Sample Interactive Lesson

The following cannot be changed; but the downloadable version is interactive.

In this lesson you will get a brief introduction to the numeric, symbolic and graphic features of Maple.

Input the following commands. They are self-explanatory. Maple response follows each command.

Thus, both input and output are listed on this Maple worksheet.

If you get an error message, correct the syntax.

First of all, we show that Maple can be used as a calculator and then show that it can be used as a computer algebra system.

Maple as a calculator:

> 2/3+8/7

Maple as a computer algebra system

Maple performs symbolic operations, like factoring, the four operations, substitution, solving sophisticated equations or systems of equations.

Factoring

> factor(x^2+5*x-6);

Multiplying two polynomials

> expand((x+6)*(x-1));

Solving an algebraic equation

> solve((x^2+5*x-6),x);

Substituting in an algebraic expression

> subs(x=3,(x^2+5*x-6));

Solving a general algebraic equation

> solve(a*x^2+b*x+c=0,x);

Solving a system of equations

> solve({-2*x+y-3*z=1,2*x-2*y+z=-3,x+y+z=-3},{x,y,z});

Solving a dependent system of equations

Maple solves a dependent system of equations. In the following, Maple chooses arbitrarily one variable and expresses the remaining variables in terms of that variable.

> solve({-2*x+y-3*z=1,2*x-2*y+z=-3},{x,y,z});

Solving a recurrence equation as a function of n

Using the rsolve command, Maple solves a recurrence equation, if possible, returning f as

a function of n.

> rsolve({f(n)=f(n-1)+f(n-2),f(0)=1,f(1)=3},{f(n)});

We can specify the number of digits for evaluating this expression.

> evalf(%,2);

Clearing memory of Maple variables

Sometimes Maple holds on to values assigned to variables. If you see any strange output, then assign the variable its own name, as below, using the single quotes around f. This frees the variable for further use.

> f:='f'; x:='x';

Defining a function

Maple distinguishes between an expression and a function. Define a function using the assignment symbol, :=. Create an arrow use a - (minus sign) followed by > (greater than) symbol.

> f:=x->x^2 + 1;

Evaluating a function

Define the function first. Then use evalf( ).

> evalf(f(4));evalf(f(3+h));

Maple does not evaluate f(3 + h) unless the value for h is given.

Creating a table of values for a function

To create a table of values (x, f(x)) from -2 to 2 with increments of 0.5 for f(x) = ,

use the following steps. We also separate x and f(x) by enclosing five spaces between two backward quote ( ` ) symbols. The backward quote symbol is the key located next to 1 on the top left corner of the keyboard.

> f:=x -> x^3;

> for x from -2 by 0.5 to 2 do print (x, ` `, f(x)) od;

Plotting graphs of functions

We plot 1) one function, 2) two functions and 3) a piecewise function.

Suppose the function is given by y = t sin t

To plot a graph of the function, with labels on the y-axis and title, enter the following.

Titles must be enclosed between two backward quote ( ` ) symbols.

> plot(t*sin(t), t=-3*Pi..3*Pi,`t*sin(t)` = -10..10,title=`My First Graph`);

The graph can be reduced or enlarged by resizing the plot window. To do this place the mouse pointer near the graph. Click the left mouse button. This makes the plot window visible. Drag the right bottom corner of the window in or out with the mouse pointer which turns into a two sided arrow.

Two or more functions are plotted with the same reference axes as follows.

> plot({t*sin(t),t^2*sin(t)},t=-3*Pi..3*Pi,title=`My Second Graph`);

To plot more complicated functions we need to activate the built-in package: plots.

We plot the piecewise function f( x ) =   for < 2

                                                    = for 2 <= < 3

                                                    = 5    for >= 3

(where <= means 'less than or equal to', >= means 'greater than or equal to')

using the plots package as follows.

> with(plots);

First we define the given piecewise function.

> f:=x->piecewise(x<2,x,x<3,x^2,5);

The graph is plotted as follows.

> plot(f(x),x=-2..6);

Maple connects the discontinuities by a vertical line.

To show discontinuites we use the condition: discont = true

> plot(f(x),x=-2..6,discont=true);

Polar functions and graphs

Polar functions are plotted using polarplot command. We plot r = sin 3t.

> polarplot(sin(3*t),t=0..2*Pi);

Parametric functions and graphs

Next we plot the parametric function x = , y = 2 -1 <= t <= 1.

> plot([t^2,2*t^3,t=-1..1]);

3D graphs

Three dimensional graphs are plotted with the plot3d command. Here we graph

      1.     z = cos xy, x = -3..3, y = -3..3,

and 2.     r = cos ( ) on the interval - <= <= and - <= <=

The graphs can be rotated by grabbing one corner by the mouse arrow. The shading changes accordingly. A choice of axes is available on the toolbar.

> plot3d(cos(x*y),x=-3..3,y=-3..3);

Plotting graphs in spherical coordinates

> sphereplot(theta*cos(phi),theta=-Pi..Pi,phi=-Pi..Pi);

Plotting the intersecting surfaces

We plot the intersection of two surfaces: z = - and x = 1

> implicitplot3d({z = x^2 - y^2, x = 1}, x = -5..5, y = -5..5,z = -10..30, title = `Intersecting Surfaces: z = x^2 - y^2, x = 1`);

Calculus using Maple

Limits, derivatives, partial derivatives, derivatives of higher order, indefinite, definite and improper integrals are calculated as follows.

limit of f(x) as x approaches 0.

> limit(sin(x)/x,x=0);

limit of f(x) as x approaches infinity.

> limit((1+1/x)^x,x=infinity);

limit of f(x) as x approaches 1.

> limit((x^2-1)/(x^3-1),x=1);

Derivative of a function

> diff(x^4/(x^3+1),x);

Partial derivative with respect to x.

> diff((x-2)*y^2,x);

Partial derivative with respect to y.

> diff((x-2)*y^2,y);

3rd Derivative of a function

> diff(x^6,x,x,x);

Indefinite Integral (antiderivative)of a function with respect to x.

> int(x^2-3,x);

Definite Integral of a function with respect to x from -2 to 2.

> int(x^2-3,x=-2..2);

Integration of an improper integral.

> int(1/x^.5,x= 0..1);

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MSIP Grant #P120A80089-98: "Three Urban Calculus Reform programs: Adopting the Best" 1998-2001