M23Fall02b

# M23Fall02b - faces in the coordinate planes and one vertex...

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Mathematics 23 Midterm Exam, November 5, 2002 1

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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 . b. Find ∂z ∂s when s = 2, t = 1. 2
2. Let z = f ( x,y ) = ( x 2 + y 2 ) 1 / 2 . a. Find the total diﬀerential dz . b. Find f x (3 , 4) and f y (3 , 4). c. Use your answers to parts (a) and (b) to estimate f (3 . 1 , 3 . 9). 3

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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)? 4. Evaluate R 1 0 R 1 y x 3 + 1 dxdy by changing the order of integration. 4
5. Find the volume of the largest rectangular box in the ﬁrst octant with three

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Unformatted text preview: faces in the coordinate planes and one vertex in the plane x + 2 y + 3 z = 6. Show your work. 5 6. Let R be the triangular lamina bounded by the x-axis, the y-axis, and the line x + y = 1. Let the density be ρ ( x,y ) = xy . The mass of R is 1 / 24. Find the x coordinate of the center of mass. 6 7. Find the volume bounded by the paraboloids z = 8 x 2 +8 y 2 and z = 9-x 2-y 2 . 7...
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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 - faces in the coordinate planes and one vertex...

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