A function f and a point P are given. Let 0 correspond to the direction of the d

Question

A function f and a point P are given. Let 0 correspond to the direction of the directional derivative. Complete parts (a) through (e). f(x,y) = 13 - 4x2 - 3y2, P(3,4) Find the gradient and evaluate it at P. Find the angles 0 (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero What angle(s) isare associated with the direction of maximum increase? (Type any angles in radians between 0 and 2ti. Type an exact answ-er, using 7t as needed. Use a comma to separate answ-ers as needed.) What angle(s) is are associated with the direction of maximum decrease? (Type any angles in radians between 0 and 2ti. Type an exact answ-er, using 7t as needed. Use a comma to separate answ-ers as needed.) What angle(s) isare associated with the direction of zero change? The angle(s) associated with the directions of zero change is'are (Type any angles in radians between 0 and 2ti. Type an exact answer, using n as needed. Use a comma to separate answers as needed.) Write the directional derivative at P as a function of 0; call this function g(0). g(9) = Find the value of 0 that maximizes g(0) and find the maximum value. What value of 0 maximizes g(0)? (Type any angles in radians between 0 and 2ti. Type an exact answer, using 7t as needed.) What is the maximum value? (Type an exact answer, using radicals as needed.) Verify that the value of 0 that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient. Are the values from part d consistent with the values from parts a and b?

Explanation / Answer

a.Grad f= -8xi-6yj
and at P, grad f= <-8*3,-6*4> = <-24,-24>

b.grad f lies in the direction -i-j.
So, directional derivative will be maximum in this direction, i.e. theta=5pi/4
directional derivative will be minimum in the direction opposite to that of max. derivative, i.e. theta=pi/4
and zero change will be observed in directions i-j or -i+j because in these cases the dot product of grad f and the unit vector in these directions will be 0.
Thus, theta=3pi/4,7pi/4.

c.g(theta)= - 24[cos(theta)+sin(theta)]

d.theta=5pi/4
g(theta)=24*sqrt(2).

e.It is clear that theta = direction of gradient = 5pi/4.
And max. value = magnitude of gradient = 24sqrt(2).
Yes, the values in (d) are consistent with (a) and (b).

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