Finding the Direction of a Function

[Ki-nana18]

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?

 

[SammyS]

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?

Yes, for the first part of your question.

 

[Ki-nana18]

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?

 

[LCKurtz]

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?

What do you need the unit vector for?

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?

Your gradient isn't correct. You want ∇f = <f_x, f_y>.
Then think about this question: What does the gradient have to do with the maximum rate of increase of a function? The answer to that is surely in your text.

 

[Ki-nana18]

Sorry. \nablaf=<2x, 4 cos(4y)>. Does the gradient at P₀ tell me the direction in which the function increases most rapidly?

 

[LCKurtz]

Sorry. \nablaf=<2x, 4 cos(4y)>. Does the gradient at P₀ tell me the direction in which the function increases most rapidly?

Yes, and there's more. What does the magnitude of the gradient represent?

 

[Ki-nana18]

How fast the function increases?

 

[HallsofIvy]

In what direction? A function of several variables may have different rates of increase in different directions.