Find the gradient of f(x,y). f(x,y)=(x^2)e^-2y

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Thank you for pointing that out.In summary, the conversation is about finding the gradient of the function f(x,y)=(x^2)e^-2y. The poster provides their attempt at the gradient, but someone points out that they have made a mistake in the second part of the vector. The correct gradient is <2xe^-2y, -4xe^-2y>. The poster also clarifies that the mistakes were likely due to typos or carelessness.
  • #1
ffrpg
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Here's the problem. Find the gradient of f(x,y). f(x,y)=(x^2)e^-2y.


I don't have the solution to this and I need to know if I got the right gradient (I have more problems that depend on this gradient, points on it). I ended up getting, gradient f=<2xe^-2y, 2x-2e^-2y>. I don't think it's right, but can someone help me out here?
 
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  • #2
No.
grad f= fx(x,y)i + fy(x,y)j
fx(x,y)=(2x)e-2y
fy(x,y)=(-2*2x)e-2y
 
  • #3
Sorry, Stephen, you have fy wrong.

The derivative of e-2y with respect to y is -2 e-2y The other factor, x2 is independent of y so treat it like a constant fy= (x2)(-2e-2y)= -2x2e-2y.

The gradient of 2xe-2y is the vector <2x e-2y, -4xe-2y>.

What ffrpg wrote: f=<2x^e-2y, 2x-2e^-2y> may be typos or just carelessness: x^e-2y doesn't make much sense and in "2x-2..." you MEANT (2x) times (-2), not 2x subtract 2...
 
  • #4
Ah yes. Where is my head?
 

1. What is the definition of gradient in relation to a function?

The gradient of a function is a vector that points in the direction of the steepest increase of the function at a given point, and its magnitude represents the rate of change or slope of the function at that point.

2. How do you find the gradient of a two-variable function?

To find the gradient of a two-variable function, you need to take the partial derivatives of the function with respect to each variable. In other words, for a function f(x,y), the gradient is given by the vector [∂f/∂x, ∂f/∂y].

3. What is the formula for calculating the gradient of a function?

The formula for calculating the gradient of a function is ∇f(x,y) = [∂f/∂x, ∂f/∂y], where ∂f/∂x and ∂f/∂y are the partial derivatives of the function with respect to x and y, respectively.

4. How can you interpret the gradient of a function?

The gradient of a function represents the direction of greatest increase of the function at a given point. Its magnitude indicates the steepness of the function in that direction. A larger gradient indicates a steeper slope, while a smaller gradient indicates a gentler slope.

5. How do you find the gradient of f(x,y)=(x^2)e^-2y?

To find the gradient of f(x,y)=(x^2)e^-2y, we first take the partial derivative with respect to x: ∂f/∂x = 2xe^-2y. Then, we take the partial derivative with respect to y: ∂f/∂y = -2x^2e^-2y. Therefore, the gradient of f(x,y)=(x^2)e^-2y is given by the vector [2xe^-2y, -2x^2e^-2y].

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