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mathCalculus1-07-draft

mathCalculus1-07-draft - ARE211 Fall 2007 LECTURE#16 TUE...

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Preliminary draft only: please check for final version ARE211, Fall 2007 LECTURE #16: TUE, OCT 23, 2007 PRINT DATE: AUGUST 21, 2007 (CALCULUS1) Contents 4. Univariate and Multivariate Differentiation 1 4.1. The fundamental idea: linear approximations to nonlinear functions 2 4.2. Univariate Calculus 3 4. Univariate and Multivariate Differentiation Assume you all know how to calculate the derivative of a single variable function, i.e., given f , calculate d f ( · ) d x , denoted also f prime ( · ). Important to know the difference between f prime ( · ), which is a function and f prime ( x ), which is a number, the function evaluated at a point. I’ll try to be careful to use this notation from now on: g ( · ) is a RULE, represents a function. Since f prime ( · ) is a function, like any other, it may have a derivative; if it does, call it f primeprime ( · ). Do example f ( x ) = x 2 . Lots of standard kinds of functions you have to be able to differentiate in your sleep. Equivalent of being able to spell. Brainless activity. 1
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2 LECTURE #16: TUE, OCT 23, 2007 PRINT DATE: AUGUST 21, 2007 (CALCULUS1) x f df dx dx x * x * + dx * f ( x * ) f ( x * + dx * ) df = f prime ( x * ) dx f ( x * ) + f prime ( x * ) dx *
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