lec15 - Lecture 15 18.01 Fall 2006 Lecture 15:...

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Unformatted text preview: Lecture 15 18.01 Fall 2006 Lecture 15: Differentials and Antiderivatives Differentials New notation: dy = f ( x ) dx ( y = f ( x )) Both dy and f ( x ) dx are called differentials . You can think of dy = f ( x ) dx as a quotient of differentials. One way this is used is for linear approximations. Δ y dy Δ x ≈ dx Example 1. Approximate 65 1 / 3 Method 1 (review of linear approximation method) f ( x ) = x 1 / 3 1 f ( x ) = x- 2 / 3 3 f ( x ) ≈ f ( a ) + f ( a )( x- a ) 1 x 1 / 3 ≈ a 1 / 3 + 3 a- 2 / 3 ( x- a ) A good base point is a = 64, because 64 1 / 3 = 4. Let x = 65. 1 1 1 1 65 1 / 3 = 64 1 / 3 + 64- 2 / 3 (65- 64) = 4 + (1) = 4 + 48 ≈ 4 . 02 3 3 16 Similarly, 1 (64 . 1) 1 / 3 ≈ 4 + 480 Method 2 (review) 1 / 3 1 1 1 65 1 / 3 = (64 + 1) 1 / 3 = [64(1 + )] 1 / 3 = 64 1 / 3 [1 + ] 1 / 3 = 4 1 + 64 64 64 1 1 Next, use the approximation (1 + x ) r ≈ 1 + rx with r = 3 and x = 64 ....
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This note was uploaded on 02/27/2009 for the course MATH 155b taught by Professor Staff during the Fall '08 term at Vanderbilt.

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lec15 - Lecture 15 18.01 Fall 2006 Lecture 15:...

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