Gradient vectors and tangent lines

In summary, the gradient vector of f(x, y) = xy is <7, 3> and the tangent line to the level curve f(x, y) = 21 at the point (3, 7) is y = -7/3*x + 14. The normal to this tangent has a slope of -1/m, represented by the equation y = 3/7*x + 40/7 and the attached graph provides a visual representation of the gradient vector and tangent line.
  • #1
jcook735
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gradient vectors and tangent lines!

If f(x, y) = xy, find the gradient vector f(3, 7) and use it to find the tangent line to the level curve f(x, y) = 21 at the point (3, 7).


I already found the gradient vector to be <7, 3>, Maybe I am missing something obvious, but I have no clue how to find the tangent line.
 
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  • #2


grad(xy)*(x-3,y-7) = 0
<7,3>*(x-3,y-7) = 0
7*(x-3) + 3*(y-7) = 0
7x - 21 + 3y - 21 = 0
7x + 3y - 42 = 0
3y = -7x + 42
y = -7/3*(x) + 42/3

y= -7/3*(x) + 14 is the equation of a tangent line to f(x,y)=xy at point (3,7)

And by the way, the normal to this tangent will have a -1/m slope, so your normal to this tangent is y-7=3/7*(x-3) -> y=3/7*x+40/7

Review the attached graph
 
Last edited:

Related to Gradient vectors and tangent lines

1. What is a gradient vector?

A gradient vector is a vector that points in the direction of the greatest increase of a function. It is perpendicular to the level curves or surfaces of the function at a given point.

2. How do you find the gradient vector of a multivariable function?

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

3. What is the relationship between gradient vectors and tangent lines?

The gradient vector at a point on a function is perpendicular to the tangent line to the function at that point. This means that the gradient vector can be used to find the slope of the tangent line at that point.

4. Can gradient vectors be used to find the direction of steepest ascent?

Yes, the direction of the gradient vector is always the direction of steepest ascent for a function. This means that if you want to increase the value of a function as quickly as possible, you should move in the direction of the gradient vector.

5. How are gradient vectors used in optimization problems?

Gradient vectors are used in optimization problems to find the minimum or maximum value of a function. By setting the gradient vector equal to zero and solving for the variables, you can find the critical points of the function, which can then be used to determine the minimum or maximum value.

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