Calculus: The Language of Change

Exercise Set 17.1
The Explicit Plane in 3-D

by David Theobald

Exercise 1 : Graph a Plane

Sketch the graph of z=(2/3)x-(3/2)y+2 with paper and pencil.  Check your work using the program SurfSlices.

Solution

A helpful way to think about three dimensional equations is to break them down into 2-D parts.  This means holding one of the variables constant and looking at the remaining two to see the shape made in their respective plane.

Clear[x, y, z, s, t];
z:=(2/3) x-(3/2)y+2;
x := s;
y := t;

ParametricPlot3D[{x, y, z}, {s, -1, 1}, {t, -1, 1}, 
      ViewPoint -> {5, 2, 1}];

To check your work, copy the code from the SurfSlices notebook and enter your function z.  NOTE: you may have chosen a different view to draw than what I have .

Exercise 3 : Steepest Planes

part a

Which plane is steepest z=2x-3y or z=3x+2y?

Solution of part a

sub parts - of solution - if needed

3 4 input if Math' ca comp' s

few words

part b

Which plane is steepest z=3x-4y or z=(3/5)x+(4/5)y?

Solution of part b

explanation

sub parts of problem b - if needed

few words

part c

Verify your answers using the program PlaneLines.

Solution of part c

explanation

Exercise 4 : The Linear Gradient Vector

Let L[x,y]=2x-3y.  Let U=(({{(3/5)}, {(4/5)}})).

part a

Show that  |U|=1.

Solution of part a

This is easy to verify.  All we must do is compute the magnitude of vector U.  The formula for this is:

|U| = (Sqrt[(x^2) +  (y^2)])

sub parts

Clear[MagU, U, x, y];
x = (3/5);
y = (4/5);
U=(({{x}, {y}}));
MagU = (Sqrt[(x^2) +  (y^2)])

We have therefore proven U to be a unit vector.

part b

Plot the line on z=L[x,y] pointing in the direction of U that goes through (x,y,z)=(1,2,-4).

Solution of part b

explanation

sub parts of problem b - if needed

few words

part c

Verify your answers using the program PlaneLines.

Solution of part c

explanation