a5 - Math 237 1 Let f(x y = ln(x2 y 2 Assignment 5 Due...

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Math 237 Assignment 5 Due: Friday, Oct 22nd 1. Let f ( x, y ) = ln( x 2 + y 2 ). a) Find the directional derivative of f at ( - 1 , 2) in the direction of the vector v = (3 , - 4). b) Find the direction in which f is increasing the fastest at (1 , 1). What is the magnitude of this rate of change? c) Find the equation of the tangent line at (1 , 1) to the level curve f ( x, y ) = ln 2. 2. A space-ship cruising on the sunny side of the planet Venus starts to overheat. The space-ship is located at (1 , - 1 , 1) and the temperature of the ship’s hull when at location ( x, y, z ) will be T = 200 + e - x 2 - 2 y 2 - 3 z 2 , where x , y , z are in meters. a) In what direction should the ship proceed in order to decrease the temperature the quickest. b) If the ship travels at
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