Maxima - Using its symbolic math capabilities:
The following table develops progressively more sophisticated uses of Maxima as a symbolic math tool. Remember, you do not need to use or memorize all of this table at once - simply go up to the point that your current needs dictate, and return to this table when they become more extensive.
Note: in the third column some expressions have been typeset for greater ease of reading.
Topic Discussion Maxima Input Maxima Output
Basic typing Remember, use a semicolon and the enter key to terminate input. (2+5/6-sqrt(4))^2; 25
--
36
Using the previous line The % symbol tells Maxima to use the result of the previous calculation. %+1; 61
--
36
Using a line by name Alternatively, you can refer to a result you wish to use by the name of its output line. D2+1; 97
--
36
Numeric evaluation To tell Maxima to calculate a floating point result, use the float function. float(%); 2.694444444444445
Defining a function To define a function: give it a name, followed by its independent variable in parentheses, followed by the symbols :=, followed by its definition. f1(x):=x^2-5*x+6;           2
f1(x) := x - 5 x + 6
Using a function Once defined, you can use a function in an intuitive way. f1(5); 6
Assigning a value to a variable  The : sign assigns a value to a variable. a:5; 5
The variable assignment will now be used everywhere in place of the variable name. f1(a); 6
Defining an equation The = sign defines an equation. x=1-b*y; x = 1 - b y
Solving an equation   An equation can be readily solved for any of its variables, if a simple expression exists. solve(%,y);        x - 1
[y = - -----]
         b
Another example, yielding both roots of a quadratic. solve(x^2+2*x-3=0,x); [x = - 3, x = 1]
In order to use any of the results from a solve step, we need to "extract" them from the output list. %[2]; x = 1
Plotting a function Let's look at a plot of our function and see that the roots tally. The second argument used by the plot2d function is a list, and is indicated by square brackets ([]). This particular list specifies the x-range of interest. plot2d(x^2+2*x-3=0,[x,-4,2]);
Systems of equations Maxima can also solve systems of equations, if the equations and variables of interest are presented to it as lists. solve([x+3*y=3,2*x+5*y=5],[x,y]); [[x = 0, y = 1]]
Plotting multiple functions To plot multiple functions, we use a list of their names as the first argument in the plot2d function. Let's use this feature to check the previous result. plot2d([(3-x)/3,(5-2*x)/5],[x,-1,1]);
Solving equations numerically   Even when an analytic solution cannot be found, Maxima can in many cases solve the equation numerically. Let's first examine this seventh-order polynomial by plotting it.

plot2d(x^7-5*x^6+4*x^4-5*x^2+x+2,[x,-1,1]);
We see two roots between -1 and 1. To get their exact values, we load the newton routine, and use it with a guess to the right of the root we wish to get.

load("newton");

 /sw/share/maxima/5.9.0rc3/share/numeric/newton.mac
newton(x^7-5*x^6+4*x^4-5*x^2+x+2,1); 8.194213634964119B-1
We repeat the process for the second root. newton(x^7-5*x^6+4*x^4-5*x^2+x+2,0); - 5.763042928902195B-1
Functions of multiple variables Multivariate functions are defined as before, but separating variables with commas in the definition.

f2(x,y):=sin(x^2+y^2);

                 2   2
f2(x, y) := SIN(x + y )
f2(1,4);

SIN(17)
float(%);

- .9613974918795568
3D Plots These are created as before, but using the plot3d command, and with separate lists for the ranges of the x and y axes. plot3d(f2(x,y),[x,-1,1],[y,-1,1]);
Factoring Polynomials Maxima can factor some polynomials. factor(x^2+2*x+1);        2
(x + 1)
Expanding Polynomials Or expand them expand(%);  2
x + 2 x + 1
Simplifying Trigonometric expressions Much the same can be accomplished for trigonometric expressions. trigsimp(sin(x)^2+cos(x)^2); 1
Expanding Trigonometric expressions As with their expansion.

trigexpand(cos(x+3*y));

trigexpand(%);
expand(%);
Simplifying rational expressions In many cases, Maxima can simplify rational expressions fullratsimp((x^2+2*x+1)/(x+1)+1/(4*x+3));  
Expanding rational expressions Or expand them. ratexpand((x^2-1)/(x+2));
Substituting the Results of One Calculation Into Another  We can save ourselves quite a bit of typing by using substitutions. First, we isolate the result we wish to substitute.

solve(x^2-1=0,x);

[x = - 1, x = 1]
%[2]; x = 1
Then, we perform our substitution subst(%,y=3*x-11); y = - 8
Calculating limits  Let's create an interesting function, and calculate its limit as x approaches 0

f3(x):=sin(x)/x;

         SIN(x)
f3(x) := ------
           x
limit(f3(x),x,0); 1
To calculate the limit from above, we simply add PLUS (or MINUS from below) to our entry.

limit(tan(x),x,%PI/2,PLUS);

MINF
limit(tan(x),x,%PI/2,MINUS); INF
Simple Differentiation We can differentiate any function with respect to a given variable diff(c*x^2-sin(d*x),x); 2 c x - d COS(d x)
Higher Order Differentiation Higher order differentiation is also readily available. diff(x^3,x,3); 6
Simple Integration We can also perform indefinite integration on a function (assuming a simple closed form for the answer exists and is computable by Maxima). integrate(cos(3*x),x); SIN(3 x)
--------
   3
Definite Integration Or definite integration, if we enumerate the lower and upper limit. integrate(cos(3*x),x,0,%PI/2);   1
- -
  3
Numeric Integration In cases where a closed form integral is unavailable, Maxima can compute the result numerically.

integrate(sin(sin(x)),x,0,1);
romberg(sin(sin(x)),x,0,1); .4306059236425572
Summation  Summations can be written out. sum(1/(n^2),n,1,INF);
If desired, they can be evaluated by setting SIMPSUM to TRUE.

SIMPSUM:TRUE;

TRUE
sum(1/(n^2),n,1,INF);    2
%PI
----
 6
Products Products work in much the same way. product(1/(n^2),n,1,10);        1
--------------
13168189440000
Series Expansion   Series expansions can also be computed. Let's calculate the series
expansion of the sine function around x=0, up to the fifth power in x.		 taylor(sin(x),x,0,5);          3    5
        x    x
/T/ x - -- + --- + . . .
        6    120
Since the output of taylor has special properties, we need to convert it into a polynomial. trunc(%);      3    5
    x    x
x - -- + --- + . . .
    6    120
We can now compare our series approximation to the original function. plot2d([%,sin(x)],[x,0,%PI]);

We welcome your feedback on this workflow/tutorial - please email us at symmath@hippasus.com