Worksheet_6 - Discussion Tuesday, September 28th Subject:...

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Discussion — Tuesday, September 28th Subject: Taylor series, the second derivative test, and changing coordinates. 1. Consider f ( x , y ) = 5 ( sin 2 ( x ) + y 2 ) - 3 e - ( x + 1) y + 3 e - y + 1. (a) Show that (0,0) is a critical point for f . (b) Calculate each of f xx , f xy , f yy at (0,0) and use this to write out the 2 nd -order Taylor approximation for f at (0,0). (c) To make sure the next two problems go smoothly, check your answer to (b) with the instructor. 2. Let g ( x , y ) be the approximation you obtained for f ( x , y ) near (0,0) in 1(b). (a) It’s not clear from the formula whether g , and hence f , has a min, max, or a saddle at (0,0). Test along several lines until you are convinced you’ve determined which type it is. (b) Check that you’re right in (a) using the 2 nd -derivative test. The next problem will help explain why this test works. 3. Consider alternate coordinates on R 2 where ( u , v ) corresponds to u (1,1) + v ( - 1,1). (a) Sketch the
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Worksheet_6 - Discussion Tuesday, September 28th Subject:...

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