Sketch the gradient vector for the function

In summary, the gradient of the function will be approximately 1/(distance between the level curves in the direction of the gradient), and the run will be approximately 1/2 of the rise.
  • #1
Feodalherren
605
6

Homework Statement


Untitled.png



Homework Equations





The Attempt at a Solution


Ok so I think I know how to get the direction. It's going to be perpendicular to the tangent of the level curve and pointing in the direction where f(x,y) is increasing. So on the graph that was provided it will point in negative y and positive x.

I am completely lost on finding the length though.
 
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  • #2
Feodalherren said:

Homework Statement


Untitled.png



Homework Equations





The Attempt at a Solution


Ok so I think I know how to get the direction. It's going to be perpendicular to the tangent of the level curve and pointing in the direction where f(x,y) is increasing. So on the graph that was provided it will point in negative y and positive x.

I am completely lost on finding the length though.

The length of the gradient is going to proportional to the rate at which f(x,y) is increasing as you move along the direction of the gradient. Can't you approximate that by the distance between the level curves and the amount f(x,y) increases between them?
 
  • #3
The distance between level curves -2 and -1 = 1

Isn't that how much f(x,y) increases?
 
  • #4
Feodalherren said:
The distance between level curves -2 and -1 = 1

Isn't that how much f(x,y) increases?

Right. So the magnitude of the gradient is going be approximately 1/(distance between the level curves in the direction of the gradient). The denominator will depend on where you are on the level curves.
 
  • #5
Ok so in a simple rise/run, that means 1 is my rise, what is my run?

My professor has the right answer as "approx 2".
 
  • #6
Feodalherren said:
Ok so in a simple rise/run, that means 1 is my rise, what is my run?

My professor has the right answer as "approx 2".

At (4,6)? The run is the perpendicular distance between the level curves at that point. What's that "approx"?
 
  • #7
I guess it stands for "approximately".

This just isn't working out at all.. I'm so confused :/
 
  • #8
Feodalherren said:
I guess it stands for "approximately".

This just isn't working out at all.. I'm so confused :/

I guessed what "approx" meant, I was just asking you what your guess for the run was around the point (4,6). What's your guess for the distance between the level curves -1 and -2 near that point by reading off the graph? It is just a rise/run guesstimate.
 
  • #9
Well my guess is that the rise is 1.

As for the run that's where I get lost. Do i estimate in the same direction as the gradient, or in the X or Y direction? What's the deal here?
 
  • #10
Feodalherren said:
Well my guess is that the rise is 1.

As for the run that's where I get lost. Do i estimate in the same direction as the gradient, or in the X or Y direction? What's the deal here?

In the same direction as the gradient, i.e. perpedicular to the level curves. Looks to me like about 1/2 is a good guess around (4,6). Don't you agree?
 
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  • #11
Ah it makes sense in that case. I would have guessed that it would be in that direction too but I wasn't sure. I guess I'm just not really grasping the concept. I realize that a gradient is just essentially a derivative in any direction but for some reason I'm having a hard time wrapping my head around this.
Anyway, thanks a bunch for your patience!
 

FAQ: Sketch the gradient vector for the function

What is a gradient vector?

A gradient vector is a mathematical concept used in multivariable calculus to represent the direction and magnitude of the steepest ascent of a function at a given point. It is a vector that points in the direction of the greatest change in the function's value, and its magnitude represents the rate of change.

How is the gradient vector calculated?

The gradient vector is calculated by taking the partial derivatives of the function with respect to each of its variables and arranging them into a vector. For example, if the function is f(x, y), the gradient vector would be [∂f/∂x, ∂f/∂y].

What is the significance of the gradient vector?

The gradient vector is significant because it provides important information about the behavior of a function at a given point. Its direction indicates the direction of the steepest ascent, which is useful in optimization problems. Its magnitude represents the rate of change and can be used to determine the slope of a tangent line to the function's graph at that point.

How do you sketch the gradient vector for a function?

To sketch the gradient vector for a function, first calculate the gradient vector using the partial derivatives of the function. Then, plot the vector starting at the point of interest and in the direction indicated by the vector. The length of the vector can be scaled to represent the magnitude of the gradient.

Can the gradient vector be negative?

Yes, the gradient vector can have negative components. This indicates that the function is decreasing in the direction of that component. However, the magnitude of the gradient vector is always positive, as it represents the rate of change of the function.

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