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

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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 , 2)? Solution . We need to find the directional derivative D u ( f ) at the point (1 , 1) 1
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in the direction of the unit vector h 1 2 , 1 2 i . Now D u ( f ) = f (1 , 1) · h 1 2 , 1 2 i = h 2 , 8 i · h 1 2 , 1 2 i = 10 / 2 .
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