s10wk07 - Math 23 B. Dodson Week 7 Homework: 14.6...

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Math 23 B. Dodson Week 7 Homework: 14.6 directional derivatives 14.7 max/min 14.8 Lagrange multipliers 15.1 approximating sums for double integrals 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 , 0 > . So the rate of change is D ±u f (1 , - 3) = --→ grad · ±u = < - 3 , 0 >< - 4 5 , 3 5 > = 12 5 . We say that f is increasing at a rate of 2.4 in the direction of ±u.
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2 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 >, which at P (1 , - 3) is < - 3 1 , ln 1 > = < - 3 , 0 > . so the max rate of change is the length of the gradient, 3, which occurs in the direction of - ± i = < - 1 , 0 >, the direction of the gradient. Week 7 Homework:
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This note was uploaded on 03/26/2010 for the course MATH 23 taught by Professor Yukich during the Spring '06 term at Lehigh University .

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s10wk07 - Math 23 B. Dodson Week 7 Homework: 14.6...

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