The rate of change of a scalar function f in the direction of a unit vector is given by
This is the directional derivative
So, the maximum rate of change is given by
Hi, i need help in the following question..
Suppose I am on point (3,-8)
i am given F(x,y) = 80 - 2x^2 - y^2,
find the equation of the path i should take (in terms of x and y) if i want to move in the direction of maximum increase of F.
I know that the level curves are ellipses, and was given a hint to solve it by equating v'(u) = gradient vector of F.. where v(u)=(x(u),y(u)) is the vector representing the path.
however im stuck at this point, i wonder if anyone can help me? will very much appreciate it!
i agree this makes sense if i have a constant direction vector, and that the path is a straight line, however i believe the path is not a straight line because the path is a curve that is always tangent to the level curves, which are ellipses.
thus i do not think i am able to use this
opps haha i meant perpendicular, yeah thanks! but just wondering, how do i go about solving those equations?
am i correct to do it like this?
x(t) = -4xt + C, y(t) = -2yt + D
x(0) = 3, => C = 3,
y(0) = -8, => D = -8,
thus x = -4xt + 3, y = -2yt -8
solving t, (x-3)/(-4x) = (y+8)/(-2y)
thus i have -2yx + 6y = -4xy - 32x
=> 3y = xy -16x ?