ( x, y ) of f for all ( x, y ) within 1 / 4 of (0 , π/ 4). Solution: Assuming f has continuous second partial derivatives, Taylor’s theorem gives that there exists a point c on the line segment joining ( x, y ) to (0 , π/ 4) such that  R 1 , (0 ,π/ 4) ( x, y )  = ± ± ± ± 1 2 ² f xx ( c ) x 2 + 2 f xy ( c ) x ( yπ/ 4) + f yy ( c )( yπ/ 4) 2 ³ ± ± ± ± ≤ 1 2 ²  f xx ( c )  x 2 + 2  f xy ( c )  x  yπ/ 4  +  f yy ( c )  ( yπ/ 4) 2 ³ ≤ 1 2 ² 2 x 2 + 4  x  yπ/ 4  + 2( yπ/ 4) 2 ³ ≤ x 2 + x 2 + ( yπ/ 4) 2 + ( yπ/ 4) 2 = 2 x 2 + 2( yπ/ 4) 2...
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.
 Spring '10
 WILKIE
 Approximation, Linear Approximation

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