DISCOVERING MATHEMATICS WITH MAPLEAn Interactive Exploration for Mathematicians, Engineers, and EconometriciansChapter 1. A Tour of Maple VJust a Maple SessionIn this session we shall run through a number of examples in order to show the use of Maple as a calculator. But we shall not attempt to give a complete picture of the Maple commands we use. The details will be left to the introductory worksheets TourW1a.mws and TourW1b.mws. This session is merely meant to provide an overall view of the possibilities, without giving much attention to individual commands and all their special features. So, immediate and complete understanding is unlikely and not strictly required. In the worksheets that follow this session, more attention is paid to the details.Numerical CalculationsWe shall adopt the usual convention that Maple commands, constants and variables are recognizable by the font and the style. In the text, a Maple command, a variable, expression or parameter is usually printed in boldface, like expression. Maple input lines are preceded by a Maple prompt (>) and the Maple instructions are always given in the typewriter font. Finally, Maple variables in an output region are printed in italics, like variable. After starting a new worksheet, the cursor (|) blinks directly to the right of the Maple prompt. This means that Maple waits for an instruction to be given at this position. We might try ?trig in order to find out what Maple knows about the trigonometric functions. To indicate that we have finished typing an instruction and wait for Maple to act on it, we hit the <Enter> key.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEiP0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEldHJpZ0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJw==A help page opens with information on the trigonometric functions. From this information we learn that the Maple command sin pertains to the ordinary sine function. Verification of a few values may convince you. 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JSFHYes, we recognize the correct values corresponding to the familiar arguments of the sine function. The commas are used to separate the Maple answers and place them in a single row instead of each on an output row of its own. This Maple output row of sine values is made into a so-called 'list' by putting square brackets around it.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JSFHEvery complete Maple instruction must end in a semicolon (;) or colon (:). The semicolon causes the result of the instruction to appear as output on the screen. This output is suppressed when the colon is used. When the closing symbol (; or :) is omitted, nothing happens after hitting the <Enter> key, simply because Maple is waiting for the instruction to be completed. Newcomers to the Maple system often make the mistake of repeating the instruction in such a situation, with the most likely result of an error message (syntax error) being issued, because Maple tries to read the double instruction as a single one. The best advise is to type a ; when Maple appears to be waiting. The ditto symbol or percentage sign (%) refers to the last instruction processed by Maple. This symbol should be repeated once (%%) or twice (%%%) if you wish to refer to the one but last Maple instruction or the one before that, respectively. To go back any further requires Maple's history command; see Exercise 1 of section 1.5.Further, in the previous lines, the use of constants like LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== (=Pi) catches the eye, and also the fact that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbXNxcnRHRiQ2Iy1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbXNxcnRHRiQ2Iy1JI21uR0YkNiRRIjNGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== remain unevaluated. Apparently, Maple does not automatically replace such constants by approximating decimal fractions, but instead keeps working with the symbolic (and thus exact) representations like LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbXNxcnRHRiQ2Iy1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbXNxcnRHRiQ2Iy1JI21uR0YkNiRRIjNGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==. Consider the following symbolic expression: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JSFHIf we happen to need a numerical value for this expression, Maple obliges on request by evaluating it to any precision required, that is to say, rounded to as many decimals as we wish. 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 instruction evalf(%,50) - this means: floating point evaluation; a floating point number, 'float' briefly, is a real number represented by a decimal fraction with a fixed but freely chosen number of digits - produces a numerical value for the quantity symbolic_expression rounded to 50 decimal digits. The same result can be achieved by clicking the right mouse button on the Maple output of symbolic_expression and selecting Approximate in the menu that pops up.The instruction evalf(%, 50) has no effect on the exact value of the variable symbolic_expression.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUTRzeW1ib2xpY19leHByZXNzaW9uRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9JSFHMaple knows a few mathematical constants, like LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJmdhbW1hO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==, the Euler constant. The names of these constants are protected, so that it is impossible to accidently assign new values to them. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjUGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkktSSNtbkdGJDYkUSgzLjE0MTU5RidGMg==JSFHYou may assign new values or expressions to such protected names but only after using Maple's command unprotect; this is not really recommended. Another well-known constant is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, the base of the natural logarithm ln. This constant is not a Maple constant. In fact, Maple knows the mathematical constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= as exp(1). To avoid having to type the full expression exp(1) every time we wish to use LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= , we could define LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= to have this value throughout the Maple session. But beware, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is not a protected constant: we can redefine it if we like. PkkiZUc2Ii1JJGV4cEdGJDYjIiIiNiQtSSZldmFsZkclKnByb3RlY3RlZEc2I0kiZUc2Ii1JI2xuR0YoRiY=QyQ+SSJlRzYiJCIlPUYhIiQiIiI=LUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kiZUdGJw==JSFHIt is possible to protect a newly defined expression in the following way:QyQ+SSJlRzYiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMiIiJGLA==LUkocHJvdGVjdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLkkiZUdGJw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSNtbkdGJDYkUSYyLjcxOEYnRkBGK0YrLUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kiZUdGJQ==JSFHWithout doubt you remember that the mathematical constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is also equal to the limit value of the sequence 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 as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= tends to infinity. Let us check to see if this is known to Maple. Note that infinity is also a Maple constant.
As you will see below, we first use the Limit command with capital L. Use of a capital letter is not accidental but an indication of the so-called `inert' form of a Maple command, which is returned unevaluated, or rather, evaluated to its name. The use of the lower case command causes the value (in this case of the limit) to appear as output. We shall return to this later.
[Update Note:
Now in Standard Maple .mw worksheets, one can create a normal limit expression using the Expression and Common Symbols palette, then select the expression and right click on it and choose Format Menu, "Convert to, Inert Form" to get the inert form of an expression. If instead you select the whole input as presently written and right click on it and choose Format Menu, "Convert to, 1d Math" you will see the underlying Maple commands. You can convert it back to 2d math input in the same way. The inert form of the limit is signaled by the light gray "lim", typical of Windows menu choices that are unavailable, in this case, evaluation is not performed.] Ly1JJkxpbWl0RzYiNiQpLCYiIiJGKSokSSJuR0YlISIiRilGKy9GK0kpaW5maW5pdHlHJSpwcm90ZWN0ZWRHLUkmbGltaXRHRiVGJg==LCYtSSRyaHNHJSpwcm90ZWN0ZWRHNiNJIiVHNiIiIiJJImVHRighIiI=LUkmZXZhbGJHJSpwcm90ZWN0ZWRHNiMvLCYtSSRyaHNHRiQ2I0kjJSVHNiIiIiItSSRleHBHNiRGJEkoX3N5c2xpYkdGLDYjRi0hIiIiIiE=JSFHThis confirms that Maple knows about this limit. We repeat that Maple distinguishes between lower case and upper case, but not only in Maple commands, also in general. Therefore, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and E are totally different names with possibly totally different meanings. In fact, E is a new name, so that any value can be assigned to it, whereas e = exp(1) is still protected.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSNtbkdGJDYkUSYyLjE3OEYnRkAvJStleGVjdXRhYmxlR0ZERkBGK0ZaRkBGK0ZaRkA=JSFHIn one of the preceding input lines we used the evalb command. The letter b comes from the name Boole, so that with evalb Boolean expressions are evaluated with two possible outcomes: true and false. Both are Maple constants.The sharp (#) is used by Maple to add comments to an input line. Text following # is ignored by Maple. In the current worksheet environment this option is not as useful as it used to be in previous Maple releases, because since Release 4 plain text can be put in text regions anywhere we wish. You may have noticed that there is a difference between the equality sign (=) and the assignment symbol (:=). This is no surprise for those who are familiar with Pascal. The symbol := is used to assign a value (or a name) to an arbitrary string of symbols, but remember, no interspacing. The assigned value faces the side of the equality sign. It is this feature that allows Maple (and any other CA system for that matter) to do calculations with symbols, where purely numerical packages can only handle numbers. This distinguishing feature is an essential characteristic of a CAS and that is why it will run through all the forthcoming discussions. Symbolic CalculationsWe continue our tour with examples in which symbols are manipulated like numbers. Apart from numbers, other objects like functions, polynomials and matrices can take part in these computations. 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JSFHThis is a familiar result; you recognize Newton's Binomium, don't you? If the exponent 4 is increased to 100, precisely 101 terms will be printed to the screen instead of a mere five. This is rather a lot, acknowledging that we most likely are not really interested in every individual coefficient. Therefore, it seems best to suppress the output. 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JSFHIt is good practice not to accept any result face value, we checked a more or less random coefficient, say number 31. Realizing that we are dealing with binomial coefficients, the outcome should be - apart from the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-power of course - the binomial coefficient binomial(100,31).Let us verify this. 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 is rather convincing. 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JSFHshould give the same result, again neglecting the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-power. Factorizing this large integer into prime numbers gives: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JSFHClearly, there are exactly five factors 2 occurring in this decomposition. Did Maple do a good job? We may gain confidence by counting the number of factors 2 occurring in the binomial coefficient binomial(100,31) in a different way, and comparing the two results. 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JSFHWhat does the answer mean? According to Maple, 97 is the precise number of factors 2 occurring in 100! (100 factorial). Recall that floor(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=) is the largest integer less than or equal to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= . Hence, floor(100/LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=) is the number of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-multiples not larger than 100. As LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRjU2JFEiNkYnRjgvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjhGK0Y4 < 100 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRjU2JFEiN0YnRjgvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjhGK0Y4, which implies that floor(100/LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRImlGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjI= ) = 0 for all LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJpRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiN0YnRj5GPkYrRj4= or larger, the summation index LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= does not need to run beyond 6. As a result, the number of factors 2 occurring in binomial(100,31) is 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 in Limit before, we used the inert form of the Maple command (capital first letter) in the left-hand side of the equality to enhance the readability. As a consequence, evaluation takes place to the name, so that the command's full name is printed to the screen.
[Update Note:
We could have introduced the same limit expression on the left hand side as on the right hand side and then converted it to Inert Form as explained above. Instead we typed in the actually 1d Maple command for an inert "Limit". If you convert the whole expression to 1d Math you will see that the only difference on the right hand side is the lower case form "limit".]You probably won't be surprised to learn that Maple is able to differentiate and integrate. Let us choose a differentiable function (any will do), and let us instruct Maple to calculate its derivative and second order derivative. PkkiZkc2Ii1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjLCYqJkkieEdGJCIiIi1JI2xuR0YnNiMsJkYuRi4qJEYtIiIjRi5GLkYuRi0hIiI=PkkjZidHNiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkSSJmR0YkSSJ4R0YkPkkkZicnRzYiLUklZGlmZkclKnByb3RlY3RlZEc2JUkiZkdGJEkieEdGJEYqJSFHClearly, the form of the second order derivative is rather involved. Maple does not have any trouble plotting such complicated functions. In the figure below the graphs of the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and its derivative LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZidGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn are plotted on the interval [LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFEiM0YnRjVGNUYrRjU=,1] and displayed together in a single picture. Note that those names which have to be taken literally should be surrounded by left quotes as in `LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZidGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn ` and `LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZiJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn`. This is necessary here, because, as we shall see, the single quote ' has a special meaning in the Maple language - this is also the case for the double quote " in Release 4. So, if we wish to use quotes as part of a name (or string), we have to put left quotes around this string. LUklcGxvdEc2IjYlPCRJImZHRiRJI2YnR0YkL0kieEdGJDshIiQiIiIvSSZjb2xvckdGJEkmYmxhY2tHRiQ=JSFHIt is not so difficult to distinguish the two graphs, because LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= has positive values only and its derivative LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZidGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn has positive and negative values. Judging by the graphs, the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= has a local maximum and a local minimum on the interval [LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFEiM0YnRjVGNUYrRjU=,1]. Indeed, the derivative vanishes at two different points which are positioned symmetrically with respect to the origin. This symmetry is obvious from the analytical form of the derivative: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZidGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=)/LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=) is a function of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQUYrRkE=. LUknZnNvbHZlRzYiNiQvSSNmJ0dGJCIiIS9JInhHRiQ7ISIkIiIiJSFHWe see that fsolve (the letter f again comes from 'float') finds only one of the two zeros of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZidGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn. This is not a mistake in the Maple program as you might think, but an unavoidable consequence of the numerical process of calculating roots of equations. At a later stage we shall try to clarify this point.NiVJKG1pbmltdW1HNiIvSShtYXhpbXVtR0YkLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSVzdWJzR0YpNiQvSSJ4R0YkSSIlR0YkSSJmR0YkRic=JSFHJSFHTo verify Maple's computation of the derivative, we could try to reverse the process and calculate the indefinite integral by giving the following instruction:Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSNmJ0dGKEkieEdGKC1JJGludEdGKEYpJSFH[Update Note:
Convert to the previous input to 1d Math to see the Maple syntax for this input equation.]
Apparently, this is quite easy, as Maple recovers the original function very quickly. Indefinite integration - or finding a primitive - of the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= itself may prove much harder, if not impossible. 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JSFHAs was to be expected, Maple is unable to find a primitive function for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and shows this by echoing the integration formula. Note that the first letter of int is lower case. Therefore, the failure to produce a result can not be attributed to an inert command. In fact, Maple's lack of success does not come as a surprise, because it is impossible to express a primitive for this function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= in terms of elementary functions or other special functions. On the other hand, Maple is able to calculate the definite integral of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= over the interval [LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFEiM0YnRjVGNUYrRjU=,1] without difficulty, if instructed to proceed numerically.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JSFHThe capital letter I of Int in the right-hand side of the last input line prevents Maple from first trying to integrate symbolically before finally turning to numeric integration methods. This makes sense, because we already know that Maple is unable to integrate this function symbolically. The procedure `evalf/int` (left quotes necessary) has the same objective.`evalf/int`(f,x=0..1);Again, note the back quotes; here the forward slash (/) is not the symbol for 'division', but merely a simple connector for joining the two commands evalf and int. In other words, it is part of the name of the procedure. The actual advantage of this procedure over the sequence of the individual commands int and evalf is that it gives more control over the precision and the numerical integration procedure we wish Maple to use. In fact Maple automatically calls evalf/int when we apply evalf after Int (with a capital I). To complete this Maple session we give a few examples of the use of matrices under Maple. With this in mind, we first load the LinearAlgebra package of linear algebra procedures. A Maple package is just a collection of Maple procedures written for a common type of application. In order to avoid making the Maple system top-heavy by including too many instantly available procedures in the default Maple library, a selection was made of generally applicable routines, that is to say, of procedures that are useful in more than one specific situation. Procedures of a less general nature are grouped in special packages. The Maple command with(), in which the name of the particular package is put between the parentheses, is used to load all procedures of this package simultaneously. During the active session all procedures of the package thus loaded into the Maple library remain instantly available. QyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJLkxpbmVhckFsZ2VicmFHRiUhIiI=JSFH[Upgrade Note:
If you remove the colon from the previous input and re-execute it, you will see the list of commands loaded by the package.
You can easily input matrices with the input palette as long as the dimension does not get too large, and you can right click to get a menu to evaluate the quantities evaluated below with some of these commands from the package.]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 us compute the determinant of the matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, its trace, and its characteristic polynomial. PkkxZGV0ZXJtaW5hbnR+b2Z+QUc2Ii1JLERldGVybWluYW50R0YkNiNJIkFHRiQ=PkkrdHJhY2V+b2Z+QUc2Ii1JJlRyYWNlR0YkNiNJIkFHRiQ=Pkk/Y2hhcmFjdGVyaXN0aWN+cG9seW5vbWlhbH5vZn5BRzYiLUk5Q2hhcmFjdGVyaXN0aWNQb2x5bm9taWFsR0YkNiRJIkFHRiRJJ2xhbWJkYUdGJA==JSFHThe values of the determinant and the trace agree with the coefficients of the characteristic polynomial in as much as 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 and the coefficient of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSkmbGFtYmRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtbkdGJDYkUSIyRidGOi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGOkYrRjo= is 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. With the Maple command fsolve we can compute all zeros of the characteristic polynomial of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, including the complex ones. LUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiVJP2NoYXJhY3RlcmlzdGljfnBvbHlub21pYWx+b2Z+QUdGJ0knbGFtYmRhR0YnSShjb21wbGV4R0YlLUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiRJP2NoYXJhY3RlcmlzdGljfnBvbHlub21pYWx+b2Z+QUdGJ0knbGFtYmRhR0YnJSFHIt is clear that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= has one real and one pair of complex conjugate eigenvalues. Be warned that the Maple constant I is not used as the standard notation for the general identity matrix, but instead denotes the complex number LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbXNxcnRHRiQ2Iy1GIzYlLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkgtSSNtbkdGJDYkUSIxRidGNEY0RjQ=.The matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is invertible, because its determinant does not vanish, and hence LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is non-singular. Let us verify that Maple computes the inverse correctly.LUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJUkoX3N5c2xpYkdGKDYlSSJBR0YoKiRGLSEiIkkldHJ1ZUdGJQ==JSFHJSFHWe have come to the end of this Maple session. We hope that this introductory Maple session has whetted your appetite. You can now work through the worksheets TourW1a.mws and TourW1b.mws, and try your hand at the corresponding assignments in worksheets TourA1a.mws and TourA1b.mws.