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a5_soln

# a5_soln - Math 237 1 Let f(x y = ln(x2 y 2 Assignment 5...

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Math 237 Assignment 5 Solutions 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). Solution: We ﬁrst ﬁnd the unit vector ˆ v in the direction of ~v . We have ˆ v = ~v k ~v k = (3 , - 4) 3 2 + 4 2 = ± 3 5 , - 4 5 ² . The partial derivatives of f are f x = 2 x x 2 + y 2 , f y = 2 y x 2 + y 2 , which are both continuous at ( - 1 , 2). So, we get D (3 , - 4) f ( - 1 , 2) = f ( - 1 , 2) · ˆ v = ± - 2 5 , 4 5 ² · ± 3 5 , - 4 5 ² = - 22 25 b) Find the direction in which f is increasing the fastest at (1 , 1). What is the magnitude of this rate of change? Solution: We have f = ³ 2 x x 2 + y 2 , 2 y x 2 + y 2 ´ so f (1 , 1) = (1 , 1) . Therefore, f is increasing the fastest in the direction of f (1 , 1) = (1 , 1) and the magnitude of this rate of change is k∇ f (1 , 1) k = k (1 , 1) k = 2. c) Find the equation of the tangent line at (1 , 1) to the level curve f ( x, y ) = ln 2. Solution: Since

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a5_soln - Math 237 1 Let f(x y = ln(x2 y 2 Assignment 5...

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