# a8 - k is a constant for each k = 1 2,n and n X k =1 a 2 k...

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University of Toronto at Scarborough MATA33 Assignment 8 Winter 2008 Quiz 8 is based on this assignment and the relevant text readings and lecture notes. Quiz 8 will be written in the 9th tutorial, which takes place during the week of March 17 - 21. Study: Sections 17.5 and 17.6 for this assignment. Read ahead in Section 17.7. Problems: 1. Section 17.5, Pages 765 - 766 # 1, 2, 5, 9, 10, 16, 17, 19, 20, 22 - 24. 2. Section 17.6, Page 768 - 769 # 1 - 4, 9 - 13, 17, 20. 3. Page 796 # 20 - 22. 4. For the following two functions ﬁnd all point(s) ( x,y ) such that f x ( x,y ) = f y ( x,y ) = 0 (a) f ( x,y ) = x 2 + 2 y 2 - x 2 y (b) f ( x,y ) = xy + a 3 x + b 3 y where a and b are non-zero constants. 5. Verify that the function z = xe y + ye x is a solution to the equation z xxx + z yyy = xz xyy + yz xxy 6. Throughout this problem consider n real variables x 1 , x 2 ,..., x n and let u ( x 1 ,x 2 ,..., x n ) = n X k =1 a k x k where a
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Unformatted text preview: k is a constant for each k = 1 , 2 ,...,n and n X k =1 a 2 k = 1. Let z = e u ( x 1 , x 2 ,..., x n ) and verify that n X k =1 ∂ 2 z ∂x 2 k = z . 7. Let z = x 2 + xy + y 2 , x = s + t, and y = st . Find ∂z ∂s and ∂z ∂t two ways: (a) By ﬁrst substituting x and y as functions of s and t and diﬀerentiating directly. (b) By the chain rule. 8. Repeat problem 7 for the functions z = x y , x = se t , and y = 1 + se-t NOTES: 1. The FINAL EXAMINATION is on Tuesday, April 22, 9am-12noon in the GYM. A Review Sheet for the Exam will be posted at the Website in due course. 2. Friday, March 21 is a holiday and thus there are no MATA33 lectures on that day. That missed lecture will essentially be made-up on Tuesday, April 7 in accordance with the UTSC Calendar....
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