- Graphs
- Solving Equations
- Computations
- Help

plot( x^2, x = -1..1 );

plot( { sin(x), (1/3)*x }, x = -Pi..Pi );

By looking at the graph we can solve the equation sin(x) = x/3. The roots are determined by the places where the two curves cross.

We can graph surfaces using the `plot3d` command. For
example, to graph z = xy, where x and y run from -1 to 1, we
do this:

plot3d( x*y, x=-1..1, y=-1..1, axes = BOXED, style = PATCH);

For more information use the `?plot3d` command. No
semicolon needed for this one.

solve(x^3 = 27);1/2 1/2 3, - 3/2 + 3/2 I 3 , - 3/2 - 3/2 I 3

Note that there are three solutions, two of which are complex. Now let's try something more complicated:

solve(x^3 + 1.5*x = 27);2.83, - 1.42 + 2.74 I, - 1.42 - 2.74 I

**Note:** There is a good reason why we wrote "1.5
" instead of "(3/2)" as the coefficient of x
in the last equation. If one of the numbers in the equation
is in decimal form, then Maple tries to find an approximate
solution in decimal form. If none of the numbers are in
decimal form, as in the first example, then Maple tries to
find an exact solution. This may fail, since there is no
algebraic formula for the roots polynomial equations of
degree five or more (
Galois
).

solve( { 2*x + 3*y = 1, 3*x + 5*y = 1} ); {y = -1, x = 2}

These can contain literal as well as numerical coefficients:

solve( { a*x + 5*y = 1, 3*x + b*y = c}, { x, y } ); - 3 + a c - b + 5 c {y = ----------, x = - ----------} - 15 + a b - 15 + a b

In the second example we have to tell Maple that x and y are the variables to be solved for. Otherwise it wouldn't know.

You can solve systems of two equations in two unknowns of the form f(x) = 0, g(x) = 0 by graphing the functions f(x) and g(x) and seeing where the curves cross.

Maple does arithmetic pretty much as you would expect it to:

3*(1.3 + 1.7)^2/2 - 0.1; 13.400000000

It has built-in commands which can do a lot of work quickly. For example, to add up the numbers 1, 1/2, 1/3, ... 1/10, we do this:

> sum(1/n, n= 1..10); 7381 ---- 2520

Note that Maple gave us the *exact* answer as a
fraction in lowest terms. For an approximate answer in
decimal form, do this:

> sum(1.0/n, n= 1..10); 2.928968254

The only difference was the `1.0` in place of `1
`. Note the decimal point. We can also things like
factor numbers:

> ifactor(123456789); 2 (3) (3803) (3607)

> p := (a+b)^2; # define p to be the square of (a + b) 2 p := (a + b) > expand(p); # expand it 2 2 a + 2 a b + b > factor(a^2 + 2*a*b + b^2); 2 (a + b) > a := 1; # define a to be 1 a := 1 > p; # re-evaluate p 2 (1 + b) > a := 'a'; # define a to be a again a := a > p; # check it out 2 (a + b)

> sin(Pi/2); 1 > arcsin(1); 1/2 PiWe can set things up for conversions like this:

> deg := evalf(Pi/180); # use evalf to convert to decimal form deg := .01745329252 > rad := 1/deg; rad := 57.29577951 > sin(90*deg); 1. > arcsin(1)*rad; 28.64788976 Pi > evalf("); # " stands for the result of the preceding computation 90.00000002

Note two things. Sometimes we need to use the `evalf
` function to convert results from exact to floating
point (decimal) form. Sometimes it is convenient to use the
quote(") sign: it stands for the result of the
preceding computation.

> f := x -> sqrt( 1 + x^2 ); 2 f := x -> sqrt(1 + x ) > f(1); 1/2 2 > f(a+3); 2 1/2 (10 + a + 6 a)These can have more than one variable:

> g := (x,y) -> sqrt(x^2 + y^2); 2 2 g := (x,y) -> sqrt(x + y ) > g(3,4); 5

> diff( sin(cos(x)) + x^3 + 1, x ); 2 - cos(cos(x)) sin(x) + 3 xIt can do both definite and indefinite integrals:

> int( x^2, x ); 3 1/3 x > int( x^2, x = 0..1); 1/3 > evalf( int( sqrt( 1 + x^3 ), x = 0..1 ) ); 1.111447971

The last computation deserves comment. Suppose we just do the obvious thing (try it!).

> int( sqrt( 1 + x^3 ), x = 0..1 ):

Maple does not give us a numerical answer because the
integral of this function cannot be expressed in terms of
elementary functions. In particular it cannot be integrated
by the usual techniques. However, note that we have
surrounded our computation with `evalf( ... )`.
This forces Maple to evaluate the integral numerically: `
evalf` stands for *evaluate in floating point form*

Here are some examples of matrix calculations. Be sure to
use `with(linalg)` before doing them. Also be sure
to try these out to see how they work!

> with(linalg): > a := matrix([ [1, 2], [3, 4] ]);matrix is displayed ........> b := matrix([ [0, 1], [1, 0] ]); ........ > multiply( a, b ); ........ > multiply( b, a ); ........ > evalm( b &* a ); # another way of computing ba .......... > ?evalm # consult help on evalm > aa := inverse( a ); ........ > c := randmatrix(2,2); # 2x2 random matrix > evalm( 1/c ); # invert it > print( aa ); # redisplay aa ........

Did you notice the difference between the product `ab
` and the product `ba`?

For information and examples on a particular Maple " function", use the "?" command. For example,

> ?solve

gives information on the `solve` command. Often it
is helpful to scroll to the end of the help window and look
at the examples, bypassing the technical discussion that
precedes it. You can also try the command with no keyword:

> ?

This gives additonal information on how to use the help system.

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Last modified by
jac
March 27, 1995

Copyright © 1995 Department of Mathematics, University
of Utah