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Unformatted text preview: Math 23 Sections 110-113 B. Dodson Week 7 Homework: 14.6 directional derivatives 14.7 max/min 14.8 Lagrange multipliers Problem 14.6.8: (a) Find the gradient of f ( x, y ) = y ln x . (b) Evaluate the gradient at P (1 ,- 3) . (c) Find the rate of change of f at P in the direction of the unit vector u = <- 4 5 , 3 5 > . Solution: The gradient is < y x , ln x >, which at P (1 ,- 3) is <- 3 1 , ln 1 > = <- 3 , > . So the rate of change is D u f (1 ,- 3) =-- grad u = <- 3 , ><- 4 5 , 3 5 > = 12 5 . We say that f is increasing at a rate of 2.4 in the direction of u. Problem 14.6.8d: Find the maximum rate of change of f at P and the direction in which it occurs. Solution. We had the gradient < y x , ln x >, 2 . which at P (1 ,- 3) is <- 3 1 , ln 1 > = <- 3 , > . so the max rate of change is the length of the gradient, 3, which occurs in the direction of- j = <- 1 , >, the direction of the gradient. Week 7 Homework: 14.7 max/min 14.8 Lagrange multipliers Problem 14.7.28: Find the absolute maximum and minumum values of the function f ( x, y...
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