(1 point) consider a function f(r,y) at the point (2,1) At that point the functi

Question

(1 point) consider a function f(r,y) at the point (2,1) At that point the function has directional derivatives: rvio in the direction (parallel to) (3,1), and in the direction (parallel to) (2,2). The gradient of f at the point (2,1) is Note: You can earn partial credit on this problem. Preview My Answers Submit Answers You have attempted this problem 0 times. You have unlimited attempts remaining. Email instructor Page generated at 02/28/2017 at 08:36pm MST WeBWork 1996-2016 l theme: math4 l ww version: 2.11 l pg version 2 11l The WeBWork

Explanation / Answer

Directional derivative in the direction parallel to vector s is given by formula:

f/s = gradf · s/|s|, where gradf = (f/x , f/y ).

We are given two directional derivatives of the function f(x,y) at the point (2,1):

gradf ·(3, 1)/|(3, 1)| = 6/10

gradf · (2, 2)/|(2, 2)| = 2/8

Set gradf = (u, v) and substitute the length of the vectors:

(u,v) · (3, 1)/(3² + 1²) = 6/10
(u,v) · (2, 2)/(2² + 2²) = 2/8

(u,v) · (3, 1)/10 = 6/10
(u,v) · (2, 2)/8 = 2/8

(u,v) · (3, 1) = 6
(u,v) · (2, 2) = 2

3u + v = 6
u + v = 1

Solve this system of equations and you'll get that u = 5/2 , v = -3/2

Therefore the gradient of the f(x,y) at the point (2, 1) is (5/2, -3/2) <== ANSWER

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