Given f(x, y) = 3x^2 + xy + y^2
Find the gradient ∇f(1, 1).
Then Find the rate of change of f(x, y) at the point (1, 1) in the direction of (1, 2)
0. Let's find the partial derivatives of f.
(d/dx)f(x,y) = 6x + y,(d/dy)f(x,y) = x + 2y.
So (d/dx)f(1,1) = 7, (d/dy)f(1,1) = 3.
1. Gradient is a vector formed from the partial derivatives. Therefore the answer is<7, 3>
2. The rate of change is equal to a*(d/dx)f(x,y) + b*(d/dy)f(x,y)where <a,b> is the direction vector of the length 1.
We have direction vector of the form <1, 2>. Its length is sqrt(5) so the corresponding unit vector (of the length 1) is <1/sqrt(5), 2/sqrt(5)>.
The rate of change at the point (1,1) is equal to(1/sqrt(5))*7 + (2/sqrt(5))*3 = 13/sqrt(5) = 13*sqrt(5)/5 which is approx = 5.81.
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