DISP Mathematics (Scie 1500R)

MAPLE Lab. Tutorial 2 Functions and Plotting
Monday January 24.

** 1** EXPRESSIONS AND FUNCTIONS.

There is a difference between an * expression* like
and the function that assigns to each the expression
.
This difference is very clear in the MAPLE syntax and you need to be aware
of the difference.
For functions we often write
In MAPLE we make
an arrow by typing ` ->` ( a minus sign -, followed by a greater than ).

Type the following commands:

` > f:= x3 - 4*x2 + 1;` This defines as an * expression*.
` > g:=x-> x3 - 4*x2 + 1;` This defines as a * function*.
` > f(2);` This gives nonsense.
` > subs(x=2, f);` This is the way to do it for expressions.
` > g(2);` This is the way to do it for functions.
` > subs(x=a, f);`
` > g(a);`
` > (f(x+h) - f(x))/h;` More nonsense.
` > (g(x+h) - g(x))/h;`
` > simplify(%);`

In this list is a function, , and are expressions and is a number.

In expressions, the formula is ``static'' and the variable is important, we can
change the variable in MAPLE with the ` >subs( )` command.

A function is a more ``active'' object. It assigns an expression to each and we can easily change to other variables, numbers or expressions.

MAPLE has lots of built in functions that it knows about. They include:

` sin, cos, tan, cosec, sec, cotan, exp, log, (or ln) sqrt, abs`
(absolute value) and many more.

Try commands like ` > sin(Pi/4); > exp(Pi); >log(a2); >exp(%);
> abs(-3), > sqrt(5)` and so on.

** 2** PLOTTING.

MAPLE has a huge variety of plotting routines available. To see them all
type ` >with(plots);`. Usually one doesn't want to see this list.
Go back and change the ; to a :. You need to use this command to call up
all those sophisticated plotting programs. Just plain graphs are available
without it. You can plot both expressions and functions but the commands are
slightly different.

Type the following commands:

` > plot(f, x= -3..3);` Here is an expression that involves .
You give the range of you want with ` x= a..b`. Just * two* dots.
` > plot(g(x), x= -3..3);` Here we make an expression like .
` > plot(g, -3..3);` Here g is a function so we don't specify the
variable.

We can plot two (or more) functions on the same graph by making a * set*
of expressions (or functions). We do this with * curly* brackets .
Thus,
` > plot({f, 3*x2}, x = -3..3);`

We can make * parametric* plots (remember the lecture from last term!) by
using vector notation. In MAPLE vectors (and other * lists* are made
with * square* brackets .

` > plot([sin(2*t), cos(3*t), t=0..2*Pi]);`
Experiment with other values than 2 and 3.

You can animate your graphs:
Try the following simple animation:
` > animate(t*x2,x=-1..1,t=1..2,frames=30);`

To view an animation, you must click on the plot and then on the play button.

** 3** MORE COMPLICATED FUNCTIONS

You can program in MAPLE. One way to do this is with a * procedure*.
This begins with ` proc( )` and one fills in the variable(s) that one
needs inside the brackets. It ends with ` end;`. One can insert
conditions in the procedure with ` if ... fi` (think of them like brackets).

Complicated functions that are defined by two or more expressions need this
format. Try the following examples:
` >G:= proc(x) x2 +2*x - 3 end;`
` >plot(G, -4..4);`
` >H:= proc(x) if x<=0 then 2*x+1 else x2 + 1 fi end;`
` >plot(H, -4..4);`
` >K:= proc(x) if x < -2 then x+3
elif x <= 2 then 5 - x2
else 3-x fi end;`

Use the Return key to put commands on several lines, and then Enter after the ;

** 4** EXERCISES

** NAME STUDENT # **

Before you start these exercises type ` restart`.

1. Define the function
.

Write the MAPLE commands here:

2. Plot on the interval . Estimte (as best you can) the places
where .

3. Plot on the interval . Estimte (as best you can) the places
where .

4. Solve the equation using ` fsolve`

5. What does the following MAPLE command do?
` > f_1:= proc(x) f(x-1) end;`

If you are not sure, type it and then graph both and on

6. What MAPLE commands would you use to get a function whose graph is the
one for reflected about the -axis? the -axis?

7. Type the commands ` > g:= x-> sin(x); > h:= x->x2;`

Plot and on the same axes and on the interval .

Estimate the places where .

8. Type ` fsolve(g(x) = h(x), x);` and write what MAPLE responds here...

9. Plot and on the interval . Estimate the place in this
interval where .

Type ` fsolve(g(x) = h(x), x, -0.5..0.5);` and enter what MAPLE responds,

10. Use a parametric plot for
on . How
many times does the curve intersect itself?