Drawing Lines - A Slippery SlopeAn Introduction to MapleJeremy JohnsonThe purpose of this worksheet is to introduce some of the functionality of Maple (simple mathematical computations such as those seen in High School mathematics, plotting, and simple programming constructs such as assignment). The material is presented in the context of computing with lines, solving linear equations, and interpolation. Recall that through any two distinct points there exist a unique line, and that it is easy to determine the line by computing the slope of the two points and determining an arbitray point (x,y) on the line such that the slope between it and the other two points is the same as the slope between the two given points. The first problem uses Maple to find and plot the line that goes though two given points. The fact that there is a unique line through any two distinct points can be generalized to higher degree curves. More generally, given 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 points [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] with distinct LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2OVEhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSp1bmRlcmxpbmVHRjcvJSpzdWJzY3JpcHRHRjcvJSxzdXBlcnNjcmlwdEdGNy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR0ZDLyUnb3BhcXVlR0Y3LyUrZXhlY3V0YWJsZUdGNy8lKXJlYWRvbmx5R0Y3LyUpY29tcG9zZWRHRjcvJSpjb252ZXJ0ZWRHRjcvJStpbXNlbGVjdGVkR0Y3LyUscGxhY2Vob2xkZXJHRjcvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRicvJSptYXRoY29sb3JHRkMvJS9tYXRoYmFja2dyb3VuZEdGQy8lK2ZvbnRmYW1pbHlHRjEvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUpbWF0aHNpemVHRjQtSSVtc3ViR0YkNiYtRiw2OVEieEYnRi9GMkY1RjhGO0Y9Rj9GQUZERkZGSEZKRkxGTkZQRlJGVEZXRllGZW5GZ25Gam4tRiM2Iy1GLDY5USJpRidGL0YyRjVGOEY7Rj1GP0ZBRkRGRkZIRkpGTEZORlBGUkZURldGWUZlbkZnbkZqbi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLyUscGxhY2Vob2xkZXJHRjdGKw==, there exists a unique polynomial of degree 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 that goes through the given points. The problem of finding the polynomial is called interpolation, and the second problem uses Maple to find the unique quadratic that goes through 3 such points. The third problem involves finding the best line that goes through a cloud of points that appear to cluster about a line. There is a bonus problem involving circles. Through any three points not on the same line, there exists a unique circle. The same function, solve, used to solve the first two problems can be used to find the circle on three given, non-colinear points. Before working on these problems the basic functionality of Maple is introduced and the specific commands and features necessary for the lab are reviewed.
Maple has a wealth of mathematical knowledge built into it and can be used throughout your college and professional career whenever mathematical computations are required. Maple "knows" all or almost all of the Mathematics you will see in your Mathematics, Science, and Engineering courses at Drexel. In particular, Maple has commands to perform all of the computations you will learn about in your calculus course (e.g. limits, differentiation, integration) and provides functionality, such as plotting, numeric and symbolic computation, and scripting that will allow you to explore these concepts. The purpose of this course is to help students become familiar with the capabilities (and limitations) of Maple and to teach students how to effectively use Maple to explore mathematics, science, and engineering concepts. Maple is a very powerful tool (it is the result of many years of development by Computer Scientists and Mathematicians) and can be complicated to use; however, we hope that after this course you will become sophisticated enough in its use that you can overcome these difficulties and can benefit from its power. In order to become an effective user, you must be aware of some concepts from Computer Science (e.g. evaluation - names vs. values, programming, data structures, algorithms, graphics, user interface design, theory of computation - what can and can not be computed) and this course will introduce you to these concepts in the context of computations that arise in some of your math, science and CS courses. Hopefully the course will help you learn/relearn some of the material in your previous math and CS courses and will enable you to more easily use the material in the future.
<Text-field style="Heading 1" layout="Heading 1">Introduction to Maple</Text-field>This section introduces some of the functionality of Maple. Maple provides a worksheet interface which combines computation, text, and graphics. Maple provides a large collection of commands for performing various mathematical computations. Computations can be performed with numbers (integers, fractions, square roots, etc.) and symbolic expressions (variables, mathematical symbols such as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2OVEjUGlGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlc2l6ZUdRIzEyRicvJSVib2xkR1EmZmFsc2VGJy8lJ2l0YWxpY0dGNy8lKnVuZGVybGluZUdGNy8lKnN1YnNjcmlwdEdGNy8lLHN1cGVyc2NyaXB0R0Y3LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lK2JhY2tncm91bmRHRkIvJSdvcGFxdWVHRjcvJStleGVjdXRhYmxlR0Y3LyUpcmVhZG9ubHlHRjcvJSljb21wb3NlZEdGNy8lKmNvbnZlcnRlZEdGNy8lK2ltc2VsZWN0ZWRHRjcvJSxwbGFjZWhvbGRlckdGNy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lKm1hdGhjb2xvckdGQi8lL21hdGhiYWNrZ3JvdW5kR0ZCLyUrZm9udGZhbWlseUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSltYXRoc2l6ZUdGNA==, mathematical functions such as sin(x), etc.).
Maple commands are entered within an execution group. All commands must be terminated with a semicolon or colon. If terminated with a semicolon, then, after the enter key entered, the command is evaluated and the result is printed. If terminated with a colon, the command is evaluated but NOT printed. The value of the previous command can be referred by % (note that the value of % refers to the last dynamically executed command - this may not be the command located at the previous execution group, if the user entered a command in another execution group). Multiple commands can be put in one execution group provided they are separated by semicolons or colons. Execution groups can be inserted into a worksheet before or after the cursor (look under the insert menu).You can insert text areasText can be placed in an execution group by inserting a paragraph (go to the insert menu) before or after the cursor.2+3*5;2+3*5:%;2*x + x^2 -x + 3*x*x + x^3/x;The result of a command can be assigned to a variable, which can be used in later commands.v := 2+3*5;Observe that once a variable is assigned a value, the corresponding value is used in expressions containing the variable (i.e. it can no longer be used as a symbol like we did with x above. 2*v;v+1;In the following assignment, the variable v is assigned the value obtained from the expression v+1 using the current value of v. If v is used in another expression after the updated assignment, the new value is usedv := v + 1;2*v;v := 'v';v+1;In the remainder of the section we show how Maple can perform arithmetic with integers, rational numbers, polynomials, and rational functions. Maple can perform basic arithmetic and there are functions for finding greatest common divisors, least common multiples, and prime factorizations of both integers and polynomials. Maple can also perform arithmetic with mathematical symbols such as Pi and functions such as sin(x).Often it is necessary to store a collection of numbers or other mathematical objects (such as a list of points). Maple provides "data structures" such as sequences, lists, and sets for this purpose. The basic notation for these data structures is introduced so that we can plot a list of points.Next we introduce the maple command solve which can be used to solve a large class of equations. This command is the key tool for this lab and will be used again throughout the course. The introduction is concluded with a few common errors that you may encounter and things you need to watch out for.
<Text-field style="Heading 2" layout="Heading 2">Getting Help</Text-field>The most important thing to learn in this lab, is how to get help. Maple provides an extensive array of resources to help you use Maple. There are resources for new users, for students, for quickly looking up syntax or how to accomplish basic tasks, detailed documentation with examples for all commands, and even materials reviewing different mathematical concepts. The various help resources can be accessed through the Help menu. You should familiarize yourself with these features. In particular, observe the item for new users, which contains an introduction, quick tour, and summary of how to do basic tasks, the Math Dictionary, which defines over 5000 mathematical terms, the search facility which allows you to search for topics or words, and a table of contents which allows you to peruse all of the features of Maple.The item, Maple on the Web, provides easy access to a variety of resources availble from the Maple website, including a student resource center.
<Text-field style="Heading 2" layout="Heading 2">Arithmetic</Text-field>Maple can compute exactly with very large integers. The next command computes 100!, where 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.. Note that 100! has 158 digits. The resulting number can be factored using the command ifactor. Maple can also perform arithmetic with fractions. Fractions are always simplified by removing common factors in the numerator and denominator. The functions ilcm and igcd computes the least common multiple and greatest common divisor of two or more integers.3! = 3*2*1;100!;ifactor(%);length(100!);1 + 1/2 + 1/3;11/6 + 1/4;ilcm(6,4) = 6*4/igcd(6,4);Sum(1/i,i=1..10) = sum(1/i,i=1..10);
<Text-field style="Heading 2" layout="Heading 2">Polynomials and Rational Functions</Text-field>Maple can also compute with polynomials and rational functions (quotient of two polynomials). Polynomials are entered as expressions involving powers of any symbol. In these examples we use the symbol x, though any symbol can be used. Just like the integer examples, we show how to factor polynomials and compute greatest common divisors. We also show how to divide polynomials to obtain a quotient and remainder. Note that these command require that the name of the variable be passed as an argument.1/(1-x) + 1/(1+x);simplify(%);convert(%,parfrac);factor(x^2-1);x^12-1;factor(%)expand(%);subs(x=1,%);gcd(x^15-1,x^12-1);rem(x^15-1,x^3-1,x);quo(x^15-1,x^3-1,x);
<Text-field style="Heading 2" layout="Heading 2">Symbolic and Numeric Computation</Text-field>Maple is aware of symbolic expressions such as Pi and can compute symbolically with them. The function evalf can be used to obtain numeric approximations, to any desired precision, of many mathematical expressions.Pi;evalf(Pi);evalf(Pi,100);evalf(Pi,500);sin(Pi);
<Text-field style="Heading 2" layout="Heading 2">Sequences, Sets, and Lists</Text-field>The command seq can be used to generate a sequence of elements, where the ith element of the sequences is given as a function of the index i. Elements in a sequence are separated by commas. An arbitray sequence can be constructed by listing element separated by commas. The ith elements of a sequence S can be accessed with S[i]. Note that sequences are indexed starting at 1.seq(i,i=1..10);seq(3*i+1,i=1..10);S := 2,7,1;S[1]; S[2]; S[3];A list can either be empty, denoted by [], or it contains a sequence of elements [S]. Elements of a list are accessed in the same way as elements in a sequence. The number of elements in a list can be determined using the function nops. An empty list has zero elements.L := [];nops(L);L := [seq(i,i=1..3)];nops(L);p1 := [1,3];p2 := [3,2];p1[1]; p1[2];L := [p1,p2];L[1];L[1][2];L := [1,2,2,3];A set is similar to a list except that duplicate elements are removed. Sets are designated in Maple using curly braces.S := {1,2,2,3};
<Text-field style="Heading 2" layout="Heading 2">Plotting Functions and Points</Text-field>Simple two-dimensional plots can be obtained using the plot function. Three dimensional plots are also available with the plot3d command. Additional plotting functions are available in the plots package. plot(sin(x),x=-Pi..Pi);Additional plotting facilities are available in the plots package. Functions in the plot package can be used by specifying the function name and package name with the syntax package[name], e.g. plots[pointplot]. To avoid having to specify the package everytime you want to call functions in a given package, use the with(package) command.with(plots);