Determine gradient of a function f(x,y)

In summary, the conversation is about finding a vector normal to a curve using the gradient and determining the equation for the tangent line at a specific point on the curve. The gradient is found by expressing the curve as f(x,y)=C and evaluating it at a given point. The tangent line is perpendicular to the normal, making it vertical in this case.
  • #1
-EquinoX-
564
1

Homework Statement


View the curve below as a contour of f(x,y).
(y-x)^2 + 2 = xy - 3

Use gradf (2,3) to find a vector normal to the curve at (2,3).


Homework Equations





The Attempt at a Solution


I am not sure how do I get the vector normal to the curve, is it using a cross product>?
 
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  • #2


No. Express your curve as f(x,y)=C and use the gradient. That's what they told you to do.
 
  • #3


Dick said:
No. Express your curve as f(x,y)=C and use the gradient. That's what they told you to do.

what's c here?
 
  • #4


The constant you get by putting x= 2, y= 3 into the function. Since it will drop out of the derivative, its value is not important and you can just leave it as "c". Since the gradient always points in the direction of fastest increase it is always normal to level curves.
 
  • #5


so the gradient will be the normal to the level curves??
 
  • #6


-EquinoX- said:
so the gradient will be the normal to the level curves??

That's exactly what Halls said, isn't it? C doesn't matter. The gradient will depend only on x and y.
 
  • #7


well the gradient I have is:

(-3y + 2x) i + (2y - 3x) j

is this true?
 
  • #8


-EquinoX- said:
well the gradient I have is:

(-3y + 2x) i + (2y - 3x) j

is this true?

Sure. Now just evaluate it at (2,3).
 
  • #9


okay I get -5i , if then I am asked to find an equation for the tangent line to the curve at (2,3) and determine whether it's vertical , diagonal, or horizontal. How do I do this?
 
  • #10


The tangent is perpendicular to the normal, isn't it?
 
  • #11


so then I am assuming it's vertical?
 
  • #12


I wouldn't say your "assuming" anything. It is.
 

1. What is the gradient of a function?

The gradient of a function is a vector that shows the rate of change of the function at a specific point. It is a measure of the steepness or slope of the function at that point.

2. How do you determine the gradient of a function?

To determine the gradient of a function, you need to find the partial derivatives of the function with respect to each variable. These partial derivatives are then used to calculate the gradient vector at a specific point.

3. Why is the gradient important?

The gradient provides important information about the behavior of a function. It can help determine the direction in which the function is increasing or decreasing, and can be used to find the maximum or minimum values of the function.

4. What is the relationship between the gradient and the level curves of a function?

The gradient vector is always perpendicular to the level curves of a function. This means that the gradient shows the direction in which the function is changing the fastest, while the level curves show points where the function has the same value.

5. Can the gradient of a function be negative?

Yes, the gradient of a function can be negative. This happens when the function is decreasing in the direction of the gradient vector. However, the magnitude of the gradient is always positive as it represents the rate of change of the function.

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