Unformatted text preview: The linear approximation to y = x 2 . EXAMPLE 6.22 Let f ( x ) = √ x + 4. Then f ± ( x ) = 1 / (2 √ x + 4). The linear approximation to f at x = 5 is L ( x ) = 1 / (2 √ 5 + 4)( x5) + √ 5 + 4 = ( x5) / 6 + 3. As an immediate application we can approximate square roots of numbers near 9 by hand. To estimate √ 10, we substitute 6 into the linear approximation instead of into f ( x ), so √ 6 + 4 ≈ (65) / 6 + 3 = 19 / 6 ≈ 3 . 1 6. This rounds to 3 . 17 while the square root of 10 is actually 3 . 16 to two decimal places, so this estimate is only accurate to one decimal place. This is not too surprising, as 10 is really not very close to 9; on the other hand, for many calculations, 3 . 2 would be accurate enough....
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 Fall '07
 JonathanRogawski
 Math, Calculus, Numerical Analysis, Approximation, Linear Approximation, Vector Space

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