- 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:

Find the plane tangent to the given surface at . 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 . - Use implicit methods to find the plane(s) tangent to the ellipsoid

at the point . Plot the tangent plane(s) on the same graph as the ellipsoid and rotate to see the point of tangency. Use plotting ranges , , and . - There is only one point at which the plane tangent to the surface is horizontal. Find it and plot it along with the function on the same graph. Be sure to use plotting option axes=none so that you can rotate the graph and see that the plane is tangent and horizontal.

2017-10-30