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 examples below show all 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[1,2](f)(x,y);Note in the above D command that the

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

> g := x-> sin(x)-x^3/7+x^2; > tl := D(g)(5)*(x-5)+g(5); > plot({g(x),tl},x=-2..8);

The next example shows how to find the tangent plane to the function
at . You could write the partials with `diff` or `D`. This example uses `D` as it is easier to plug in the the point with this syntax; with `diff` the `subs` command would be used.

> 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,style=patchnogrid);

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});

There are two plot commands for three-dimensional graphs **plot3d** and **implicitplot3d**. The first assumes the **=z** and is therefore not included in the command.

>plot3d(g(x,y),x=-10..10,y=-10..10,axes=boxed);Note the difference in the syntax for the second plot command. An equal sign must be included in the equation. This gives the flexibility of being able to graph equations without having to solve for z first.

>with(plots): >implicitplot3d([x^2+y^2=1-z^2,x=y],x=-1.1..1.1,y=-1.1..1.1,z=-1.1..1.1,axes=normal,color=[black,magenta],style=[wireframe,patchnogrid],thickness=2);Three-dimensional plots have many options. Some have been used inthe above command. To see more information try these two commands.

>?plot,colornames >?plot3d,options

- Compute the three distinct second order partial derivatives of

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

- a)
- Plot the function and the plane on the same graph. Use intervals , , .
- b)
- Find the derivative of in the plane.
- c)
- Graph the two-dimensional intersection of the plane and .
- d)
- Does your two-dimensional graph look like the intersection from your three-dimensional graph? Be sure to use the same ranges to properly compare and rotate the 3-D graph.

- Find the equation of the plane tangent to the surface at the point and find the equation of the plane tangent to the graph at . Plot both tangent planes on the same graph as the surface over the intervals and . Be sure to rotate the graph to see that the planes are tangent to the surface.

2010-11-08