This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ....
View
Full
Document
This note was uploaded on 02/27/2009 for the course MATH 155b taught by Professor Staff during the Fall '08 term at Vanderbilt.
 Fall '08
 Staff
 Antiderivatives, Derivative

Click to edit the document details