# lec15 - MIT OpenCourseWare http/ocw.mit.edu 18.01 Single...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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± 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) ² ³ 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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lec15 - MIT OpenCourseWare http/ocw.mit.edu 18.01 Single...

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