# Directional derivative and gradient concepts

1. May 3, 2010

### clairez93

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

A series of true/false questions. I guess I don't understand the concepts of this very well:

1. If you know the directional derivative of f(x,y) in two different directions at a point P, we can find the derivative with respect to the x and y axes and thus we can determine the derivatives at this point P.

2. If $$f(x,y) = x^{2} + y^{2}$$ then
$$\nabla f$$$$\bot$$ $$graph(f)$$.

For the level curves in the figure and point P:
a) If u is a unit vector and the level curves of f(x,y) are given as shown, then at the point P, we have

$$f_{u} = D_{u}f = \nabla f$$.

b) For the same f and the unit vector v shown,
$$f_{v} = D_{v}f =$$(I can't get this to come out right, but it's supposed to say that the magnitude of the gradient of f times cos theta)

c) There is a function z = f(x,y) and a point P so that the maximum rate of change in f as you move away from P is 7 and the minimum rate of chang ein f as you move from P is -5.

2. Relevant equations

3. The attempt at a solution

1. Not sure about this, but I want to guess that it has something to do with the chain rule for many variables?

2. I want to say true, because I know that the gradient is always perpendicular to the level curve, but I'm not sure if that is what "graph(f)" refers to?

a) Not a clue really here. I want to say false because I don't think that the directional derivative is equal to the gradient, unless they're parallel? In any case I don't think that they would be equal to the partial derivative...

b) I kind of want to say false again because, because where did the unit vector go? The dot product of the gradient and the unit vector u should be equal to the directional derivative, but the u has disappeared, unless u is equal to 1? Which I don't think it is.

c) So they're trying to say that the magnitude of the gradient is 7 and the negative of that is -5? I think that's wrong, because isn't the maximum and minimum the same except that the minimum is the negative of the magnitude? So it can't be 7 and -5?

Most of my answers here are complete guesswork. Help would be appreciated.

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Last edited: May 3, 2010
2. May 3, 2010

### lanedance

do most of these assume a conitinuous differentiable function?

start with the directional derivative using the dot product, assuming f is differentiable at the point, then think vectors & linear independence
not sure what graph(f) is either....
once again use the dot product form of the directional derivative? note that u looks perpindicular to the level curves at that point... (i'd make that assumption)
doesn't make sense.... aren't we onto v & cos(theta)? anyway write out the directional derivative...