Part 1: Elementary Calculations
Each of the command entry instructions below is sent to Maple for processing by pressing the RETURN or ENTER key on your keyboard.
1. In the Maple worksheet,
type
2+3
and then press
RETURN or ENTER. (From now on, we will not add this last part of the
instruction.)
2. Enter
2^3
Note where that the cursor remains in the exponent until you move
it.
3.
Enter
2*(8-2^3)
Part 2: Variables
1.
Enter
x:=3;
y:=2
Note where that the semicolon
allows you to put two commands on the same line
2. Enter
2*x+y
3. The result of the preceding step
follows from the fact that the variable x has been assigned the value
3 and the variable y has been assigned the value 2. Check
this by entering
x;
y
4. Enter the following line. Note that it ends with
a colon
z:=5:
5. Enter
z+y
Note that entering
a line with a colon has the effect as entering it without a colon, except that
nothing is displayed afterwards.
6. Use pencil and paper to decide
what number xy2 + 3z represents, and then check your answer by
entering
x*y^2+3*z
Use
your mouse to return to the expression just entered, replace the 3 by
4, and press RETURN.
7. Suppose we want to remove
the identification of x with 3.
Enter
x:='x'
Check this by entering
x
Now remove the identifications of y
with 2 and z with 5. Check by entering
x*y*z
You
should see xyz
as the output.
8. Maple
distinguishes between := and =. The first is used for definitions. The equals
sign alone is used to enter equations.
Check this by entering
solve(s^2-s-1=0,s)
(Here the s after the comma in the
command tells Maple what to solve for.)
9. Enter
2+3
and then
% +
5
The percent sign has the
value of the last quantity calculated. Check this by entering
x*y
and
then
% +
z
Part 3: Text
1. You may enter text as you would in any word processor by clicking on the word "Text" on the toolbar (next to the word "Math").
2. You can return to Maple Notation input by clicking on the word "Math".
Part 4: What Maple remembers
1. Create a new region and
type
x:=15
On a new line
type
x+5
2. Go back into the region, delete the
line
x:=15
and press
RETURN. In the new region,
enter
x
Notice that Maple still remembers that
x has been assigned the value 15 -- even though that line no
longer appears in the worksheet.
Maple remembers what has been entered in the order it was entered.
If you save a
Maple file without removing the output, then reload it later, it will
look the same as when you saved it. But Maple will not remember any of
the displayed commands. All of those commands must be reentered before you
are again in the same state as when you saved and closed the
file.
3. Enter
z:=3*s*t+2
Now assign
the value 7 to s and the value 1 to t.
s:=7;
t:=1
Then enter
z
to see the
current value of z.
4. Go back up to the line
�
s:=7;
and change
the 7 to 8. Then press RETURN while the cursor is still in this
line. Again note that an altered line must be entered for the
� altered
value to be acknowledged by Maple. Similarly, change the value of t
to -1. Now enter
z
again.
� Maple remembers what has been entered in the order it was entered.
5. A very useful command is the
restart command. If you enter
restart
Maple clears the memory. It is the same as loading the current worksheet.
All Maple commands that you need, including loading
of
packages, must be reentered.
Part 5: Functions
1. Next we define a function to assign the value
10 sin x to each x. Enter
x:='x'
f:=x->10*sin(x)
and then
f(1)
You may be surprised to see 10
sin(1) rather than a decimal approximation.
2. To obtain
a decimal approximation, enter
evalf(10*sin(1)) or
evalf(%)
The command "evalf" stands for "floating point
evaluation."
3. To be sure that the action of evalf is clear, enter
2/3
and then
evalf(2/3)
and then
evalf(2/3,14)
4. Let's evaluate f at pi/6. First,
we need the value of pi. Enter
Pi
Alter this line to
read
evalf(Pi)
Compare this
with
pi
evalf(pi)
Both "pi" symbols look the same, but
only the one obtained with an upper case "P" and a lower-case "i" carries the
numerical value.
Now find a decimal approximation to
f(pi/6)
Part 6: Graphing
1. In order to
plot 3 sin(2x) over the interval [0,2 pi].
Enter
plot(3*sin(2*x),
x=0..2*Pi)
We can obtain the same result by
defining the function f given by f(x) = 3 sin(2x) and plotting
f(x). Enter the following:
f:= x->3*sin(2*x)
plot(f(x), x=0..2*Pi)
2. When the plot appears, use your mouse to "select" it -- you should see a box around the graphic. Experiment with the drag buttons at the corners and sides of the box. Experiment with the icons at the top of the screen to see what they do.
3. Next we plot data given as
ordered pairs. Enter
data:=[[0,1],[2,2],[4,6],[5,1]]
Then enter
plot(data,
style=point)
Go back and see what happens
if you delete "style=point."
4. Often we wish to view several
graphs at the same time. One way to do this is to enter the desired plots
directly into the plot command.
Enter
plot([sin(x),cos(x)],x=0..2*Pi, color=[green,blue])
5. Another way to create multiple plots is
to use the display command in the plots package. To access this
package click on "Tools" on the top toolbar, then click on "Load Package" then
click on "Plots".
Now enter
graph1:=plot(sin(x),x=0..2*Pi,color=blue):
This creates a plot and names it "graph1." Create a second plot by
entering
graph2:=plot(cos(x),x=0..2*Pi,
color=green):
To see both plots
together, enter
display(graph1,graph2)
6. To plot a curve defined implicitly, like
the unit circle for example, first make sure that you have loaded the
plots package (see 5 above)
then on a new line
type implicitplot(x^2+y^2 = 1, x = -1 .. 1, y = -1 ..
1).
7. Try ploting x^2-y^2=1 with x =
-3..3 and y = -3..3.
8. In the Maple worksheet, click on the word plots (following the word loading) to see many other possible plot commands.
Part 7: Differentiation and Integration
1. Define g to be the function given
by g(x) = x2cos x.
Check
your work by evaluating g(pi). You should obtain -pi2.
If you have trouble, look at Part 5 again.
2. Now enter
diff(g(x),x)
Then
enter
diff(g(x),x,x)
How would you
calculate the third derivative?
3. If you want to calculate the derivative of an
expression that you have not yet entered, just replace g(x) by the
expression.
For example, enter
diff(x^3-x^2+2,x)
Now insert a literal constant in the expression: Enter
diff(x^3-a*x^2+2,x)
Then change the final x to an a. That is, enter
diff(x^3-a*x^2+2,a)
What is
the role of the symbol after the comma in the differentiation
expression?
4. Next we
calculate indefinite integrals. If necessary, unassign
x:
x:='x'
Then enter
int(x*sin(x),x)
You can check that the result is an
antiderivative of x sin(x) by entering
diff(%,x)
5.
Next try to find an antiderivative for
sin(x3 + x5). Maple does not know an
antiderivative of this function that may be defined in terms of functions known
to it. On the other hand, try
int(exp(-x^2),x)
Note that Maple gives an
answer involving the "erf" function , which you probably never heard of and not
that useful to you at the moment. We'll discuss this more later in the
course.
6. Next we
calculate definite integrals. To integrate x sin(x) over the interval
[0,pi/2], enter
int(x*sin(x),x=0..Pi/2)
7. Now try this method on the
integral of sin(x3+x5) over the interval
[0,pi/2]. Maple still doesn't know an antiderivative for
sin(x3+x5). To obtain a numerical estimate,
enter
evalf(int(sin(x^3+x^5),x=0..Pi/2))
If you know that all you want is a numerical estimate, you can
enter
evalf(Int(sin(x^3+x^5),x=0..Pi/2));
The significance of the upper-case I in
Int is that Maple does not try to
find a symbolic solution before starting on the numerical estimate.
8. Maple can
also compute improper integrals.
Enter
int(1/x^2,x=1..infinity)
Part 8: Algebraic Operations
1. Enter the
polynomial
P:=3*x^5-18*x^4-7*x^3+42*x^2-40*x+240
2.
Use the following command to factor P
factor(P)
Note that there is no space bewtween factor and (P)
3. Enter
expand(%)
4. We may
use the solve command to obtain a complete factorization. Enter the
following
S:=solve(P=0,x)
Notice in the complex roots that Maple
uses a capital I for the imaginary unit -- rather than just
i.
Part 9:
Limits
1. Limits are easy to calculate in
Maple. As a test we'll calculate the limit of sin(x)/x as x
approaches 0. Enter the
following:
limit(sin(x)/x, x =
0)
2. The Maple limit command also
calculates limits at infinity. Enter
limit((1 + x3)/((1 +
x)*(1+2*x2)), x = infinity)
3. The standard limit command will deal
with one-sided limits if there is no possibility of confusion. Enter
limit(sqrt(x)/sqrt(x + x^2), x=0
4. Now
try
limit(abs(x)/x, x = 0)
5. Why is this
limit ambiguous? Try
limit(abs(x)/x, x = 0, right)
Now replace "right" by
"left."