M23Fall02b-sol - Mathematics 23 Midterm Exam, November 5,...

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Unformatted text preview: Mathematics 23 Midterm Exam, November 5, 2002. Solutions Answer all questions and be sure to show all work . No notes, books, or calculators are allowed. 1. Let z = ln( xy 2 ) ,x = s + t, and y = s 2 + t 2 . a. Find z x and z y . Solution . z x = 1 x , z y = 2 y . b. Find z s when s = 2, t = 1. Solution . z s = z x x s + z y y s = 1 x (1) + 2 y (2 s ) = 29 / 15 . 2. Let z = f ( x,y ) = ( x 2 + y 2 ) 1 / 2 . a. Find the total differential dz . Solution . dz = f x ( x,y ) dx + f y ( x,y ) dy = x ( x 2 + y 2 )- 1 / 2 dx + y ( x 2 + y 2 )- 1 / 2 dy. b. Find f x (3 , 4) and f y (3 , 4). Solution . f x (3 , 4) = 3 / 5 , f y (3 , 4) = 4 / 5. c. Use your answers to parts (a) and (b) to estimate f (3 . 1 , 3 . 9). Solution . f (3 . 1 , 3 . 9) = f (3 , 4) + f x (3 , 4)(1 / 10) + f y (3 , 4)(- 1 / 10) = 5- (1 / 50) . 3. Let f ( x,y ) = 10 + x 2 + 4 y 2 . Starting from (1 , 1), what is the rate of change of f ( x,y ) in the direction of the point (2...
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This note was uploaded on 11/06/2009 for the course MATH 23 taught by Professor Yukich during the Spring '06 term at Lehigh University .

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M23Fall02b-sol - Mathematics 23 Midterm Exam, November 5,...

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