The discussion revolves around finding the direction of the function f(x,y)=x^2+sin(4y) that increases most rapidly at the point P0=(1,0) and determining the derivative of f in that direction. The subject area includes multivariable calculus, specifically gradient vectors and directional derivatives.
Some participants have provided guidance on the role of the gradient in determining the direction of increase and have prompted further exploration of its magnitude. Multiple interpretations of the gradient's significance are being considered, but there is no explicit consensus on the next steps.
Participants are working under the constraints of a homework assignment, which may limit the information they can use or the methods they can apply. There is an ongoing discussion about the correctness of the gradient calculation and its implications for the problem.
Yes, for the first part of your question.Ki-nana18 said:Homework Statement
Find the direction in which the function f(x,y)=x^2+sin(4y) increses most rapidly at the point P0=(1,0). Then find the derivative of f in this direction.
Homework Equations
The Attempt at a Solution
I think I have to find the gradient at point P0 and then find a unit vector is this right?
Ki-nana18 said:Homework Statement
Find the direction in which the function f(x,y)=x^2+sin(4y) increses most rapidly at the point P0=(1,0). Then find the derivative of f in this direction.
The Attempt at a Solution
I think I have to find the gradient at point P0 and then find a unit vector is this right?
Ki-nana18 said:Okay, I found the gradient <2x+sin(4y), x^2+4cos(4y)> and at point P0 it is
<2,5>. Now if I only have one point how do I find the unit vector wouldn't I need another point or an initial vector?
Ki-nana18 said:Sorry. \nablaf=<2x, 4 cos(4y)>. Does the gradient at P0 tell me the direction in which the function increases most rapidly?