Finding the Direction of a Function

1. Oct 23, 2011

Ki-nana18

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. 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?

2. Oct 23, 2011

SammyS

Staff Emeritus
Yes, for the first part of your question.

3. Oct 23, 2011

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?

4. Oct 23, 2011

LCKurtz

What do you need the unit vector for?

Your gradient isn't correct. You want ∇f = <fx, fy>.
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.

5. Oct 23, 2011

Ki-nana18

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

6. Oct 23, 2011

LCKurtz

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

7. Oct 24, 2011

Ki-nana18

How fast the function increases?

8. Oct 24, 2011

HallsofIvy

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

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