m2a-20C-sp2006

# m2a-20C-sp2006 - ( x, y ) at P along these directions,...

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Print Name: Section Number: TA Name: Section Time: Math 20C. Midterm Exam 2 May 26, 2006 No calculators or any other devices are allowed on this exam. Write your solutions clearly and legibly; no credit will be given for illegible solutions. Read each question carefully. If any question is not clear, ask for clariFcation. Answer each question completely, and show all your work. 1. (a) (5 points) Find and sketch the domain of the function f ( x, t ) = ln(3 x + 2 t ). (b) (5 points) Find all possible constants c such that the function f ( x, t ) above is solution of the wave equation, f tt - c 2 f xx = 0. # Score 1 2 3 4 Σ

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2. (a) (5 points) Find the direction in which f ( x, y ) increases the most rapidly, and the directions in which f ( x, y ) decreases the most rapidly at P 0 , and also fnd the value o± the directional derivative o± f

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Unformatted text preview: ( x, y ) at P along these directions, where f ( x, y ) = x 3 e-2 y , and P = (1 , 0) . (b) (5 points) Find the directional derivative o f ( x, y ) above at the point P in the direction given by v = h 1 ,-1 i . 3. (a) (5 points) Find the tangent plane approximation of f ( x, y ) = x cos( y/ 2)-y 2 e-x at the point (0 , 1). (b) (5 points) Use the linear approximation computed above to approximate the value of f (-. 1 , . 9). 4. (10 points) Find every local and absolute extrema of f ( x, y ) = x 2 + 3 y 2 + 2 y on the unit disk x 2 + y 2 1, and indicate which ones are the absolute extrema. In the case of the interior stationary points, decide whether they are local maximum, minimum of saddle points....
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## m2a-20C-sp2006 - ( x, y ) at P along these directions,...

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