A3 - 1 in the direction of the vector ~v =(1 3 b Is there a...

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Math 237 Assignment 3 Due: Friday, Feb 6th 1. Determine all points where the function is differentiable. a) f ( x, y ) = | y | b) g ( x, y ) = ± x 2 y | x | + | y | if ( x, y ) 6 = (0 , 0) 0 if ( x, y ) = (0 , 0) . c) h ( x, y ) = ± x 2 / 3 y 4 / 3 x 2 + y 2 if ( x, y ) 6 = (0 , 0) 0 if ( x, y ) = (0 , 0) . 2. Write the chain rule for ² ∂w ∂y ³ x if w = F ( u, v, y ), and u = u ( x, y ), v = v ( x, y ), where x and y are independent variables. 3. Let f ( s, t ) = x arctan( x 2 + y 2 ) where x = x ( s, t ) = e st and y = y ( t ) = t 2 + 1. Find f s and f t . 4. Dorthy’s position at any time t is given by ( x, y, z ) = ( t cos t, t sin t, 2 t ). The air temperature at any point ( x, y, z ) is given by a function T : R 3 R . If T ( - π, 0 , 2 π ) = (1 , - 2 , 1) then determine the rate of change of temperature Dorthy experiences at time t = π . 5. Let f ( x, y ) = xy ln( x + y ). a) Find the directional derivative of f at (1
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Unformatted text preview: , 1) in the direction of the vector ~v = (1 , 3). b) Is there a direction at (2 ,-1) in which the rate of change of f is equal to 3? Justify your answer. c) Find the direction at (2 ,-1) in which the rate of change of f is the greatest. 6. Let f ( x, y ) = ± xy √ x 2 + y 2 + 1 if ( x, y ) 6 = (0 , 0) 1 if ( x, y ) = (0 , 0) . a) Show that f is not differentiable at (0 , 0). b) Find the directional derivative D (1 , 1) f (0 , 0). 7. Find the equation of the tangent plane to the surface z 2 = x 2 + 4 y 2 at the point (3 , 2 , 5)....
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This note was uploaded on 04/18/2010 for the course MATH 237 taught by Professor Wolczuk during the Spring '08 term at Waterloo.

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