- Purpose
- Background

- The tangent plane for explicitly defined surfaces
- The tangent plane for implicitly defined surfaces
- Exercises

The Maple commands for computing partial derivatives are `D`
and `diff`. The `diff` command can be used on both expressions and functions whereas the `D` command can be used only on functions. The commands below show examples of first order and second order partials in Maple.

> f := (x,y) -> x^2*y^2-x*y; > diff(f(x,y),x); > diff(f(x,y),y,y); > D[1](f)(x,y); > D[2,2](f)(x,y);Note in the above D command that the

> eval(diff(f(x,y),x,y),{x=-1,y=1}); > D[1,2](f)(-1,1);

> f:=(x,y)->1/(1+x^2+y^2); > tp:=D[1](f)(1/8,1/4)*(x-1/8)+D[2](f)(1/8,1/4)*(y-1/4)+f(1/8,1/4); > plot3d({f(x,y),tp},x=-1..1,y=-1..1);

To find a point, , where the tangent plane is horizontal, you would need to solve where both first order partials are equal to zero simultaneously.

> solve({diff(f(x,y),x)=0,diff(f(x,y),y)=0},{x,y});

The horizontal plane at that point would simply be . Below is how to plot the surface and the horizontal tangent plane.

> tp:=f(0,0); > plot3d({f(x,y),f(0,0)},x=-1..1,y=-1..1);

>with(plots): >surf:=x^2+y^2+z^2=1; >implicitplot3d(surf,x=-2..2,y=-2..2,z=-2..2);The tangent plane to an implicitly defined surface is given below:

To find the tangent plane to the sphere at the point and is positive, you would first need to find the coordinating value for the ordered pair. Below is how you would do this in Maple as well as find and plot the tangent plane implicitly.

>with(plots): >F:=x^2+y^2+z^2-1; >solve(eval(F,{x=1/2,y=-1/2}),z); >a:=eval(diff(F,x),{x=1/2,y=-1/2,z=sqrt(2)/2}); >b:=eval(diff(F,y),{x=1/2,y=-1/2,z=sqrt(2)/2}); >c:=eval(diff(F,z),{x=1/2,y=-1/2,z=sqrt(2)/2}); >tp:=a*(x-1/2)+b*(y+1/2)+c*(z-sqrt(2)/2)=0; >implicitplot3d({F=0,tp},x=-2..2,y=-2..2,z=-2..2,numpoints=2000);

- Compute the three distinct second order partial derivatives of

at the point using the**diff**command and then again using the**D**command. - Given:

- a)
- Find the plane tangent to the given surface at .
- b)
- Plot the surface and the tangent plane on the same graph and rotate the 3-D plot to show the point of tangency. Use plotting ranges and .
- c)
- Find the point where the tangent plane to the given surface would be horizontal.
- d)
- Plot the surface and the horizontal tangent plane on the same graph and rotate the 3-D plot to show the point of tangency. Use the same plotting ranges as above.

- Use implicit methods to find and plot the plane tangent to the ellipsoid

at the point . Use plotting ranges , , and

2016-09-13