Tangent Planes
Math 211  Prof. Leise

Email completed labs to tleise@amherst.edu by noon next Wednesday.

The goal today is to visualize tangent planes to surfaces and see how they approximate a surface close to the point of tangency.

Let's start with our old friend the elliptic paraboloid, f(x,y)=x^2+4y^2.

Exercise 1. Plot the surface z=x^2+4y^2.

Exercise 2. Find an equation for the tangent plane to the surface z=x^2+4y^2 at the point (2,1,8).

Exercise 3. Plot the surface and the tangent plane in the same set of axes for -8<x<12, -9<y<11.  You can make a list of functions to graph within the same Plot3D command:  Plot3D[{f(x,y), g(x,y)}, {x,-8,12},{y,-9,11}].  Rotate the graph for a good view--make sure you can see where the surfaces touch.

Exercise 4. Plot the surface and the tangent plane in the same set of axes for 1<x<3, 0<y<2.  Plot them again for 1.9<x<2.1, 0.9<y<1.1.  You should see that the tangent plane is very close to the surface as you "zoom into" the point of tangency, that is, the tangent plane is a good approximation to the surface close to that point.

Exercise 5. 
(a) Find an equation for the tangent plane to the surface z=x^2-x*y-y^2 at the point (1,1,-1).  
(b) Find parametric equations for the normal line to the surface z=x^2-x*y-y^2 at the point (1,1,-1).  Hint: if you know the tangent plane equation, then you must know a normal vector to the surface.
(c) Edit the following Show command to combine plots of the surface, the tangent plane, and the normal line.

Show[Plot3D[x^2-x y-y^2,{x,0,2},{y,0,2}],Plot3D[x-y,{x,0,2},{y,0,2}],ParametricPlot3D[{t,t,t},{t,-1,1}],PlotRange->{{0,2},{0,2},{-2,0}},BoxRatios->{1,1,1}]

Exercise 6. Next let's look at the function f(x,y)=(y*x^3-x*y^3)/(x^2+y^2) from lecture.  Begin by plotting this function for -10<x<10, -10<y<10.

Exercise 7. Based on the partial derivatives of f at (0,0) that we found in lecture, give the equation of the  plane that is tangent to this surface at (0,0,0).  (It's a very simple plane equation.)  Plot the surface together with the tangent plane.

Exercise 8. Find the partial derivative ((∂f)/(∂x)) for points not at the origin.  You may use the D command: Simplify[D[f,x]].  Plot ((∂f)/(∂x))(x,y).

Exercise 9. Find the tangent plane at (0,0,0) to the surface z=((∂f)/(∂x)).  Note that this involves evaluating the partial derivatives of ((∂f)/(∂x)) at (0,0) using the definition.

Exercise 10. Find the mixed partial (((∂^2)f)/(∂y∂x))(you can use the D command in Mathematica, if you wish).  First plot z=(((∂^2)f)/(∂y∂x)) for -10<x<10, -10<y<10, then make a second plot with -1<x<1, -1<y<1, and finally a third plot with -0.1<x<0.1, -0.1<y<0.1.  Is the surface z=(((∂^2)f)/(∂y∂x)) beginning to look like a plane as we "zoom in", as happened with the previous examples?  Why can't this surface be approximated by a plane close to the origin?  Be sure to check the limit of (((∂^2)f)/(∂y∂x)) in different directions approaching (0,0).  What is going wrong?