A function f and a point P are given. Let Theta correspond to the direction of th directional derivative. Theta E [0,2π]
f(x, y) = √(12 - x2 - y2), P(-1, -1/√3).
A. Find the gradient and evaluate it at P. ~~~> I get: <√6/8, √2/8>
B. Find the angles thetha(with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change ~~~> I get: MaxIncrease: Θ = (π/6) MaxDecrease: Θ = (7π/6) ZeroChange: Θ = (2π/3).
C. Write the directional derivative at P as a function of theta; call this function g(theta).
D. Find the value of theta that maximizes g(theta) and find the maximum value.
e. Verify that the value of theta that maximizes g corresponds to the direction of the gradient.
E. Verify that the value of theta that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient