M0030M: Maple Laboration

Transkript

1 M0030M: Maple Laboration Norbert Euler This document contains the rules and instructions for the Maple computer lab as well as the Maple exercises for the course M0030M. The rules and instructions are given in both Swedish and English, whereas the Maple exercises are listed as Problems 1 to 5 in English only. If you, as a student in this course, have any questions regarding the rules and instructions, then please contact your teacher or the exminer, Norbert Euler, for further clarification. Regler och instruktioner: 1. För att bli godkänd på Maple-momentet av kursen M0030M måste studenterna lösa uppgifterna i detta dokument med hjälp av Maple. 2. Studenterna får jobba enskilt eller i en grupp bestående av högst 3 personer. En lektion om Maple kommer att ges av studenternas lärare och inkluderar en kort introduktion av Maple och information om uppgifterna. Studenterna förväntas kunna lösa uppgifterna på egen hand. På Fronter kan man även finna information om Maple av Mikael Stenlund. 3. Flera laborationer har blivit schemalagda för denna kurs i E-huset. Mer information angående schemat finns tillgängligt på LTU s hemsida. 4. Det finns bara två betyg på detta moment. Studenterna kan antingen bli godkända eller icke godkända. För att bli godkänd måste studenterna lösa alla problem korrekt och presentera resultatet väl i en skriftlig redogörelse. Det är viktigt att studenterna anger sin e-post adress i den skriftliga redogörelsen, skriver datum för inlämningen och sitt fullständiga namn, samt vilket program de studerar, t.ex. Rymdteknik. Studenternas lärare bestämmer hur rapporten skickas in, t.ex. bifogad i mail eller en utskriven på papper. Varje lärare kommer att ge instruktioner under lektionen hur studenterna ska lämna in den skriftliga redogörelsen. OBS! Skriftlig redogörelse kan ej bli inskickad via fronter. Rättningen och bedömningen av studenternas skriftlig redogörelse kommer att utföras av läraren som undervisar gruppen. Om läraren inte är nöjd med skriftlig redogörelse så kommer hen att ge dessa studenter instruktioner för förbättringar. 5. Under den sista lektionen kommer läraren med en lista med studenternas betyg för laborationen. OBS! Studenterna är ansvariga för att själva kontrollera att deras betyg skrivits in korrekt på listan under den sista lektionen.

2 6. Varje lärare har en skyldighet att rapportera in varje misstanke om fusk till disciplinnämden för vidare utredning och åtgärder. Följande är ett example på fusk. Varje form av kopiering av resultat och lösningssteg till någon av uppgifterna från laborationen. Detta gäller såväl kopiering av andra personers lösningar och lösningar som möjligen kan hittas på internet. 7. Alla studenter ska lämna in sina skriftliga redogörelser senast den 14 december Inga skriftliga redogörelser kommer att intas efter 14 december Den student som inte lämnar in sin rapport senast detta datum kommer automatiskt att bli underkänd denna omgång. Nästa omgång kommer att vara i maj Studenterna kommer då att få nya uppgifter för labben under LP4 (2016). Rules and Instructions: 1. Students must solve the problems listed in this document by using the program Maple. 2. Students can work in groups consisting of maximum three student members and hand in a joint written presentation. One lecture on Maple has been scheduled for each group during the course and this lecture will include a short introduction to Maple and some information regarding the exercises. The students are expected to make use of the online help included in the Maple package which is installed on the LTU computers, as well as the Maple notes by Mikael Stenlund that are provided on fronter. 3. Several lab sessions have been scheduled for this course in the E-building and students can attend any of those sessions, provided space is available in the computer room. Please visit the LTU schema website for details on the lab schedule. 4. Two grades are possible for the Maple computer lab, namely a Pass or a Fail. In order to pass the Maple computer lab the student must solve the listed problems correctly and present the results neatly in a written presentation. The written presentation should contain all Maple steps used in solving the problems and the solutions should moreover be described in the written presentation. Students can use Word or Latex to write their written presentation. Students must write their full name and surname, a contact address, the date of submission, as well as the programme in which they are studying, e.g. Rymdteknik, on the front page of their written presentation. The form

3 of submission, i.e. in printer form or as an attachement by , will be determined by the particular teacher. Most teachers prefer a paper printed version of the written presentation, but exact information regarding the submission will be given by the teacher during the lecture. Please note that the written presentations can not be submitted via fronter. The evaluation and grading of the written presentations will be done by the teacher who is responsible for the particular group that he/she is teaching. If the teacher is not satisfied with the solutions handed in by the student(s), then the teacher will provide the student(s) with instructions for improvements. 5. During the last lecture, the teachers will bring with them a list that will contain the grades of the written presentations for the students in their group. It is the students own responsibility to verify that their grades have been recorded correctly on that list during the last lecture in January. 6. Teachers are obliged to report any suspected foul play of the laboration to the university disciplinary office for further investigation and conviction. The following is an examples of foul play of the laboration: the copying, in any form or by any means, of results or program steps of the solutions of any hereby listed exercise, from other persons or student participants or from an internet site. 7. All students must submit their written presentations to their teacher at the latest on the 14th of December No lab results will be accepted after this date. Those students that have not handed in their written presentations by this date will automatically fail this Maple computer lab. The next opportunity to do the Maple computer lab will then be in May 2016 and different exercises will be provided during LP4 (2016) for this lab.

4 Maple Exercises: Solve the following five problems with the help of Maple and write your results and explanations clearly in a written presentation. Please follow the Rules and Instructions as listed above. Problem 1: Consider the function f(x) = x x x 1 2 and use Maple to calculate the following. a) Find the intersection that the curve y = f(x) makes with the x-axis. b) Find the coordinates of the points, where the tangent line to the curve y = f(x) is horizontal. c) Plot the curve y = f(x) in the x-interval [ 5, 3]. Problem 2: Consider the functions a) f(x) = 2x2 + 3x 5 3x b) f(x) = 5 x 1 2 x c) f(x) = 3x2 + 5x + 6 x + 1 Find now all asymptotes for each function by using the Maple command Asymptotes, by first loading the package with(student[calculus1]).

5 Problem 3: Make use of the Maple command fsolve to solve the equations i) sin(x) + 1 x 2 = 0 (two real solutions) ii) x 4 x 1 = 0 (two real roots) iii) x 5 + 4x 4 2x 3 + 6x 2 + 3x 5 = 0 (three real roots) Problem 4: Consider the following two planes in R 3 : Π 1 : 15x + 19y + 21z = 12 Π 2 : 19x + 8y + 26z = 3. Load the Maple package with(linearalgebra) and then a) Calculate the intersection of the above planes. b) Plot the two planes on the same graph with the Maple command plot3d. Problem 5: Load the Maple package with(linearalgebra) and then a) Enter, in your Maple file, any two non-zero vectors of your choice v R 3 and u R 3. Use Maple commands to i) Calculate the dot product u v ii) Calculate the cross product u v and the scalar-triple product v u v. iii) Calculate the length of u and the length of v.

6 iv) Find the equation of the plane or the line that is spanned by your vectors u and v. b) Enter, in your Maple file, any 5 4 matrix A and any 4 5 matrix B of your choice, such that A and B do not contain any zero entries. Use Maple commands to calculate i) The product AB. ii) The determinant of AB. iii) Write down any non-zero vector b R 5 and use Maple to solve x R 5 for the equation (AB)x = b. How many solutions does this equation have and how many arbitrary (free) parameters do your solutions contain? c) Generate a random 6 6 matrix P with the Maple command P := RandomMatrix(6, 6, generator = 1.. 9) i) Calculate the determinant of your generated matrix P. ii) Calculate the inverse of P, that is calculate P 1. If your generated matrix P is not invertible, then generate a new random matrix P until you find an invertible matrix and calculate then its inverse. iii) Generate a random column vector b R 6, by the Maple command b:=randomvector(6, generator = rand(1.. 30)) and use your matrix P 1 from part c ii) above to solve the equation P x = b.