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Unformatted text preview: Fall 2007 ARE211 Problem Set #10 Second Calculus Problem Set Due date: Nov 20 (1) Consider the function f ( x, y, z ) = xyz , with y = x 2 and z = x 1 / 3 . (a) Rewrite f as a function g : R R alone and compute g prime ( ). Using g prime , approximate the change in f when x increases by 0.1 units, starting from (8 , 64 , 2). (b) Compute the total derivative of f with respect to x . Using the total derivative, ap proximate the change in f when x increases by 0.1 units, starting from (8 , 64 , 2). (c) Write down the differential of f at (8 , 64 , 2). Using the differential, approximate the change in f when x increases by 0.1 units, starting from (8 , 64 , 2). (d) Identify the direction h * that ( x, y, z ) moves in, starting from (8 , 64 , 2), when x in creases. Write down the directional derivative of f in the direction h * , i.e., f h * ( , , ), and evaluate this derivative at (8 , 64 , 2). Using f h * (8 , 64 , 2), approximate the change in f when x increases by 0.1 units, starting from (8 , 64 , 2). (e) Check to see that all four of these distinct methods give you the same answer! (2) Recall that a function f : R n R m is nothing more than m functions, f 1 ...f m , each mapping R n R , and stacked on top of each other....
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 Fall '07
 Simon

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