How to graph tangent plane and surface

In summary, the problem involves graphing a surface and its tangent plane at a given point. The approach is to choose a suitable domain and viewpoint, and then zoom in until the two become indistinguishable. Some questions arise about whether a computer is necessary and how to form a plane from given points. The surface's equation is z = x^2 + xy + 3y^2 and the tangent plane's equation is z = 3x + 7y - 5.
  • #1
JosephK
40
2

Homework Statement



Graph the surface and the tangent plane at the given point.
(Choose the domain and viewpoint so that you get a good view of
both the surface and the tangent plane.) Then zoom in until the
surface and the tangent plane become indistinguishable.

Homework Equations





The Attempt at a Solution



I tried using Scilab but I didn't know how to use it. Does this problem require a computer or can I graph by hand? I tried to plug in points for x and y trying to keep points close together. By close I mean 1, -1, and 0. How do I form a plane from these points?
 
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  • #2
JosephK said:

Homework Statement



Graph the surface and the tangent plane at the given point.
(Choose the domain and viewpoint so that you get a good view of
both the surface and the tangent plane.) Then zoom in until the
surface and the tangent plane become indistinguishable.

Homework Equations





The Attempt at a Solution



I tried using Scilab but I didn't know how to use it. Does this problem require a computer or can I graph by hand? I tried to plug in points for x and y trying to keep points close together. By close I mean 1, -1, and 0. How do I form a plane from these points?

What is the equation of the surface? You forgot to include the problem description.

You should be able to sketch a graph of the surface and tangent plane without needing to use computer software.
 
  • #3
If you were to graph "by hand" you are going to have a hard time "zooming" in!
There is a free on-line 3 d graphing program at
http://fooplot.com/index3d.php?

I haven't tried it myself.
 
Last edited by a moderator:
  • #4
Mark44 said:
What is the equation of the surface? You forgot to include the problem description.

You should be able to sketch a graph of the surface and tangent plane without needing to use computer software.


z = x^2 + xy + 3y^2 (1, 1, 5)

Tangent plane I found: z = 3x +7y -5

Thank you!
 

1. How do I find the equation of a tangent plane to a surface at a given point?

To find the equation of a tangent plane, you will need the coordinates of the point on the surface and the gradient vector of the surface at that point. The equation can be written as z = f(a,b) + fx(a,b)(x-a) + fy(a,b)(y-b), where f(a,b) is the value of the surface at the given point, fx(a,b) and fy(a,b) are the partial derivatives of the surface at that point with respect to x and y, and (x-a) and (y-b) represent the distances from the given point along the x and y axes.

2. What is the relationship between the tangent plane and the surface at a given point?

The tangent plane is a flat surface that touches the surface at the given point. It shares the same slope and curvature as the surface at that point, making it a good approximation of the surface in the immediate vicinity of the point.

3. How do I graph the tangent plane and surface on the same coordinate system?

To graph both the tangent plane and the surface, you will need to plot points on the surface and then find the corresponding points on the tangent plane using the equation of the tangent plane. You can then connect these points to create a visual representation of both surfaces on the same coordinate system.

4. Can the tangent plane be used to approximate the surface at points other than the given point?

Yes, the tangent plane can be used to approximate the surface at nearby points. However, as you move further away from the given point, the accuracy of the approximation decreases.

5. What is the significance of the tangent plane in multivariable calculus?

The tangent plane is important in multivariable calculus because it allows us to approximate complex surfaces with simpler, flat surfaces. This makes it easier to perform calculations and make predictions about the behavior of the surface. The tangent plane also helps us understand the behavior of functions in three-dimensional space.

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