Numbers  (See also Numerical Calculations )

• To assign the value 3.75 to z enter this.  Notice that you must use colon-equals .  z will thereafter  be equal to 3.75.

>	restart:z:=3.5;

• The exact value of :

>

Pi;

• The floating point (decimal) approximation of ; you must capitalize Pi to obtain this approximation.

>	evalf(Pi);

• The floating point approximation of  correct to 37 digits:

>	evalf(Pi,37);

• The number , the base of the natural logarithm.

>

exp(1);

• Floating point approximation of :

>	evalf(exp(1));

•  The exact value :

>

sqrt(2);

• The floating point approximation of .

>	evalf(sqrt(2));

• The imaginary number , which is .

>

I;

• Greatest integer less than or equal to 21.3:

>	floor(21.3);

• Smallest integer greater than or equal to 15 .98:

>	ceil(15.98);

• Round 23.98 to the nearest integer:

>	round(23.98);

• The fractional part of  21.3:

>	frac(21.3);

>

Arithmetic  (See also Numerical Calculations )

• Addition and subtraction:

>

2+8-6;

• Multiplication; be sure to separate factors with the asterisk:

>

21*7;

• Division:

>

21/7;

• Exponentiation; here is :

>

2^5;

• The exact value of   :

>	(2^3/3^7)*sqrt(3);

• The floating point approximation of this number:

>	evalf((2^3/3^7)*sqrt(3));

• Order of operations:  If parentheses are not present, exponentiation is performed before multiplication and division; multiplication and division before addition and subtraction.
• Here is   :

>	2+3*5^3/7;

• If you want :

>	(2+3)*(5/7)^3;

• The floating point value of this number:

>	evalf((2+3)*(5/7)^3);

• Be careful to use enough parentheses.  For example, contrast  (2+12)/(1+6)=14/7=2  with  (2+12)/1+6=14+6=20.  Be particularly careful when dividing to enclose the entire numerator, and the entire denominator,  in parentheses.

>	(2+12)/(1+6);

>	(2+12)/1+6;

•  n  factorial, for a positive integer n, is the product n!= 1*2*3*4* . . . * n:

>

5!;

>

Algebra  (See also   Algebraic Calculations )

• To expand an expression:

>	expand((x+4)^5);

• To factor an expression:

>	factor(x^5+20*x^4+160*x^3+640*x^2+1280*x+1024);

• To simplify an expression:

>	simplify((x+3*x+50*x^2)/2 );

•  The normal  command gets a common denominator and removes common factors in the numerator and denominator. Here we normalize :

>	normal(4/(3*(x+1))-x/(3*(x-2)*x));

• Solving equations:  For more information, see   solve   and   fsolve .

>	solve(x^2+x=6,x);

Note :  An equation  contains an equals sign.  If you omit the equals sign and the right-hand-side of the equation, for example by typing solve(x^2+x,x);  then Maple assumes that you mean solve(x^2+x=0,x);

>	solve(x^2+x=3,x);

>	solve(x^2+x=0,x);

>	solve(x^2+x,x);

• Solving a system of equations for a set of unknowns:

>	solve({x+y=3,x-2*y=7},{x,y});

• Use fsolve  to numerically solve an equation.  (This may not find all  solutions).   fsolve  stands for "floating point solve".

>	fsolve(x^3-7*x+1=0,x);

• Solve numerically for a solution in a given interval.  Here we find the solution that lies between 0 and 1:

>	fsolve(x^3-7*x+1=0,x=0..1);

• To add a number of terms, where you know a formula for the typical nth term:  Here we calculate +. . . + :

>	restart:sum(i^2,i=1..n);

>

Functions (See also Function Notation )

• To define your own function:  (Here, we define f(x)=x^2-3).  Be sure to use   the colon equals:  :=.    Make the arrow with the hyphen and greater than symbols.

>	f:=x->x^2-3;

• Maple already knows many functions.  Some examples are the trig functions:

>	[sin(x),cos(x),tan(x),tan(x),cot(x),sec(x),csc(x)];

the inverse trig functions:

>	[arctan(x),arcsin(x),arccos(x),arccot(x),arcsec(x),arccsc(x)];

and the exponential,  logarithmic, square root, and absolute value  functions:

>	[exp(x),ln(x),sqrt(x),abs(x)];

• To evaluate a function:
  Note:   Evaluating a function is different from solving an equation.  Do not attempt to use the  solve or fsolve  command to evaluate a function.

First define the function:

>	f:=x->x^2*exp(-x);

>

f(3);

>	f(103.65*sqrt(2));

If you want to evaluate any of these, use the evalf command:

>	evalf(f(103.65*sqrt(2)));

>

Graphing  (See also   Graphics )

• Open up the Plots package by entering the with(plots)  command, then use the following:

>	with(plots):

>	plot(sin(x),x=2..13); #to plot the sine function from x=2 to x=13

or define a function y=f(x) then plot it by referring to it as f(x):

>	f:=x->x^2*exp(-x);

>	plot(f(x),x=-1..8);

>

• To plot a pair of functions:

>	with(plots): plot([3*sin(x),ln(x)],x=1..14);

>

• You can include options, such as color, line  style, line thickness, and so forth.  For more options see   plot options  .

>	with(plots): plot([3*sin(x),ln(x)],x=1..14,color=[green,magenta],linestyle=[3,1],   thickness=[2,4]);

>

• A 3-D plot:

>	with(plots);#The semicolon displays the commands in this package. plot3d(x^2*exp(-x)*sin(y),x=0..2,y=0..5);

>

• The limits on one variable may depend upon the other. Here x goes from  to   and y goes from 0 to 5.

>	with(plots): plot3d(9-x^2*y,x=-sqrt(y)..sqrt(y),y=0..5);

>

• Plot several points:  Here we plot (0,2), (3,-1), and (5,7) and join them with straight line segments::

>	with(plots): pointplot([[0,2],[3,-1],[5,7]],connect=true);

>

• Display several objects (including plots and geometric objects) on one graph:

>	with(plots):with(plottools); #See what is in the plottools package p1:=plot(x^2,x=-2..2,color=red): aline:=line([0,0],[2,4],color=blue): circ:=circle([0,0],1,color=green): display([p1,aline,circ],scaling=constrained);

>

Calculus  (See also Calculus )

• Limit of f(x) as x approaches 2:

>	f:=x->(x^2-7*x+10)/(x-2); #Defining the function limit(f(x),x=2);

• Differentiation:

>	f:=x->x^2*sin(x); diff(f(x),x);

or

>	f:=x->x^2*sin(x); D(f)(x);

• antidifferentiation:

>	g:=x->2*x*sin(x)+x^2*cos(x);int(g(x),x);

• The definite integral:

>	g:=x->2*x*sin(x)+x^2*cos(x); int(g(x),x=0..2);

>