Arthur Caswell VII 

Footnote 18- click here for the Maple 11 fVersion

 

Three-Dimensional Graphing in Maple 

 

Calculus III covers many concepts pertaining to three-dimensional figures.  From planes to spheres, the class focuses on methods to analyze these three-dimensional figures. For this project I analyzed three-dimensional figures that were presented on the homework assignments and used maple to graph examples of the figures from the assignments.  This was in hopes to better associate the mathematic tools used with the objects the functions represented.  By visually representing the functions, it is easier to comprehend values of area and volume.  Ultimately it better relates the math with real life situations such as finding the mass of a baseball given the density function or finding the volume of a given container that is a simple geometric shape. 

Maple is able to graph most three-dimensional functions sufficiently, using two commands; plot3d and implicitplot3d.  These commands have a simple formula, which can be seen in the accompanying examples, which makes it simple to define a function and plot it within a certain range.  These commands can plot an array of figures from ellipsoids to paraboloids and can be quite useful when attempting to form a visual of the functions and figures that the problems in Calculus III present. 

Through the use of the visual representation, it is much easier to understand the positions and intersections that many of the problems describe.  The graphing tools in Maple make it possible to see tangent planes, critical points of a functions and even regions that are being integrated to find volume or surface area.  The simple methods of graphing in Maple combined with the vivid representations make three-dimensional graphing an imperative aid in familiarizing yourself with the common shapes and functions used in the subject matter and leads to a better understanding of the more complex functions in the future.   

In conclusion, the graphing tools in Maple can be tremendously helpful in visualizing the concepts covered in Calculus III.  It is extremely difficult to mentally picture saddle points and surface areas; however, the use of Maple enables it to be a much smoother learning process.  Therefore, Maple's graphing commands should be utilized by calculus students to better familiarize themselves with the functions and figures covered in the material throughout the course. 

 

 

 

 

with(plots); 1; with(plottools); 1 

You need to load these tools in order to use the plot commands. 

 

Homework 1 

5) 

c := sphere([4, 9, -7], 2); 1; display(c, axes = boxed, transparency = 0, labels = [x, y, z])
c := sphere([4, 9, -7], 2); 1; display(c, axes = boxed, transparency = 0, labels = [x, y, z])
c := sphere([4, 9, -7], 2); 1; display(c, axes = boxed, transparency = 0, labels = [x, y, z]) 

 

Plot 

 

Homework 6 

12) 

implicitplot3d(`+`(-x-5, `*`(5, y-5))+`*`(2, z+5) = 0, x = -20 .. 20, y = -20 .. 20, z = -20 .. 20, axes = normal)
implicitplot3d(`+`(-x-5, `*`(5, y-5))+`*`(2, z+5) = 0, x = -20 .. 20, y = -20 .. 20, z = -20 .. 20, axes = normal) 

 

Plot 

 

13) 

implicitplot3d(8*x+5*y+7*z = -17, x = -30 .. 30, y = -30 .. 30, z = -30 .. 30, axes = normal)
implicitplot3d(8*x+5*y+7*z = -17, x = -30 .. 30, y = -30 .. 30, z = -30 .. 30, axes = normal) 

 

Plot 

 

Homework 7 

1) 

implicitplot3d(z = x^2-2*y^2, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal) 

 

Plot 

 

implicitplot3d(x^2+y^2 = 5, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal) 

 

Plot 

 

implicitplot3d(x^2+2*y^2+3*z^2 = 1, x = -1 .. 1, y = -1 .. 1, z = -1 .. 1, axes = normal) 

 

Plot 

 

implicitplot3d(z = 4, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal) 

 

Plot 

 

implicitplot3d(z = 2*x^2+3*y^2, x = -10 .. 10, y = -10 .. 10, z = -40 .. 40, axes = normal)
implicitplot3d(z = 2*x^2+3*y^2, x = -10 .. 10, y = -10 .. 10, z = -40 .. 40, axes = normal) 

 

Plot 

 

implicitplot3d(z = x^2, x = -20 .. 20, y = -20 .. 20, z = -20 .. 20, axes = normal) 

 

Plot 

 

implicitplot3d(z = 2*x+3*y, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal) 

 

Plot 

 

Homework 9 

5) 

plot3d([cos(2*t), sin(2*t), 4*t], s = -10 .. 10, t = -10 .. 10, axes = normal) 

 

Plot 

 

Homework 10 

3) 

implicitplot3d(x^2-y^2-z^2 = 0, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal)
implicitplot3d(x^2-y^2-z^2 = 0, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal) 

 

Plot 

 

implicitplot3d(sqrt(x^2+2*y^2+3*z^2) = 9, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal)
implicitplot3d(sqrt(x^2+2*y^2+3*z^2) = 9, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal) 

 

Plot 

 

implicitplot3d(sqrt(x+2*y+3*z) = 4, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal)
implicitplot3d(sqrt(x+2*y+3*z) = 4, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal) 

 

Plot 

 

implicitplot3d(x^2+2*y^2+3*z^2 = 81, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal)
implicitplot3d(x^2+2*y^2+3*z^2 = 81, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal) 

 

Plot 

 

implicitplot3d(sqrt(x^2+y^2+z^2) = 4, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal)
implicitplot3d(sqrt(x^2+y^2+z^2) = 4, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal) 

 

Plot 

 

implicitplot3d(x^2+y^2+z^2 = 36, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal)
implicitplot3d(x^2+y^2+z^2 = 36, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal) 

 

Plot 

 

implicitplot3d(x+2*y+3*z = 4, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal)
implicitplot3d(x+2*y+3*z = 4, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal) 

 

Plot 

 

7) 

implicitplot3d(z = cos(2*x)*cos(5*y), x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal)
implicitplot3d(z = cos(2*x)*cos(5*y), x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal) 

 

Plot 

 

plot3d({[x, y, cos(2*x)*sin(5*y)], [x, y, 1]}, x = -10 .. 10, y = -10 .. 10, axes = normal)
plot3d({[x, y, cos(2*x)*sin(5*y)], [x, y, 1]}, x = -10 .. 10, y = -10 .. 10, axes = normal) 

 

Plot 

 

Homework 12 

8) 

implicitplot3d({3*x^2+5*y^2+3*z^2 = 1, 2*x-4*y-3*z = -1}, x = -1 .. 1, y = -1 .. 1, z = -1 .. 1, axes = normal)
implicitplot3d({3*x^2+5*y^2+3*z^2 = 1, 2*x-4*y-3*z = -1}, x = -1 .. 1, y = -1 .. 1, z = -1 .. 1, axes = normal)
implicitplot3d({3*x^2+5*y^2+3*z^2 = 1, 2*x-4*y-3*z = -1}, x = -1 .. 1, y = -1 .. 1, z = -1 .. 1, axes = normal) 

 

Plot 

 

Homework 13 

1) 

implicitplot3d(z = 36-x^2-2*y^2, x = 0 .. 2, y = 0 .. 2, z = -10 .. 50, axes = normal) 

 

Plot 

 

4) 

implicitplot3d(z = 100-x^2-4*y^2, x = 0 .. 2, y = 0 .. 2, z = -10 .. 100, axes = normal) 

 

Plot 

 

Homework 14 

1) 

implicitplot3d(z = x^11*y+11*x^10-y, x = -30 .. 30, y = -30 .. 30, z = -30 .. 30, axes = normal)
implicitplot3d(z = x^11*y+11*x^10-y, x = -30 .. 30, y = -30 .. 30, z = -30 .. 30, axes = normal) 

 

Plot 

 

2) 

implicitplot3d(z = x^2+y^2-10*x-4*y+4, x = -10 .. 10, y = -10 .. 10, z = -100 .. 100, axes = normal)
implicitplot3d(z = x^2+y^2-10*x-4*y+4, x = -10 .. 10, y = -10 .. 10, z = -100 .. 100, axes = normal) 

 

Plot 

 

5) 

implicitplot3d(z = 1-4*x+7*y, x = 0 .. 7, y = 0 .. 12, z = -100 .. 100, axes = normal) 

 

Plot 

 

Homework 15 

5) 

plot3d({[x, y, `*`(4, x^2+y^2)], [x, y, 32-`*`(4, x^2+y^2)]}, x = -10 .. 10, y = -10 .. 10, axes = normal)
plot3d({[x, y, `*`(4, x^2+y^2)], [x, y, 32-`*`(4, x^2+y^2)]}, x = -10 .. 10, y = -10 .. 10, axes = normal) 

 

Plot 

 

Homework 17 

3) 

implicitplot3d({z = x+8*y, y = 3*x^2, z = x, y = 18*x, z = 0}, x = 0 .. 10, y = -10 .. 10, z = 0 .. 10, axes = normal)
implicitplot3d({z = x+8*y, y = 3*x^2, z = x, y = 18*x, z = 0}, x = 0 .. 10, y = -10 .. 10, z = 0 .. 10, axes = normal) 

 

Plot 

 

4) 

implicitplot3d({z, y^2+z^2 = 16}, x = 0 .. 10, y = 0 .. 40, z = 0 .. 10, axes = normal) 

 

Plot 

 

Homework 18 

2) 

plot3d({[x, y, `*`(25, x^2+y^2)], [x, y, 18-`*`(25, x^2+y^2)]}, x = -10 .. 10, y = -10 .. 10, axes = normal)
plot3d({[x, y, `*`(25, x^2+y^2)], [x, y, 18-`*`(25, x^2+y^2)]}, x = -10 .. 10, y = -10 .. 10, axes = normal) 

 

Plot 

 

Homework 21 

6) 

implicitplot3d({3*x+8*y+z = 9, 1/36*x^2+1/16*y^2 = 1}, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal)
implicitplot3d({3*x+8*y+z = 9, 1/36*x^2+1/16*y^2 = 1}, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, axes = normal) 

 

Plot