Directional derivative and gradient concepts

In summary, the conversation discusses various concepts related to directional derivatives, gradients, and level curves of functions in two variables. The first question asks about the relationship between directional derivatives and derivatives with respect to the x and y axes, while the second question involves the perpendicularity of the gradient and level curves. The following questions involve the application of these concepts to specific situations, including the use of unit vectors and the maximum and minimum rates of change. The conversation also mentions the use of the chain rule and the dot product form of the directional derivative.
  • #1
clairez93
114
0

Homework Statement



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 [tex]f(x,y) = x^{2} + y^{2}[/tex] then
[tex]\nabla f[/tex][tex]\bot [/tex] [tex]graph(f)[/tex].


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

[tex]f_{u} = D_{u}f = \nabla f[/tex].

b) For the same f and the unit vector v shown,
[tex]f_{v} = D_{v}f =[/tex](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.

Homework Equations





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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  • #2
do most of these assume a conitinuous differentiable function?

clairez93 said:

Homework Statement



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 [tex]f(x,y) = x^{2} + y^{2}[/tex] then
[tex]\nabla f[/tex][tex]\bot [/tex] [tex]graph(f)[/tex].


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

[tex]f_{u} = D_{u}f = \nabla f[/tex].


b) For the same f and the unit vector v shown,
[tex]f_{v} = D_{v}f =[/tex](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.

Homework Equations





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?
start with the directional derivative using the dot product, assuming f is differentiable at the point, then think vectors & linear independence
clairez93 said:
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?
not sure what graph(f) is either...
clairez93 said:
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...
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)
clairez93 said:
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.
doesn't make sense... aren't we onto v & cos(theta)? anyway write out the directional derivative...
clairez93 said:
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?
your reasoning is good here,

It helps me to think of the plane that is tangent to the function at a given point. The rate of maximum change on the plane is the direction of the gradient, and as you point out moving in the negative direction will have the same magnitude but a negtive rate of change (think of the dot product form of the directional derivative again)

when you move perpindicular to the gradient you're moving horizontally along the plane & the rate of change is zero (think dot product again)
clairez93 said:
Most of my answers here are complete guesswork. Help would be appreciated.
 

1. What is a directional derivative?

A directional derivative is a measure of how much a function changes in a specific direction at a given point. It is represented by a vector and can be thought of as the slope of the function in that direction.

2. How is the directional derivative calculated?

The directional derivative is calculated by taking the dot product of the gradient vector and a unit vector in the desired direction. This can also be expressed using partial derivatives of the function.

3. What is the significance of the gradient vector in directional derivatives?

The gradient vector represents the direction of steepest ascent of a function at a specific point. It is used to calculate the directional derivative in a specific direction and provides important information about the behavior of the function at that point.

4. Can the directional derivative be negative?

Yes, the directional derivative can be negative. This indicates that the function is decreasing in the specified direction at that point. A positive directional derivative indicates an increase in the function in that direction.

5. How is the gradient vector related to the level curves of a function?

The gradient vector is perpendicular to the level curves of a function. This means that the directional derivative in the direction of the gradient vector is 0, as there is no change in the function in that direction. The level curves can also be used to visualize the behavior of the function and its gradient at a given point.

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