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3D Graphing - Calculus III - Laboratory 1 | MATH 2210, Lab Reports of Advanced Calculus

Material Type: Lab; Class: Calculus III; Subject: Mathematics; University: University of Wyoming; Term: Unknown 1989;

Typology: Lab Reports

Pre 2010

Uploaded on 08/19/2009

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..... Calculus III
MATH 2210—Spring 2009 .....
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Lab #1(a) In-Class: 3D Graphing
Name (Print) .......................................................................
Instructor/Time ..................... Due Date ............................
The goal of this lab is to acquaint you with the 3D graphing possibili-
ties of MAPLE. This software will be useful to you for checking worked
examples from the textbook, lectures and especially homework.
1. New User’s Tour
After starting MAPLE 10 [some classes have found the Classic Worksheet Maple 10
best], select Help −→ New Users −→ Full Tour from the main menu at the top of the
screen. Click on (1) Working through the New User’s Tour and follow the on-screen
instructions to familiarize yourself with worksheets.
You will need to click on the first (red ) executable command, then press EnterEnter
to watch MAPLE execute the commands, displaying the answers in blue.
Click on the Return to Overview link at the bottom of the page to return to the Main
Menu of the New User’s Tour.
2. Graphics Demo
Click on (5) 3-D Graphics and execute the examples provided on-screen. For each 3D
plot, experiment with
rotating the image by dragging it with the left mouse button;
enlarging/shrinking each plot by clicking on each plot image and dragging its
corners using the mouse; and
varying the style, color and axes using the buttons;
also by selecting Plot −→ Color −→ various options.
Note that the animate3d command helps to visualize a function of three variables,
z=f(x, y, t), by varying tas time; each frame gives the plot of zas a function of x
Calc III, Lab 1a Page 1 of 3 Spring 2009
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Calculus III

MATH 2210—Spring 2009

Lab #1(a) In-Class: 3D Graphing

Name (Print).......................................................................

Instructor/Time..................... Due Date............................

The goal of this lab is to acquaint you with the 3D graphing possibili-

ties of MAPLE. This software will be useful to you for checking worked

examples from the textbook, lectures and especially homework.

1. New User’s Tour

After starting MAPLE 10 [some classes have found the Classic Worksheet Maple 10

best], select

Help

New Users

Full Tour

from the main menu at the top of the

screen. Click on (1) Working through the New User’s Tour and follow the on-screen

instructions to familiarize yourself with worksheets.

You will need to click on the first (red) executable command, then press 〈Enter〉〈Enter〉

to watch MAPLE execute the commands, displaying the answers in blue.

Click on the Return to Overview link at the bottom of the page to return to the Main

Menu of the New User’s Tour.

2. Graphics Demo

Click on (5) 3-D Graphics and execute the examples provided on-screen. For each 3D

plot, experiment with

• rotating the image by dragging it with the left mouse button;

• enlarging/shrinking each plot by clicking on each plot image and dragging its

corners using the mouse; and

• varying the style, color and axes using the buttons;

also by selecting

Plot

Color

−→various options.

Note that the animate3d command helps to visualize a function of three variables,

z = f(x, y, t), by varying t as time; each frame gives the plot of z as a function of x

Calc III, Lab 1a Page 1 of 3 Spring 2009

and y for fixed time t. These frames form a ‘movie’ displayed using a bar of buttons

similar to those on your CD/tape player.

After investigating and experimenting with the worksheet examples, exit by repeatedly

clicking the second ×

button in the upper-right screen corner (the first ×

button will

kill your MAPLE session!)

  1. You should now see the [> prompt. Enter

f:=(x,y)->x^2/(x^2+y^2);

plot3d(f(x,y),x=-2..2,y=-2..2,axes=BOXED,

title="Discontinuous Surface");

to view a graph of the surface z = f(x, y) = x

/(x

  • y

).

According to this graph, what is the value of f along the y-axis (i.e. for x = 0)?

and along the x-axis (i.e. for y = 0)?

  1. Enter

f(x,0);

f(0,y);

and record the results here:

f(x, 0) =

f(0, y) =

Does this agree with the values computed in Step 3?

  1. Scroll back to your 3D plot in Step 3 using the scrollbar on the right screen, and click

on the plot.

Select first

Plot

−→

Style

−→

Patch and Contour

and then

Plot

−→

Style

−→

Contour

to

see level curves or contour lines plotted on the surface. Describe in words the shape

of the level curves for z = f(x, y).

Calc III, Lab 1a Page 2 of 3 Spring 2009

Calculus III

MATH 2210—Spring 2009

Lab #1(b) Take-Home: Multiple 3D Graphing

Name (Print).......................................................................

Instructor/Time..................... Due Date............................

This lab demonstrates the plotting of several functions of two variables

on the same set of axes.

1. Plotting a Sphere

Solving x

+ y

+ z

= 1 for z gives z = ±

1 − x

− y

. Here the graph of z =

1 − x

− y

gives the upper (‘northern’) hemisphere, while z = −

1 − x

− y

gives the lower (‘southern’) hemisphere.

Plot the upper hemisphere by entering

f:=sqrt(1-x^2-y^2);

plot3d(f,x=-1..1,y=-1..1,axes=BOXED);

Rotate the resulting image by dragging with the left mouse button. Describe the

surface you see.

The surface may not appear spherical due to different scales in the vertical and hori-

zontal directions. If so, correct this by pressing the

button above the screen, or

selecting

Plot

Scaling Constrained

from the main menu (top of the screen).

Now view both halves of the sphere together by entering

plot3d({f,-f},x=-1..1,y=-1..1,axes=BOXED);

(Rather than entering a new plot3d command, it is easier to click on your previous

plot3d command with the left mouse button, edit appropriately, and press 〈Enter〉.)

Rotate as before. Does the visible image appear to be a sphere? If not, try to explain

any defect in the graphical depiction of this surface.

Calc III, Lab 1b Page 1 of 2 Spring 2009

  1. Repeat Step 1 with z = ±

1 − x

  • y

instead of z = ±

1 − x

− y

. Try using

− 2 ≤ x ≤ 2 and − 2 ≤ y ≤ 2 (rather than − 1 ≤ x ≤ 1 and − 1 ≤ y ≤ 1 as in Step 1).

What is the resulting surface called? (You may refer to the table on p.691 of the

textbook. Pay attention to the form of the equation. )

  1. Graph also both halves of the surface z

= x

  • y

in the same way and give the

common name for this surface.

  1. Graph also both halves of the surface z

= 1 + x

  • y

and give the common name

for this surface.

  1. Intersection of Two Surfaces

View the upper half-cone z =

x

  • y

(compare with Step 3) and the plane z = x+

simultaneously by entering

plot3d({sqrt(x^2+y^2),x+1},x=-2..2,y=-2..2,axes=BOXED);

Examine the intersection of these two surfaces, rotating as necessary, and experiment-

ing with different lighting using

Plot

−→

Lighting

. What is the common name for the

intersection of these two surfaces?

Calc III, Lab 1b Page 2 of 2 Spring 2009