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# a4 - Dorthy experiences at time t = π 5 Let f R 2 → R...

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Math 237 Assignment 4 Due: Friday, Oct 16th 1. Determine all points where the function is diﬀerentiable. a) f ( x, y ) = ± x 4 - y 4 x 2 + y 2 , if ( x, y ) 6 = (0 , 0) 0 , if ( x, y ) = (0 , 0) . b) g ( x, y ) = | xy | . c) h ( x, y ) = ± x 4 / 3 y 1 / 3 x 2 + y 2 , if ( x, y ) 6 = (0 , 0) 0 , if ( x, y ) = (0 , 0) . 2. Let F = F ( x, y, t ), x = x ( s, t ) and y = y ( s, t ) and deﬁne w ( s, t ) = F ( x ( s, t ) , y ( s, t ) , t ). Write the chain rule for w s and w t . State any assumptions you need to make. 3. Let f ( s, t ) = xe xy 2 with x = x ( s, t ) = ( s + 2 t ) 2 and y = y ( s, t ) = cos( st ). Find f s and f t . 4. Dorthy’s position at any time t 0 is given by ( x, y, z ) = ( t cos t, 2 t sin t, t 2 ). The air temperature at any point ( x, y, z ) is given by a function T : R 3 R . If T ( - π, 0 , π 2 ) = (1 , - 1 , 2) then determine the rate of change of temperature
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Unformatted text preview: Dorthy experiences at time t = π . 5. Let f : R 2 → R and deﬁne g ( x, y ) = f (sin y, cos x ). Find g xx and g yy . State any assumptions you needed to make. 6. Let f, g : R → R where f and g are twice diﬀerentiable. Show that u ( x, t ) = f ( x-at ) + g ( x + at ) is a solution of the wave equation: u tt = a 2 u xx . 7. Let g : R → R and let f ( x, y ) = g ( u 2 v ), where u = e x and v = x 2 + y 3 . Use the chain rule to ﬁnd ∂ 2 f ∂x∂y . State any assumptions you needed to make....
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