A Quick Introduction to MapleTim McLarnan

Maple is a tool for doing nearly anything you think of as computational in mathematics, as well as many things you may be surprised to find a machine doing. Examples include:Anything you can do with a calculator.Exact computations with fractions.Adding, multiplying, and factoring polynomials.Solving equations, either exactly or approximately.Simplifying algebraic expressions.Plotting functions.This leaflet is a quick introduction to some of the salient things Maple can do. The idea has been to use the most common Maple functions in order to give you examples of their syntax.

(1) The Maple prompt is the symbol >.(2) Maple commands always end with a semicolon ;(3) Maple commands are always followed by ENTER. If you want to type a command with more than one line, use SHIFT-ENTER to go down a line without executing a command.(4) Maple is case sensitive. Pi is not the same as pi.(5) Save early and often. Maple seems to crash most often when you are trying to print. So a recipe for disaster is to do 2 hours of work without ever saving, then try to print.

+, -, * and / do what you expect:2+5;

NiMiIig=

Unlike most calculators, however, Maple does operations with fractions exactly:1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7;

NiMjIiRCIyIkUyI=

Other arithmetic operations are a^b, which means a to the power b, n!, which means n factorial (1*2*3*4*...*n).2^1000;

NiMiaV1sdyRwIW9jP1BvUUNFbEopSC9tbkBhJ294az5yWCJIMXhnJVJhbCJmdncpUm43JT5lJClcWllJOiM9YXooPTlyQjEwWTJAYSk0QkNbeG5YJFIhW2QpcHV2ZCUpZlZTSiNwJkdYcj1EbzlgdiJIbiVmRiJlZXA6KXkkKT4kXEFoJFw3XjVOcSQpUV1QVzJPYDA8IltTaDAiPStnIVxdVVs0S25pPTInMzoyIg==

(355/113)^10;

NiMjIjtEY14uNigzJEcheVsqeUoiNlxbSkFBKipRblhSJA==

4!;

NiMiI0M=

125!;

NiMiXXgrKysrKysrKysrKysrKyshKTN4SnIrbGJQRVZfYnFhZXZYJCkpPXlmIVFUJD4+O0kxdWQrJGZXMGtrTEojZlwqKUckSClvJykpZjg7TGNKTjM6QilmKCllVSNccltPUyZmZDNnIlwteHdWKCo0RSopKW88eEUpPQ==

Is it surprise to you that 125! ends in so many zeros? Can you find a way to compute how many zeros are at the end of n! for any n? Warning: Trying anything too huge (like 10000!) may lock up your computer and needlessly antagonize your professor. ifactor(n) factors the integer n into primes:ifactor(315);

NiMqKC1JIUc2IjYjIiIkIiIjLUYlNiMiIiYiIiItRiU2IyIiKEYt

20! - 12!;

NiMiNCslUXdwMj8hSFYj

The most recent result (2432902007697638400) can be easily included in any subsequent calculation without having to type it. The percent character (%) is used to refer to the last expression computed by Maple. (The last in time, not necessarily the one just above on the page.) Similarly, %% is the expression before %, and %%% is the expression before %%.ifactor(%);

NiMqLi1JIUc2IjYjIiIjIiM1LUYlNiMiIiQiIiYtRiU2I0YtRigtRiU2IyIiKCIiIi1GJTYjIiM2RiwtRiU2IyIpPmgoPiVGMw==

Maple's normal mode of computation is exact arithmetic, with no roundoff or truncation errors.(2^30/3^20)*sqrt(2);

NiMsJCokIiIjIyIiIkYlIyIrQz11dDUiKyxXeSdbJA==

The function evalf() gives a numerical approximation to this exact expression:evalf(%);

NiMkIislPTtdTiUhIzU=

If 10 digits aren't enough, give evalf an optional second argument to tell it how many digits you need:evalf(%%,60);

NiMkImduKFIsRi1TXm43VyE+OSlcdCEpKipwTSt5RWEocChcPTtdTiUhI2c=

Maple uses Pi, not pi, to represent NiNJI3BpRzYi. Notice what this means: variable names in Maple are case-sensitive. Would you like to know the first few digits of NiNJI3BpRzYi?evalf(Pi,200);

NiMkImN3P1FJXCYqW0hpVydmYjVAJlE+cS1UR111NiJbRyIzJWZgc0pBZV0mNFklUTRabUkjRzhsM1tAKXoxPEBNRFsuRycpKiozaUcxayJ5SSNmV1woNCNlNXYkKlJwcj4lKUddektRVkVZUUt6KmVgRWZUSiEkKj4=

Maple also works with variables:(2*x+7) * (x^2+3) * (x+2)^5;

NiMqKCwmSSJ4RzYiIiIjIiIoIiIiRiksJiokRiVGJ0YpIiIkRilGKSwmRiVGKUYnRikiIiY=

expand(%);

NiMsNCokSSJ4RzYiIiIpIiIjKiRGJSIiKCIjRiokRiUiIiciJGMiKiRGJSIiJiIkQCYqJEYlIiIlIiVxNiokRiUiIiQiJVc+KiRGJUYoIiUlUSNGJSIlcz0iJHMnIiIi

At this point, Maple has forgotten where it got this polynomial. It's just a random polynomial of degree 8. But Maple can factor it:factor(%);

NiMqKCwmSSJ4RzYiIiIjIiIoIiIiRiksJiokRiVGJ0YpIiIkRilGKSwmRiVGKUYnRikiIiY=

Could you have factored this polynomial as fast as Maple did? Could you have factored it at all? (If you think about it a bit, your answers to these questions should be "no" and "yes.")Maple doesn't always get the terms of a polynomial in the order you would expect:expand((1+x)^3*(x+2)^2);

NiMsLiokSSJ4RzYiIiIjIiNERiUiIzsiIiUiIiIqJEYlIiIkIiM+KiRGJUYqIiIoKiRGJSIiJkYr

In this case, you can use the sort command to get things arranged.sort(%);

NiMsLiokSSJ4RzYiIiImIiIiKiRGJSIiJSIiKCokRiUiIiQiIz4qJEYlIiIjIiNERiUiIztGKkYo

Often we need to assign a name to the result of a computation. Maple does this using the syntax variable := value; In making an assignment, Maple does some obvious simplifications first.e1 := (x+y)^3 * (x+y)^2;

NiM+SSNlMUc2IiokLCZJInhHRiUiIiJJInlHRiVGKSIiJg==

Maple has a built in function called simplify, which tries to simplify expressions. It does not always find the simplest form of an expression, but it is a start. Here is an example:e2 := (x^3-y^3)/(x^2+x-y-y^2);

NiM+SSNlMkc2IiomLCYqJEkieEdGJSIiJCIiIiokSSJ5R0YlRiohIiJGKywqKiRGKSIiI0YrRilGK0YtRi4qJEYtRjFGLkYu

simplify(e2);

NiMqJiwoKiRJInhHNiIiIiMiIiIqJkkieUdGJ0YpRiZGKUYpKiRGK0YoRilGKSwoRiZGKUYpRilGK0YpISIi

To understand what went on here, we can use the commands numer and denom (to take the numerator and denominator of a fraction) together with factor:factor(numer(e2));

NiMqJiwmSSJ5RzYiISIiSSJ4R0YmIiIiRiksKCokRigiIiNGKSomRiVGKUYoRilGKSokRiVGLEYpRik=

factor(denom(e2));

NiMqJiwoSSJ4RzYiIiIiRidGJ0kieUdGJkYnRicsJkYoISIiRiVGJ0Yn

So what simplify did was to find and remove the common factor of (x - y). In this particular case, the commands factor and simplify do the same thing:factor(e2);

NiMqJiwoKiRJInhHNiIiIiMiIiIqJkkieUdGJ0YpRiZGKUYpKiRGK0YoRilGKSwoRiZGKUYpRilGK0YpISIi

Maple can also solve equations, even symbolic ones. Notice the syntax of the commands below. You have to specify first the equation, then the variables you want to solve for.solve(x^2+5*x+2 = 0, x);

NiQsJiMhIiYiIiMiIiIqJCIjPCNGJ0YmRiosJkYkRidGKCMhIiJGJg==

This quadratic equation had two solutions (separated by a comma), just as you would expect. If you want a decimal approximation instead of an exact solution to an equation, use fsolve instead of solve.fsolve(x^2+5*x+2 = 0, x);

NiQkISs4R2JoWCEiKiQhK3M9WiVRJSEjNQ==

solve and fsolve can also deal with systems of more than one equation. You need to give them a set of equations (in set brackets { }) followed by a set of variables to solve for. Here's an example:solve({x+y=5, x-y=2}, {x,y});

NiM8JC9JInlHNiIjIiIkIiIjL0kieEdGJiMiIihGKQ==

Now let's get more adventuresome and try to solve an equation containing symbolic constants:solve(a*x^2+b*x+c=0, x);

NiQsJComLCZJImJHNiIiIiIqJCwmKiRGJiIiI0YoKiZJImFHRidGKEkiY0dGJ0YoISIlI0YoRiwhIiJGKEYuRjIjRjJGLCwkKiYsJkYmRihGKUYoRihGLkYyRjM=

Do you recognize this answer? It's the quadratic formula!Did you know that there was a similar formula for solving cubic equations? It's a bit more complicated: its output will fill the screen.solve(a*x^3 + b*x^2 + c*x + d = 0, x);

NiUsKComSSJhRzYiISIiLCoqKEkiY0dGJiIiIkkiYkdGJkYrRiVGKyIjTyomSSJkR0YmRitGJSIiIyEkMyIqJEYsIiIkISIpKihGMyNGK0YwLCwqJkYlRitGKkYzIiIlKiZGKkYwRixGMEYnKipGKkYrRixGK0YlRitGL0YrISM9KiZGL0YwRiVGMCIjRiomRi9GK0YsRjNGOUY2RiVGKyIjNyNGK0YzI0YrIiInKigsJiomRiVGK0YqRitGMyokRixGMEYnRitGJUYnRigjRidGMyMhIiNGMyomRixGK0YlRidGSCwqRiQjRidGQEZERkFGS0ZIKiheI0Y2RitGM0Y2LCZGJEZCRkQjRjBGM0YrRissKkYkRk1GREZBRktGSCooXiMjRidGMEYrRjNGNkZQRitGKw==

Look at this answer a bit until you undertand the notation. There are three roots, which are written separated by commas. The second and third roots also involve the square root of -1, which Maple writes as I. Warning again: I isn't i. Case matters to Maple. Maple can also solve the general fourth degree equation (add in an NiMqJEkieEc2IiIiJQ== term), though the output now fills several screens. Can it do more?solve(a*x^5 + b*x^4 + c*x^3 + d*x^2 + e*x + f = 0, x);

NiMtSSdSb290T2ZHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYjLC4qJkkiYUdGKCIiIkkjX1pHRiUiIiZGLSomSSJiR0YoRi1GLiIiJUYtKiZJImNHRihGLUYuIiIkRi0qJkkiZEdGKEYtRi4iIiNGLSomSSJlR0YoRi1GLkYtRi1JImZHRihGLQ==

Why can't Maple solve this equation? Because \311variste Galois proved shortly before his death at the age of 20 that there is not general formula like the quadratic formula which solves polynomial equations of degree 5 and higher. (That is, there is no formula solving all such equations. For particular values of the coefficients, there are often simple solutions.)fsolve still works to find approximate solutions to equations of high degree, though:fsolve(x^5+x+1=0, x);

NiMkIStpbXhbdiEjNQ==

Maple only found one solution to the last equation. How can we convince ourselves there is only one? One way to begin to build evidence might be to graph the function.plot(x^5+x+1, x=-2..2);

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

This at least suggests there is only one root between x=2 and x=2.Maple sometimes makes an unhelpful choice of axes:plot(x^3 - 1/x^2, x=-2..2);

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

The function has a vertical asymptote. Maple's plot shows y values down to -2000000 (which Maple shortens as -2E6). You can fix this problem by selecting your own scale for the y-axis:plot(x^3 - 1/x^2, x=-2..2, y=-10..10);

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

There are many options for the plot command, and many other types of plots. (Try typing ?plot[options]; or ?plots; to learn more about these.)Here is an example of a plot of several functions at once.plot({x, x^2, x^3, x^4, x^5}, x=-1.1 .. 1.1);

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

One problem with this plot is that you may not be sure which curve is which. To make matters worse, Maple doesn't always assign the same colors to curves: the next time you run it, the colors might change. Here's a way to deal with that: make the functions an ordered list (in square brackets) instead of an unordered set (in curly braces), and give an additional argument listing the colors you want for each curve.To plot sin(x) in black, cos(x) in red, and arctan(x) in blue, you could sayplot([sin(x),cos(x),arctan(x)], x=-Pi..3*Pi, y=-2..2, color=[black, red, blue]);

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If you're color blind or if you're printing in black and white, you can use linestyles instead of colors. Again, ?plot[options]; will give useful advice.plot([sin(x),cos(x),arctan(x)], x=-Pi..3*Pi, y=-2..2, color=black, linestyle=[SOLID, DASH, DOT]);

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3D plots are also possible.plot3d(x^2-y^2, x=-2..2, y=-2..2);

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(To obtain this picture, I ran the Maple command, clicked on the picture, and then played with the choices on the Tool Bar until I got the view I wanted.) One can also view this 3-dimensional surface as a contour plot. You can make contour plots from 3D plots using the Tool Bar.

In order to use Maple effectively, you'll need to know where to get help. An excellent and convenient source is Maple's online help, which is available under the Help menu item. You can also access help directly from the Maple prompt, like this:?coeffs;The output appears in another window, but here it is. It includes a calling sequence, usually some obtuse discussion. Most usefully, it ends with examples and with suggestions of other related functions to look at. All functions and libraries which Maple knows about have online help items. The online help is arranged hierarchically; so if you don't know the name of the command you're looking for, you can often find it by browsing through the help system

coeffs - extract all coefficients of a multivariate polynomialCalling Sequence coeffs(p, x, 't')Parameters p - multivariate polynomial x - (optional) indeterminate or list/set of indeterminates t - (optional) an unevaluated name

The coeffs function returns an expression sequence of all the coefficients of the polynomial p with respect to the indeterminate(s) x. If x is not specified, coeffs computes the coefficients with respect to all the indeterminates of p (see the indets function). If a third argument t is specified (call by name), it is assigned an expression sequence of the terms of p. There is a one-to-one correspondence between the coefficients and the terms of p. Note that p must be collected with respect to the appropriate indeterminates.

s := 3*v^2*y^2+2*v*y^3;

NiM+JSJzRywmKiYlInZHIiIjJSJ5R0YoIiIkKiZGJyIiIkYpRipGKA==

coeffs( s );

NiQiIiQiIiM=

coeffs( s, v, 't' );

NiQsJCokJSJ5RyIiJCIiIywkKiRGJUYnRiY=

t;

NiQlInZHKiRGIyIiIw==

collect, coeff, tcoeff, lcoeff, indets, PolynomialTools[CoefficientVector]

The bookshelf in the Math/CS Lounge on the second floor of Dennis contains a fair number of manuals for various versions of Maple. The basics of the program haven't changed much, so any of these manuals should get you started. Other students, faculty, etc. are also good resources. Finally, the Maple 9.5 directory contains 2 manuals in pdf form: gsg.pdf is a 42 page Getting Started Guide; lrnguide.pdf is a 332 page Learning Guide.

Here are a few more commands which are useful in calculus. We begin by defining an expression, making some substitutions, and computing its derivative directly from the definition:f := x^2+x;

NiM+SSJmRzYiLCYqJEkieEdGJSIiIyIiIkYoRio=

subs(x=3, f);

NiMiIzc=

subs(x=x+h, f);

NiMsKCokLCZJInhHNiIiIiJJImhHRidGKCIiI0YoRiZGKEYpRig=

expand(%);

NiMsLCokSSJ4RzYiIiIjIiIiKiZGJUYoSSJoR0YmRihGJyokRipGJ0YoRiVGKEYqRig=

% - f;

NiMsKComSSJ4RzYiIiIiSSJoR0YmRiciIiMqJEYoRilGJ0YoRic=

%/h;

NiMqJiwoKiZJInhHNiIiIiJJImhHRidGKCIiIyokRilGKkYoRilGKEYoRikhIiI=

simplify(%);

NiMsKEkieEc2IiIiI0kiaEdGJSIiIkYoRig=

limit(%, h=0);

NiMsJkkieEc2IiIiIyIiIkYn

An irritating feature of this calculation was using subs(x=3, f) to compute f when x=3. We'd like to be able just to say f(3), but we can't, because f is just an expression, not a function. So how would we define a function in Maple? There are 2 ways. For simple one-line functions, we can do this:g := x -> x^2;

NiM+SSJnRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiQ5JCIiI0YlRiVGJQ==

This defines g as the function taking x to NiMqJEkieEc2IiIiIw==. We can now use g just like any other function:g(x);

NiMqJEkieEc2IiIiIw==

g(3);

NiMiIio=

g(x+h);

NiMqJCwmSSJ4RzYiIiIiSSJoR0YmRiciIiM=

More complicated functions are defined in a more complicated way. Begin the function with proc(variables), and end it with end proc;. Here, for instance, is a function which takes 2 arguments, and returns the larger of the two, assuming both are numbers:h := proc(x,y) if x>y then x else y end if end proc;

NiM+SSJoRzYiZio2JEkieEdGJUkieUdGJUYlRiVGJUAlMjklOSRGLUYsRiVGJUYl

h(3,5);

NiMiIiY=

h(5,3);

NiMiIiY=

Maple can directly compute derivatives, sums, integrals and limits. Here are some examples:diff(sin(x^2),x);

NiMsJComLUkkY29zRzYkSSpwcm90ZWN0ZWRHRihJKF9zeXNsaWJHNiI2IyokSSJ4R0YqIiIjIiIiRi1GL0Yu

int(x^2, x);

NiMsJCokSSJ4RzYiIiIkIyIiIkYn

int(x^2, x=1..4);

NiMiI0A=

sum(k^2, k=1..10);

NiMiJCZR

sum(1/k^2, k=1..infinity);

NiMsJCokSSNQaUdJKnByb3RlY3RlZEdGJiIiIyMiIiIiIic=

y := limit((x^2-3*x+2)/(x-1), x=1);

NiM+SSJ5RzYiISIi

Particularly when you are working with second and third derivatives and higher, it is often useful to define a function (rather than an expression) and then to use D to compute a derivative function. Here's a function and its first, second, and third derivatives, at x and at a numerical point:f := x -> x^5+1/x;

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJDkkIiImIiIiKiRGLiEiIkYwRiVGJUYl

f(x);

NiMsJiokSSJ4RzYiIiImIiIiKiRGJSEiIkYo

f(1);

NiMiIiM=

D(f);

NiNmKjYjSSJ4RzYiRiY2JEkpb3BlcmF0b3JHRiZJJmFycm93R0YmRiYsJiokOSQiIiUiIiYqJEYsISIjISIiRiZGJkYm

D(f)(x);

NiMsJiokSSJ4RzYiIiIlIiImKiRGJSEiIyEiIg==

D(f)(1);

NiMiIiU=

(D@@2)(f)(x);

NiMsJiokSSJ4RzYiIiIkIiM/KiRGJSEiJCIiIw==

(D@@2)(f)(1);

NiMiI0E=

(D@@3)(f)(x);

NiMsJiokSSJ4RzYiIiIjIiNnKiRGJSEiJSEiJw==

(D@@3)(f)(1);

NiMiI2E=

Like any powerful tool, Maple offers any number of ways for you to make mistakes. Here are some particularly popular ones

Right now, for instance, y has a value: it is -1. I'll therefore get into loads of trouble if I try to use y as a variable:plot(x^2, x=-1..1, y=-1..2);

Error, (in plot) invalid arguments

To fix this, I need to tell Maple that y is now just the variable y again. I do that by saying y:='y';y;

NiMhIiI=

y := 'y';

NiM+SSJ5RzYiRiQ=

y;

NiNJInlHNiI=

If you do this, Maple thinks the expression you want it to evaluate is not over. The current version of Maple, however, will go ahead and put in the semicolon after a warning:2+6

Warning, inserted missing semicolon at end of statement, 2+6;

NiMiIik=

Maple does ^ first, then * and /, then + and -. Notice the difference:x^1/2;

NiMsJEkieEc2IiMiIiIiIiM=

Since Maple does exponentiation before division, it reads this as NiMpSSJ4RzYiSSIxR0Yl/2.x^(1/2);

NiMqJEkieEc2IiMiIiIiIiM=

This is a possible way to write NiMtSSVzcXJ0RzYkSSpwcm90ZWN0ZWRHRiZJKF9zeXNsaWJHNiI2I0kieEdGKA==, as is sqrt(x).If you are plotting or solving a complicated expression, it is often useful to get Maple to print it first, so you can be sure you have the parentheses in the right place. Then either give the expression a name or use copy and paste or use % to get the correct expression into your command.-b+sqrt(b^2-4*a*c)/2*a;

NiMsJkkiYkc2IiEiIiomLCYqJEYkIiIjIiIiKiZJImFHRiVGK0kiY0dGJUYrISIlI0YrRipGLUYrRjA=

Nuts. That's not the root of a quadratic. Let's get the parentheses right.(-b+sqrt(b^2-4*a*c))/(2*a);

NiMsJComLCZJImJHNiIhIiIqJCwmKiRGJiIiIyIiIiomSSJhR0YnRi1JImNHRidGLSEiJSNGLUYsRi1GLUYvRihGMg==

That looks right. Now I can give it a name and work with it:root1 := %;

NiM+SSZyb290MUc2IiwkKiYsJkkiYkdGJSEiIiokLCYqJEYpIiIjIiIiKiZJImFHRiVGL0kiY0dGJUYvISIlI0YvRi5GL0YvRjFGKkY0

root2 := (-b-sqrt(b^2-4*a*c))/(2*a);

NiM+SSZyb290Mkc2IiwkKiYsJkkiYkdGJSEiIiokLCYqJEYpIiIjIiIiKiZJImFHRiVGL0kiY0dGJUYvISIlI0YvRi5GKkYvRjFGKkY0

root1*root2;

NiMsJCooLCZJImJHNiIhIiIqJCwmKiRGJiIiIyIiIiomSSJhR0YnRi1JImNHRidGLSEiJSNGLUYsRi1GLUYvISIjLCZGJkYoRilGKEYtI0YtIiIl

simplify(%);

NiMqJkkiYUc2IiEiIkkiY0dGJSIiIg==

I do this all the time.x^2+2x;

Error, missing operator or `;`

solve(x^2+bx+c=0, x);

NiQqJCwmSSNieEc2IiEiIkkiY0dGJkYnIyIiIiIiIywkRiNGJw==

Here we got the wrong answer because we wrote bx, which Maple reads as a new variable, rather than b*x, which is what we meant.Worst of all, if you write something like 2(x), Maple interprets this as a function 2 evaluated at the point x, and returns just 2. This is obviously not what you intend, and represents a stupid error in the Maple parser, but you need to be aware of it.1+2(x^2);

NiMiIiQ=

Here are some examples:evalf(pi);

NiNJI3BpRzYi

evalf(Pi);

NiMkIithRWZUSiEiKg==

lim(sin(x)/x, x=0);

NiMtSSRsaW1HNiI2JComLUkkc2luRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHRiU2I0kieEdGJSIiIkYuISIiL0YuIiIh

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

NiMiIiI=

This is really just a flaw in normal mathematical notation. When we write NiMqJkkkc2luRzYkSSpwcm90ZWN0ZWRHRiZJKF9zeXNsaWJHNiIiIiNJInhHRigiIiI=, what we mean is NiMqJEkpKHNpbih4KSlHNiIiIiM=. This shorthand is horribly misleading, but it is universal. Maple is only capable of understanding the second expression, though it will write meaningless strings that may look sensible if you try to use the first expression.sin^2*x;

NiMqJkkkc2luRzYkSSpwcm90ZWN0ZWRHRiZJKF9zeXNsaWJHNiIiIiNJInhHRigiIiI=

evalf(sin^2*1);

NiMqJEkkc2luRzYkSSpwcm90ZWN0ZWRHRiZJKF9zeXNsaWJHNiIiIiM=

sin(x)^2;

NiMqJC1JJHNpbkc2JEkqcHJvdGVjdGVkR0YnSShfc3lzbGliRzYiNiNJInhHRikiIiM=

evalf(sin(1)^2);

NiMkIiskPU0yMyghIzU=

Maple can be used as a sort of mathematical word-processor, though it's a bit rough. The basic commands you need to know are:To type text into Maple, hit the little T button near the top of the Maple window, or use Insert -> Text or its keyboard shortcut. This will turn the current group into text instead of a Maple command.To insert a new Maple command or a new block of text, go to Insert -> Execution Group, and either insert a new block before or after the one containing the cursor. There are also keyboard shortcuts for this.To insert superscripts or other mathematical expressions into Maple text, go to Insert -> Standard Math, then use the input area near the top of the window to type the mathematical expression you want in Maple syntax. Maple should now stick that expression into your worksheet. This may take you a bit of playing around. It's pretty clunky, but I can usually get it to work.You can also insert things like paragraphs, sections, and hyperlinks.To use a variable name like NiMmSSJhRzYiNiMiIiM=, use a[2].a[0] + a[1]*x + a[2]*x^2;

NiMsKCZJImFHNiI2IyIiISIiIiomJkYlNiNGKUYpSSJ4R0YmRilGKSomJkYlNiMiIiNGKUYtRjFGKQ==

It's worth mentioning at least 3 other tidbits related to the Maple user interface.First, if you have trouble remembering the syntax for common Maple commands, you can click on View -> Palette -> Show All to get a collection of palettes you can click on to insert templates for Maple commands in your worksheet. By default, these palettes are available to the left side of your Maple window.Second, the Windows and Linux platforms offer another, simpler user interface for Maple, which was the standard interface before Maple 9, and which runs much faster on slower machines with less memory. To launch Maple with this "classic worksheet" interface, either double-click on the CW-Maple ("Classic Worksheet Maple") icon or, in Linux, type maple -cw at the command prompt. You lose a few bells and whistles like the ability to specify the transparency of a 3D plot, but you gain a lot of responsiveness.Finally, there is also a command-line version of Maple. You might want this, for instance, if you were connecting to a Linux host via ssh, and you wanted to run Maple without fancy graphics but without the overhead of an X-windows session. To run command-line Maple, type maple at the Linux shell prompt, or cmaple at the Windows command prompt. On the Mac, assuming Maple lives in the Applications directory, you want to run /Applications/Maple\ 9.5/Maple\ 9.5.app/Contents/MacOS/bin/maple. For the record, the ordinary graphical version of Maple is launched under Linux by typing xmaple at the shell prompt.

Want a spreadsheet in which the cells can contain any Maple expression? You can Insert->Spreadsheet. There is more information available from the Help system.

Many of Maple's commands live in library packages that are not loaded at startup. You can load one of the packages using the with() command. Loading a package lists the new functions that have been defined. A list of packages can be found by going to Help->Table of Contents and clicking on Packages. Each package has an associated help file.with(plots);

Warning, the name changecoords has been redefined

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

fieldplot([y,-x], x=-2..2, y=-2..2);

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with(Student[Calculus1]);

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

RiemannSum(x*(x - 2)*(x - 3), x=0..5, method = upper, output = plot);

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

Packages you may find yourself wanting include:1. plots, for all kinds of fancy plots.2. linalg, for linear algebra. There is also a LinearAlgebra package.3. Student[Calculus1] and various other parts of the Student package, for tools aimed at helping students work with new concepts.4. DEtools, for solving and visualizing differential equations.

Is there more to learn about Maple? Yes! Maple is a very broad and powerful system for mathematical computation. It has more than 2000 built-in functions. The large majority of these have never been used by anyone at Earlham. Here is a very quick glossary of the most important functions for a calculus student. More information on all these is available through Maple's online help, or in the written and human sources mentioned above:+, -, *, / Add, subtract, multiply, divide. ^ Raise a number to a power. ! Factorial. -> Used in procedure definitions. = Used in inputting equations. := Used to assign a value to a variable. <, >, <=, >=, <> Mean <, >, , , and , resp. %, %%, %%% Previous expressions. .. An interval. [ ] List. Lists are ordered. { } Set. Sets are unordered. abs Computes the absolute value of a number. coeff Get one coefficient of a polynomial. collect Collect coefficients of like powers. combine Combine terms into a single term. convert Convert from one data type to another. cos Cosine. Arguments are in radians. D Differential operator. denom Denominator. diff Differentiate. exp(1) e, The base for natural logs. evalf Evaluate an expression as a decimal approximation. exp Exponential function. exp(x) = E^x. expand Expand out an expression. factor Factor a polynomial. for...do...end do Repeat commands. fsolve Find approximate solutions to one or more equations. help Get help on a function. if...then...else...end if Conditionals. ifactor Factor an integer. int Integrate a function. Int Write down the integral, but don't evaluate. Use with changevar. limit Compute limits. Limit Write doen but don't evaluate a limit. ln, log Compute natural logs. ln(x)=log(x)log10(x). max Compute the maximum of 2 or more numbers. min Compute the minimum of 2 or more numbers. normal Put a fraction in normal form. A useful special case of simplify. numer Numerator. op Pick out one part of an expression. plot Plot a graph. plot3d Plot a 3D graph. print Pretty-print an expression. proc Define a procedure. product Product of finitely or infinitely many terms. quo Quotient of 2 polynomials. rem Remainder of 2 polynomials. simplify Try to simplify an expression. sin Sine. The arguments to trig functions should be in radians. solve Solve 1 or more equations exactly. sort Sort a list or the terms in a polynomial. sqrt Square root. student A student calculus package. Student[Calculus1] is a related package. student[changevar] Do a change of variables (integration by substitution). sum Sum of finitely or infinitely many terms. Sum Sum written down, but not worked out. Taylor Taylor series expansion. value evaluate a Sum, Limit, or Int. with Load a library package. Example: with(plots);