final_sp04

# final_sp04 - , 1), he is anxious to move in the direction...

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Name: Section Number: TA Name: Section Time: Math 10C. Final Examination June 7, 2004 Read each question carefully, and answer each question completely. Show all of your work. No credit will be given for unsupported answers. Write your solutions clearly and legibly. No credit will be given for illegible solutions. 1. (4 points) Find a vector equation for the line passing through the origin (0 , 0 , 0) and the center of the sphere ( x - 2) 2 + ( y - 3) 2 + ( z - 1) 2 = 10. # Score 1 2 3 4 5 6 7 8 Σ

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2. (4 points) Find an equation for the plane which passes through the point P (1 , 2 , 3) and is orthogonal (perpendicular) to the line x = 30 + t , y = 40 + 2 t , z = 100 - t .
3. (4 points) Compute ∂f ∂x for the following functions. (a) f ( x, y ) = x 2 e y cos x (b) f ( x, y ) = sin(3 xy 2 ) + y x

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4. (6 points) (a) Find an equation for the plane tangent to the graph of f ( x, y ) = ln(2 x + 3 y ) at the point ( - 1 , 1 , 0). (b) Find a linear approximation for f ( - 0 . 9 , 1 . 2)
5. (6 points) An ant is on a metal plate whose temperature at ( x, y ) is 3 x 2 y - y 3 degrees Celsius. When he is at the point (5

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Unformatted text preview: , 1), he is anxious to move in the direction in which the temperature drops the most rapidly. (a) Find the unit vector in the direction in which the temperature drops most rapidly at the instant he departs (5 , 1). (b) If the ant mistakenly moves toward the point (0 , 1), what rate of change will he experience at the instant he departs (5 , 1)? 6. (6 points) Let f ( x, y ) = 12 xy-3 x 2-2 y 3 . (a) Find all the critical points of f . (b) For each critical point of f , determine whether f has a local maximum, local minimum, or saddle point at that point. 7. (8 points) Find the absolute maximum and absolute minimum values of f ( x, y ) = 2 xy + y 2 + 8 x-4 y on the set D = { ( x, y ) | ≤ x ≤ 2 , ≤ y ≤ 1 } 8. (6 points) Use Lagrange multipliers to ﬁnd the maximum and minimum values of f ( x, y ) = x 2 y subject to the constraint x 2 + y 2 = 1....
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## This note was uploaded on 06/12/2008 for the course MATH 10C taught by Professor Hohnhold during the Spring '07 term at UCSD.

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final_sp04 - , 1), he is anxious to move in the direction...

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