Determine gradient of a function f(x,y)

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Homework Help Overview

The problem involves determining the gradient of a function defined implicitly by a contour equation, specifically (y-x)^2 + 2 = xy - 3, and finding a vector normal to the curve at the point (2,3).

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss expressing the curve as f(x,y)=C and using the gradient to find the normal vector. Questions arise about the meaning of the constant C and the relationship between the gradient and the normal to level curves.

Discussion Status

Some participants have confirmed the relationship between the gradient and the normal vector, while others are exploring how to evaluate the gradient at a specific point and its implications for the tangent line. There is ongoing inquiry about the nature of the tangent line at the given point.

Contextual Notes

Participants are navigating the implications of the gradient's direction and its relationship to the tangent line, with some uncertainty about the characteristics of the tangent line at the specified point.

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


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?
 


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.
 


so the gradient will be the normal to the level curves??
 


-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.
 


well the gradient I have is:

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

is this true?
 


-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).
 


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.
 

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