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Unformatted text preview: Math 237 Assignment 5 Due: Friday, Feb 11th * 1. The temperature in a region of space is given by T ( x,y,z ) = e- 2 x (1 + 2 y )( 1 1+3 z ). A fly moves along the path ( x,y,z ) = (2 t, sin t,e t- 1). a) Find dT dt at t = 0. b) Observe that the direction of the flys path at t = 0 is (2 , 1 , 1). Find the directional derivative of T in the direction of the flys path at t = 0. c) Explain the physical difference between a) and b). * 2. Let f ( x,y,z ) = x 2 + 2 y 2- 3 z 2 . Use the gradient vector to find the equation of the tangent plane to the surface f ( x,y,z ) = 3 at the point (2 , 1 , 1). * 3. Compute the directional derivative D ~u f (0 , 0) in the direction ~u = ( u 1 ,u 2 ) of f ( x,y ) = ( x | y | x 2 + y 2 if ( x,y ) 6 = (0 , 0) if ( x,y ) = (0 , 0) * 4. A particle travels along the path ( x,y ) = ( t 2- t,e 3 t ) in a plane where the temperature at position ( x,y ) and time t is given by T ( x,y,t ) = 2 x 2 y sin t . Calculate the rate of change of temperature along the particles path with respect to time at any time...
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