Finding Horizontal Tangent Planes on S

[TranscendArcu]

Homework Statement

S is the surface with equation $$z = x^2 +2xy+2y$$a) Find an equation for the tangent plane to S at the point (1,2,9).
b) At what points on S, in any, does S have a horizontal tangent plane?

The Attempt at a Solution

$$F(x,y,z): z = x^2 +2xy+2y$$
$$F_x = 2x + 2y$$
$$F_y = 2x + 2$$

Evaluated at (1,2) gives answers 6 and 4, respectively. My equation for a plane is:

$$z-9=6(x-1) + 4(y-1)$$.

I think any horizontal plane should have normal vector <0,0,k>, where k is some scalar. I'm pretty sure that S has no such normal vector. But if
$$F(x,y,z): 0 = x^2 +2xy+2y - z$$
then
$$grad F = <2x + 2y,2x + 2,-1>$$ It seems like I can let (x,y) = (-1,1) to zero the x-, y-components of the gradient. Plugging (-1,1) into the definition of z gives z = 1. This suggests to me that there is a point (-1,1,1), at which there is a horizontal tangent plane. Yet I feel pretty sure that this isn't true!

 

[ehild]

You made a little mistake when writing out the equation of the tangent plane. The y coordinate of the fixed point is 2, you wrote 1.

A surface in 3D is of the form F(x,y,z) = constant. For this surface, x²+2xy+2y-z=0. That means F(x,y,z)=x²+2xy+y-z. The gradient of F is normal to the surface, and the tangent plane of the surface at a given point. You want a horizontal tangent plane, so a vertical gradient:(0,0,a). That means F_x=2x+2y=0, F_y=2x+2=0 --->x=-1, y=1, so your result for the x,y coordinates are correct. Plugging into the original equation for x and y, you got z=x²+2xy+2y=1, it is correct. Why do you feel it is not?

ehild

 

[TranscendArcu]

When I graphed F(x,y,z) in MatLab (and it's possible I graphed it incorrectly), I observed that the the resulting paraboloid is always "tilted". Below is a picture from my plot:
http://img440.imageshack.us/img440/687/skjermbilde20111203kl85.png
How can this surface have a horizontal tangent anywhere when it is tilted like this?

 

[ehild]

Try to plot z out for -2<x<0 and 0<y<2

ehild

 

[TranscendArcu]

http://img7.imageshack.us/img7/139/skjermbilde20111203kl10.png
Hmm. I'm not seeing the a point in this picture where the gradient is pointing directly upwards. Everything still looks kind of tilted.

 

[ehild]

The function is equivalent with z=(x+2y-1)(x+1)+1 and z=1 along the lines x=-1 and y=(1-x)/2. I attach a plot of the surface near the point (-1,1)

ehild

 

[HallsofIvy]

I presume that by "horizontal" you mean perpendicular to the z-axis.

The simplest way to find a tangent planes for a surface is to write it in the form F(x,y,z)= constant. Then the normal to the tangent plane at any point is given by $\nabla F$. Here, you can write $F(x, y,z)= x^2+ 2xy+ 2y- z= 0$.

What is $\nabla F$? That will be vertical (and so tangent plane horizontal) when its x and y components are 0.

 

[ehild]

@HallsofIvy: The OP has shown the solution in his first post, he only can not believe it, as the surfaces he got with MatLab look tilted. If you could give advice how to plot surfaces with MatLab, that would be real help for him.


ehild

 

[TranscendArcu]

If anyone is familiar with MatLab, this is the code I've been using:

[x,y] = meshgrid(-2:.1:0, 0:.1:2);

%this makes a surface z evaluated at all the points in the x and y matrices
%note that the dot operator has to be used to indicate element by element multiplication
%if this is not done matrix style row by column multiplication is done and
%this is not what we wanted for this problem

z = x.^2 + 2*x*y +2*y;

%this makes the surface plot
%first we clear the figure window

figure(1);cla;

%then we make the plot

surf(x,y,z);

%this labels the axes
xlabel('X axis')
ylabel('Y axis')
zlabel('Z axis');

%this makes a title
title('Bizarre Graph');

%this sets the view angle in degrees (done by trial and error)
view(24,38);

Note that the "%" mark my annotations. I included them so that hopefully you can follow what I'm doing more easily.

 

[TranscendArcu]

Ha! I figured it out. I forgot a "."

I should have written

z = x.^2 + 2*x.*y +2*y;

Everything makes sense now.

 

[ehild]

You see: it is worth typing something out again and again. Is your plot similar to my one? It was made with Origin. I would like to see your final plot... Please...

ehild

 

[TranscendArcu]

http://img259.imageshack.us/img259/2104/skjermbilde20111204kl10.png It looks like it could have a horizontal tangent plane right around (-1,1,1)

 

[ehild]

It is really nice! And a missing dot made you sceptical about the truth of Maths??!:uhh:

ehild