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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!)
- 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
).
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)?
- 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?
- 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
- Repeat Step 1 with z = ±
√
1 − x
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. )
- Graph also both halves of the surface z
= x
in the same way and give the
common name for this surface.
- Graph also both halves of the surface z
= 1 + x
and give the common name
for this surface.
- Intersection of Two Surfaces
View the upper half-cone z =
√
x
(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
