MAT2615 EXAM PACK 2025

INTRODUCTION Dear Student Welcome to MAT2615. We trust that you will find the mathematics studied in this module interesting and useful, and that you will enjoy doing it. THIS TUTORIAL LETTER CONTAINS INFORMATION TO FACILITATE YOUR STUDIES. PLEASE READ IT CAREFULLY AND KEEP IT FOR FUTURE REFERENCE. The delivery mode for this module is “blended”, which means that only some study material will be printed and posted to you. All study material which may include additional material will be available on the university’s online campus, myUnisa (see below). The Department of Despatch will supply you with the following study matter for this module. Study Guides (Parts 1, 2 and 3), this tutorial letter, tutorial letters containing solutions to the assignments which will be sent out after the relevant closing dates. Apart from this letter you will receive the other tutorial letters during the semester. Some of the study matter mentioned above may not be available when you register. Tutorial matter that is not available when you register will be posted to you as soon as possible. It will also be available on myUnisa. If you have access to the internet, you can view the study guides and tutorial letters for the modules for which you are registered on the University’s online campus, myUnisa, at . In general, solutions will be available at this webpage sooner than you will receive them in the post. We therefore advise you to check here from time to time if possible. Note that you need to register on myUnisa. Through myUnisa you can submit assignments, access library resources, download study material as well as communicate with the university, your lecturers and other students. Since UNISA is moving towards online delivery only, it is very important to register on myUnisa and to access all your modules on myUnisa on a regular basis. 2 PURPOSE OF AND OUTCOMES FOR THE MODULE 2.1 Purpose The main purpose of this module is to extend concepts such as limits, continuity, differentiation and integration, studied in first year calculus, to functions of several variables. Furthermore, the purpose includes improving the problem solving skills of students and forming a basis of knowledge that is necessary for further studies in Mathematics and applications in Physics. 4 Downloaded by Thomas Mboya () lOMoARcPSD| MAT2615/101 2.2 Outcomes On completion of this module, you should be able to: 1. Investigate continuity of functions of several variables. 2. Determine limits, partial derivatives, gradients, directional derivatives, divergence and curl and apply these concepts to problem solving. 3. Determine the nature of extrema and solve optimization problems using Lagrange multipliers. 4. Determine double and triple integrals and use them to calculate areas and volumes. 5. Determine line, surface and flux integrals and apply the theorems of Green, Stokes and Gauss, which relate these types of integrals. 3 LECTURER(S) AND CONTACT DETAILS 3.1 Lecturer(s) The lecturer responsible for this module is Dr. J.E. Singleton. You can contact her at: Dr. J.E. Singleton Room no: 6–29 Block C, Florida Campus e-mail: A tutorial letter will be sent to you or an announcement made on myUnisa giving the telephone numbers of lecturers and any changes to this module. Please do not hesitate to consult your lecturer whenever you experience difficulties with your studies. You may contact your lecturer by phone or through correspondence or by making a personal visit to his/her office. Please arrange an appointment in advance (by telephone, letter or e– mail) to ensure that your lecturer will be available when you arrive. If you should experience any problems with the exercises in the study guide, your lecturer will gladly help you with them, provided that you send in your bona fide attempts. When sending in any queries or problems, please do so separately from your assignments and address them directly to your lecturer. 3.2 Department Department of Mathematical Sciences e-mail: Postal address: P O Box 392 UNISA 0003 5 Downloaded by Thomas Mboya () lOMoARcPSD| 3.3 University Consult the brochure myStudies @ Unisa for information on how to contact the university. Always use your student number when you contact the University. 4 MODULE RELATED RESOURCES 4.1 Prescribed books There is no prescribed textbook for this module. The study guide contains all the material that you need for exam purposes. 4.2 Recommended books Students who are interested in reading more about the subject matter or who seek additional exercises, may consult any of the recommended books listed below. (i) Anton, H.A, Bivens, I., Davis, S. with contributions by Polaski, T: Calculus (Ninth edition), John Wiley and Sons, Inc., 2010. (ii) Schey, H.M : Div, Grad, Curl and all that: an informal text on vector calculus (Fourth edition), W.W. Norton and Company, 2005. (iii) Stewart, J: Calculus: Early Transcendentals (Seventh edition), Brooks Cole, 2012. (vii) Zill, D.G., and Wright, W.S.: Calculus, early transcendentals (Fourth edition), Jones and Bartlett Publishers, 2011. 4.3 Electronic Reserves (e-Reserves) There are no e-Reserves for this module. 5 STUDENT SUPPORT SERVICES FOR THE MODULE For information on the various student support systems and services available at Unisa (e.g. student counselling, tutorial classes, language support), please consult the publication my Studies @ Unisa that you received with your study material. 6 MODULE SPECIFIC STUDY PLAN This module has 19 learning units (units for short) and we recommend that you work through at least two units per week. After studying units 1–3 attempt the self assessment task on these units which is contained in Addendum A. Then continue with your study of Units 4–11 and do Assignment 01.No assignment or exam questions will be asked on Unit 11– this is just for self study. Then study the remaining units (Units 12-19) and do Assignment 02. 6 Downloaded by Thomas Mboya () lOMoARcPSD| MAT2615/101 The following table gives a suggested study plan. Study elements Semester 1 Semester 2 Work through Units 1–3 and do the 5 February 2014 23 July 2014 self assessment task Work through Units 4–11 and do 7 March 2014 22 August 2014 Assignment 01 Work through Units 12–19 and do 4 April 2014 19 September 2014 Assignment 02 Work through previous exam 20 April 2014 3 October 2014 paper, and revision See the brochure myStudies@Unisa for general time management and planning skills. 7 MODULE PRACTICAL WORK AND WORK INTEGRATED LEARNING There are no practicals for this module. 8 ASSESSMENT 8.1 Assessment plan We have set two written assignments. For Assignments 01 and 02 you must plan to submit your answers by the dates listed below. You will receive the solutions for Assignments 01 and 02 automatically, even if you did not submit the relevant assignment. These solutions will be posted to ALL the students registered for this module about one week after the closing date of the relevant assignment, so it is important to submit your assignments so that they reach the Assignment Department at Unisa by the closing date. Markers will comment on the work that you submit in your assignments. The assignments and the comments constitute an important part of your learning and should help you to be better prepared for the examination. N.B. Please don’t wait for an assignment to be returned to you before starting to work on the next assignment. Assignments will be assessed not only on the mathematical correctness of your work, but also on whether you use mathematical notation and language to communicate your ideas clearly. Examination admission Please note that lecturers are not responsible for examination admission, and ALL enquiries about examination admission should be directed by e-mail to You will be admitted to the examination if and only if Assignment 01 reaches the Assignment Section by 7 March 2014 if you are registered for Semester 1, or by 22 August 2014 if you are registered for Semester 2. 7 Downloaded by Thomas Mboya () lOMoARcPSD| Note that your marks for the assignments contribute 20% to your final mark (the remaining 80% is contributed by the final examinations). There are two written assignments for this module. Note that not all the questions in the assignments will be marked and you will not be informed beforehand which questions will be marked.The reason for this is that Mathematics is learned by “doing Mathematics”, and it is therefore extremely important to do as many problems as possible. ASSIGNMENTS THAT REACH UNISA AFTER THE CLOSING DATE WILL NOT BE MARKED. If you have to mail an assignment from a remote area or from overseas, please mail it at least two weeks before the closing date. The assignments are designed to lead you through the study material and to help you gain an understanding of the concepts and techniques. You will note that the problems in the assignments are not the type of problems that you will be able to do by simply searching for similar worked examples. We are not interested in checking whether you can copy examples mechanically – our aim is teach you to solve problems independently so that, after having completed this module, you will be able to apply the skills that you have gained to solve problems in the workplace or in your further studies which may look very different from the worked examples in this module. Before attempting an assignment you should first study all the units which are covered by that assignment and try to get a global view of the study material. Then, when you attempt a specific question, you should study all the sections that are relevant to that question again. Pay special attention to the sketches in the guide and try to visualize the various concepts. You benefit a lot more by attempting problems yourself (even if you make mistakes) than by simply studying the solutions. You will also benefit from studying your lecturer’s comments on your answers. Semester Exam and Semester Mark Your semester mark for MAT2615 is calculated as follows: Semester mark = 0:5 Assignment 01 mark (in %) + 0:5 Assignment 02 mark (in %) The final mark for the module as a whole will then be calculated according to the following formula: Final Mark = 0:8 ( Exam mark) + 0:2 ( Semester mark) : So the semester mark will contribute 20%to the final mark and the exam mark 80%: If you obtain less than 40%in the exam, then your semester mark does not contribute to the final mark. 8.2 General assignment numbers The assignments are numbered as 01 and 02 for each semester. 8.2.1 Unique assignment numbers Please note that each assignment has a unique assignment number which must be written on the cover of your assignment. See the table in Section 8.2.2 for the unique numbers. 8 Downloaded by Thomas Mboya () lOMoARcPSD| MAT2615/101 8.2.2 Due dates for assignments The closing dates for submission of the assignments are: Semester 1 Semester 2 Assignment Nr. Unique Nr. Due date 7 March 2014 4 April 2014 22 Aug 2014 19 Sept 2014 8.3 Submission of assignments You may submit your assignments either by post or electronically via myUnisa. Assignments may not be submitted by fax or e–mail.For detailed information and requirements as far as assignments are concerned, see the brochure my Studies @ Unisa that you received with your study material. Assignments should be addressed to: The Registrar P O Box 392 UNISA 0003 To submit an assignment via myUnisa: Go to myUnisa. Log in with your student number and password. Select the module. Click on "Assignments" in the menu on the left–hand side of the screen. Click on the assignment number you wish to submit. Follow the instructions. 8.4 Assignments The assignment questions for Semester 1 are contained in Addendum B and those for Semester 2 in Addendum C. Make sure that you do the correct assignments. Solutions to the assignments will be posted to ALL students registered for this module a while after the closing date of the relevant assignment. Solutions will also be available on myUnisa. 9 OTHER ASSESSMENT METHODS Addendum A contains a self assessment task for Units 1–3. Attempt the self assessment task before you do Assignment 01 and compare your answers with those provided. 9 Downloaded by Thomas Mboya () lOMoARcPSD| 10 EXAMINATION If you are registered for the first semester, you will write the examination in May/ June 2014 and the supplementary examination will be written in October/ November 2014. If you are registered for the second semester you will write the examination in October/ November 2014 and the supplementary examination will be written in May/ June 2015. During the relevant semester, the Examination Section will provide you with information regarding the examination in general, examination venues, examination dates and examination times. The exam is a two hour exam. You are allowed to use a non-programmable calculator in the exam. The examination questions will be similar to the questions asked in the study guide and in the assignments. You need not know any proofs of theorems in this module; however, you have to be able to apply the formulas stated in the theorems and definitions. This is not the type of module that you can master by “cramming” just before the exam. You will need to work consistently throughout the semester, since you need to thoroughly understand each unit before studying the next one. This module is primarily geared towards problem solving, but in order to be able to solve the problems you need a thorough understanding of the theory. Please study carefully all the concepts that are dealt with in the study guide. You must know and understand the wording of theorems and definitions given in the guide and be able to apply these theorems and use the definitions. You should also know the conditions under which the theorems may be applied. The proofs of theorems are not required for exam purposes but we recommend that you do read the proofs and try to understand them, since this will improve your insight into the subject. DOING THE ASSIGNMENTS IS THE MOST IMPORTANT PART OF YOUR STUDY PROGRAMME. Preparing for the exam without having done the assignments would be like training for the Comrades Marathon without ever actually jogging. Just as one cannot get fit by watching other people exercise, one cannot master mathematics by only studying worked examples. The assignments are designed to lead you through the study material. Students who work hard at all their assignments and submit them in time and then study their lecturer’s comments on their marked assignments to sort out their mistakes usually find that, by the time they have submitted their last assignment, they are just about ready to write the exam. All they need to do then is to go through the work once more to study definitions of concepts and the wording of theorems and then, for further practice, to do the exercises in the study guides that they had not yet done. Answers and hints to the exercises in each of the study guides appear at the back of the guide. Complete solutions to the exercises will not be supplied, but your lecturer will gladly assist you if you have difficulties with specific exercises or if you have doubts about the correctness of your solutions. Consult the brochure my Studies @ Unisa for general exam guidelines as well as advice on exam preparation. 11 FREQUENTLY ASKED QUESTIONS The my Studies @ Unisa brochure contains an A-Z guide of the most relevant study information. 10 Downloaded by Thomas Mboya () lOMoARcPSD| MAT2615/101 12 SOURCES CONSULTED No books were consulted in preparing this tutorial letter. 13 CONCLUSION Remember that there are no "short cuts" to studying and understanding mathematics. You need to be dedicated, work consistently and practise, practise and practise some more! We hope that you will enjoy studying this module and we wish you success in your studies. Your MAT2615 lecturer 11 Downloaded by Thomas Mboya () lOMoARcPSD| ADDENDUM A: SELF ASSESSMENT TASK FOR UNITS 1–3 In each of the following 20 questions, there is only one correct option. Work through each of the questions and write down the option that you think is correct. Then score your answers out of 20 by comparing them with the table of correct options given in the solution section. You will need to enter your score in Assignment 01. Also work through the solutions that are provided. If you score below 16, we suggest that you work through units 1–3 once again before proceeding to study guide 2. A.1 QUESTIONS Question 1 (Section 2.5) Which of the following is/are true? A. k(x; y) + (a; b)k k(x; y)k + k(a; b)k for all x; y; a; b 2 R. B. k(x; y) (a; b)k k(x; y)k k(a; b)kfor all x; y; a; b 2 R. 1. Only A 2. Only B 3. Both A and B 4. Neither A nor B. Question 2 (Section 2.5) Which of the following is/are true? A. kx ak kx + ak for all x; a 2 Rn . B. kx ak kxk + kak for all x; a 2 Rn . 1. Only A 2. Only B 3. Both A and B 4. Neither A nor B. Question 3 (Sections 2.11 and 2.10) Which of the following is/are true? A. A line in R2 is uniquely determined by a point on the line and a vector parallel to the line. B. A line in R2 is uniquely determined by a point on the line and a vector perpendicular to the line. 1. Only A 2. Only B 3. Both A and B 4. Neither A nor B. Question 4 (Sections 2.11 and 2.10) Which of the following is/are true? A. A line in R3 is uniquely determined by a point on the line and a vector parallel to the line. B. A line in R3 is uniquely determined by a point on the line and a vector perpendicular to the line. 1. Only A 2. Only B 3. Both A and B 4. Neither A nor B. 12 Downloaded by Thomas Mboya () lOMoARcPSD| MAT2615/101 Question 5 (Section 2.12) Which of the following is/are true? A. A plane in R3 is uniquely determined by a point on the plane and a vector parallel to the plane. B. A plane in R3 is uniquely determined by a point on the plane and a vector perpendicular to the plane. 1. Only A 2. Only B 3. Both A and B 4. Neither A nor B. Question 6 (Section 2.12) Let each of V1 and V2 be a plane in R3: Which one of the following is FALSE? 1. It is possible for V1 and V2 not to intersect. 2. It is possible for V1 and V2 to intersect in a plane. 3. It is possible for V1 and V2 to intersect in a line. 4. It is possible for V1 and V2 to intersect in only one point. Question 7 (Section 2.12) Which of the following is/are true? A. A plane in R3 is uniquely determined by two non–parallel lines that lie on the plane. B. A plane in R3 is uniquely determined by two different parallel lines that lie on the plane. 1. Only A 2. Only B 3. Both A and B 4. Neither A nor B. Question 8 (Sections 2.11 and 2.12) Which of the following is/are true? A. If two lines, L1 and L2, in R3 are perpendicular to the same vector, thenL1 and L2 are parallel to one another. B. If two planes, V1 and V2, in R3 are both parallel to the same vector, thenV1 and V2 are parallel to one another. 1. Only A 2. Only B 3. Both A and B 4. Neither A nor B. Question 9 (Sections 2.11 and 2.10) Let L be the line in R3 that passes through the point (0; 1; 2) and is perpendicular to the vectors ( 1; 1; 0)and (1; 0; 2). An equation for L is 1. (x; y; z) = (0; 1; 2) + t ( 1; 1; 0) ;t 2 R: 2. (x; y; z) = (0; 1; 2) + t (2; 1; 2) ;t 2 R: 3. (x; y; z) = (0; 1; 2) + t (2; 2; 1) ; t 2 R: 4. (2; 2; 1) (x; y; z) = (0; 1; 2) (2; 2; 1) : 13 Downloaded by Thomas Mboya () lOMoARcPSD| Question 10 (Sections 2.11 and 2.12) Suppose the line L in R3 is defined by (x; y; z) = (1; 2; 3) + t (1; 1; 2) ;t 2 R and is perpendicular to the plane V which contains the point (3; 1; 4). The vector u = ( 2; 0; 1)is perpendicular to the line L. An equation for V is 1. (x; y; z) = (3; 1; 4) + t (1; 1; 2) ;t 2 R: 2. (x; y; z) = (3; 1; 4) + t ( 2; 0; 1) ;t 2 R: 3. 2x + 0y + z = 2: 4. x y + 2z = 12: Question 11 (Sections 2.11 and 2.12) Let L be the line in R3 that passes through the points ( 1; 1; 2)and (2; 3; 6) : Let V be the plane that contains the line L and is perpendicular to the plane 2x + y 3z + 4 = 0: An equation for V is 1. 10x 17y + z + 25 = 0: 2. 0x + 10y 5z = 0: 3. 2x + y 3z + 7 = 0: 4. None of the above. 14 Downloaded by Thomas Mboya () lOMoARcPSD| MAT2615/101 Options for Question 12–15 (Section 3.2) Consider the sketches in the four options below: Option 1 2 _ 2 2 Z Y X Option 2 _ 2 Z 2 Y X 2 Option 3 2 2 2 Z Y _ X Option 4 Y 2 Z X For each of questions 12-15 choose the option that best represents the given surfaceS in R3: Question 12 S = n (x; y; z) 2 R3 j z = p 2 x2 y2 o Question 13 S = n (x; y; z) 2 R3 j z = 2 + p x2 + y 2 o Question 14 S = f(x; y; z) j z = 2 x2 y2g Question 15 S = f(x; y; z) j x 2 + y 2 = 4 and jzj 2g 15 Downloaded by Thomas Mboya () lOMoARcPSD| Question 16 (Section 3.2) Suppose f is the R2 R function defined by f (x; y) = 9 x2 y2 and that the contour curve C of f has equation x 2 + y 2 = 4: The level of C is 1. 0 2. 2 3. 4 4. 5 Question 17 (Section 3.2) Suppose C is the semi–circle in R2 with initial point (0; 2) and endpoint (0; 2) with the origin as centre. A parametric equation for C is 1. (x; y) = (cos 2t; sin 2t) ;t 2 [0; ] : 2. (x; y) = (cos 2t; sin 2t) ;t 2 2 ; 3 2 3. (x; y) = (2 cos t; 2 sin t) ;t 2 [0; ] 4. (x; y) = (2 cos t; 2 sin t) ;t 2 2 ; 3 2 Question 18 (Section 3.2) Let C be the curve from ( 2; 2) to (3; 3)along the parabola y = x 2 6: A parametric equation for C is 1. (x; y) = t (1; t) + (0; 6) ; t 2 [0; 1] : 2. (x; y) = t (1; t) + (0; 6) ; t 2 [ 2; 3] 3. (x; y) = t (5; 5) + ( 2; 2) ; t 2 [0; 1] 4. (x; y) = t (5; 5) + ( 2; 2) ; t 2 [ 2; 3] Question 19 (Sections 3.3 and 3.2) Suppose f : R n ! R m and g : Rm ! R p are functions. The image of g f lies in 1. R p 2. R m 3. R n 4. Rn+p 16 Downloaded by Thomas Mboya () lOMoARcPSD| MAT2615/101 Question 20 (Section 3.3) Assume n; p; qand m are all different positive integers. Consider the following functions: f maps R p into R q ; g maps R q into R m ; and h maps R n into R p : Which one of the following is defined? 1. f g h 2. f h g 3. h f g 4. g f h A.2 ANSWERS AND SOLUTIONS The correct answers are: 1. 1 6. 4 11. 1 16. 4 2. 2 7. 3 12. 3 17. 4 3. 3 8. 4 13. 4 18. 2 4. 1 9. 3 14. 1 19. 1 5. 2 10. 4 15. 2 20. 4 Reasons Question 1 A. According to the Triangle Inequality N.4 (on page 27) this statement is true. B. This statement is false. Consider the following counterexample. Let (x; y) = (1; 0) and (a; b) = (0; 1) : Then k(x; y) (a; b)k = k(1 ; 0) (0; 1)k = k(1; 1)k = p 2 and k(x; y)k k(a; b)k = k(1 ; 0)k k(0; 1)k = 1 1 = 0: Hence, in this case, k(x; y) (a; b)k k(x; y)k k(a; b)k : Thus 1 is the correct answer. 17 Downloaded by Thomas Mboya () lOMoARcPSD| Question 2 A. This statement is false. Consider the following counterexample. Let x = (1; 1) and a = (0; 1). Then kx ak = k(1; 1) (0; 1)k = k(1; 2)k = p 5 and kx + ak = k(1; 1) + (0; 1)k = k(1; 0)k = 1: Thus, in this case, kx ak kx + ak : B. This statement is true. If x and a are arbitrary elements of R n , then kx ak = kx + ( a)k kxk + kak by N.4 = kxk + j1j kak by N.3 = kxk + kak : Hence kx ak kxk + kak for all x; a 2 Rn : Thus 2 is the correct answer. Question 3 A. The statement is true. See Section 2.11 and in particular Definition 2.11.1. B. The statement is true. Suppose L is a line inR2 that contains the pointp and is perpendicular to the vector v. Then all lines that are perpendicular to v are parallel to L and only one of these lines goes through the point p: v p L Y X Thus option 3 is correct. 18 Downloaded by Thomas Mboya () lOMoARcPSD| MAT2615/101 Question 4 A. The statement is true. See Section 2.11 and in particular Definition 2.11.1. B. The statement is false. For example, the X –axis and Y–axis are both lines that contain the point (0; 0; 0)and are perpendicular to the vector (0; 0; 1)(see Fig. 2.13). In general, if a is any non–zero vector in R3 and p is any point inR3 , there are infinitely many lines that containp and are perpendicular to a. See Fig. 2.15(b) and the sketch below. a p L2 L1 Z Y X Hence 1 is the correct option. Question 5 A. The statement is false. A plane is said to be parallel to a given vector if some line in the plane is parallel to that vector. (Read the first paragraph of Section 2.12.) Given a nonzero vector v and a point p in R3 , there is only one line that contains the point p and is parallel to v, but there are infinitely many planes containing that line; hence there are infinitely many planes that contain the point p and are parallel to v. The sketch below shows two such planes, V1 19 Downloaded by Thomas Mboya () lOMoARcPSD| and V2. v p V2 V1 X Y Z B. The statement is true. See Section 2.12 and in particular Definition 2.12.1. Thus the correct answer is 2. Question 6 If V1 and V2 are the same plane, then they intersect in the planeV1 (or V2). If V1 and V2 are not the same plane, then either they are parallel, in which case they do not intersect, or they are not parallel, in which case they intersect in a line (see the sketch in Question 5). Two planes cannot intersect in only one point. (Three or more planes, or a line and a plane may intersect in only one point.) Thus the correct FALSE option is4: Question 7 Both A and B are true. Suppose L1 and L2 are two different lines in the plane V. Choose two points a and b on L1 and a point c on L2 (where c is not the point of intersection of L1 and L2 in the case where L1and L2 are not parallel.) Then the three points are not collinear (i.e. do not lie on the same line) and hence determine a unique plane. We note that b a and b c are both parallel to the plane V, but not parallel to one another. Thus the cross product(b a) (b c) is perpendicular toV, and we know that a plane is uniquely determined by a point on the plane and a vector perpendicular to the plane (see Question 5B). Thus 3 is the correct answer. Question 8 A. The statement is false. See the sketch in Question 4B. The lines L1 and L2 are both perpendicular to the vector a, but they are not parallel to one another. 20 Downloaded by Thomas Mboya () lOMoARcPSD| MAT2615/101 B. The statement is false. See the sketch in Question 5A . The planes V1 and V2 are both parallel to the vector v, but they are not parallel to one another. Hence the correct answer is 4. Question 9 A vector perpendicular to the vectors ( 1; 1; 0)and (1; 0; 2)is u = ( 1; 1; 0) (1; 0; 2) = i j k 1 1 0 1 0 2 = (2; 2; 1) : Thus (2; 2; 1) is parallel to L. Thus the equation for L is (x; y; z) = (0; 1; 2) + t (2; 2; 1) ; t 2 R: Hence 3 is the correct answer. Question 10 A vector parallel to L, and hence perpendicular to V is (1; 1; 2). Since V contains the point (3; 1; 4)and is perpendicular to (1; 1; 2) ;an equation forV is (x; y; z) (1; 1; 2) = (3; 1; 4) (1; 1; 2) i.e. x y + 2z = 3 + 1 + 8 i.e. x y + 2z = 12: Hence the correct answer is 4. Question 11 A vector parallel to L and hence to V is u = (2; 3; 6) ( 1; 1; 2) = (3; 2; 4) : A vector perpendicular to the plane 2x + y 3z + 4 = 0 and hence parallel to V is (2; 1; 3) : Thus a vector perpendicular to V is (3; 2; 4) (2; 1; 3) = i j k 3 2 4 2 1 3 = ( 10; 17; 1) : Hence (10; 17; 1)is a vector perpendicular to V and thus an equation for V is (10; 17; 1) ( x; y; z) = (10; 17; 1) ( 1; 1; 2) i.e. 10x 17y + z = 10 17 + 2 i.e. 10x 17y + z + 25 = 0: Hence 1 is the correct answer. 21 lOMoARcPSD| Question 12 The surface S is the upper half of the sphere centred at the origin with radius p 2. See Remark 3.2.8. Thus the correct answer is 3. Question 13 The surface S is a cone, opening upwards with apex (vertex) at(0; 0; 2) :Study Example 3.2.7(2). Thus the correct answer is 4. Question 14 The surface S is a paraboloid, opening downwards, with turning point (0; 0; 2) : Study Example 3.2.7(1). Thus the correct answer is 1: Question 15 The surfaceS is obtained by sketching the circlex2+y 2 = 4 in the planez = 2 and then translating it vertically up to the plane z = 2 (because if (x; y; z) 2 S; then x2 + y 2 = 4 for every value of z between 2 and 2, including 2 and 2). Thus S is the cylinder in option 2. Thus the correct answer is 2. Question 16 If x2 + y 2 = 4; then f (x; y) = 9 x2 y2 = 9 4 = 5: Thus according to Definition 3.2.9, the level ofC is 5: Hence, option 4 is correct. Question 17 The semi–circle C is sketched below. (0, 2) (0, 2) _ C 2 t ( , ) x y 22 Downloaded by Thomas Mboya () lOMoARcPSD| MAT2615/101 Any point (x; y) on C can be described by x = 2 cos t; y = 2 sin t where t lies between 2 and 3 2 : Hence a parametric equation for C is (x; y) = (2 cos t; 2 sin t) ;t 2 2 ; 3 2 and hence option 4 is correct. Question 18 Consider the parabola y = x 2 6: Let x = t: Then the parabola can be described by (x; y) = (t; t2 6) : Now when (x; y) = ( 2; 2) ; t = 2 and when (x; y) = (3; 3) ; t = 3: Thus a parametric representation of C is (x; y) = (t; t 2 6) ; t 2 [ 2; 3] i.e. (x; y) = (t; t 2 ) + (0; 6) ; t 2 [ 2; 3] i.e. (x; y) = t (1; t) + (0; 6) ; t 2 [ 2; 3] : Thus 2 is the correct option. Question 19 Note that (g f ) (x) = g (f (x)) , and that f maps R n into R m and g maps R m into R p : R n f R g m R p gof x f( )x g( ) f()x = ( ) g of ( )x Thus g f maps R n into R p : Hence, according to Definition 3.2.1(a), the image ofg f lies in R p ; and so option 1 is correct. Question 20 We consider each of the options: 1. (f g h) (x) = f (g h (x)) is undefined since g h is undefined because h (x) 2 R p and Dg = R q ; but p 6= q(see the first sentence of Section 3.3 of Study Guide 1). 2. (f h g) (x) = f (h g (x)) is undefined since h g is undefined because g (x) 2 R m and Dh = R n ; but m 6= n: 3. (h f g) (x) = h (f g (x)) is undefined since f g is undefined because g (x) 2 R m and Df = R p ; but m 6= p: 4. (g f h) (x) = g (f h (x)) is defined. Firstly, f h is defined because h (x) 2 R p and Df = R p : Then since (f h) (x) 2 R q and Dg = R q ; it follows that g (f h) is defined. Thus 4 is the correct option. 23 Downloaded by Thomas Mboya () lOMoARcPSD| ADDENDUM B: FIRST SEMESTER ASSIGNMENTS B.1 ASSIGNMENT 01 YOU MUST DO THE SELF ASSESSMENT TASK IN ADDENDUM A BEFORE YOU ATTEMPT THIS ASSIGNMENT ONLY FOR SEMESTER 1 STUDENTS ASSIGNMENT 01 Units 4-10: Limits, Continuity, Differentiation, Applications of Derivatives FIXED CLOSING DATE: 7 March 2014 UNIQUE ASSIGNMENT NUMBER: Section A of this assignment will be marked but the answers to only some of the questions in Section B will be marked. Section A (This section will be marked.) 1. What score, out of 20, did you obtain for the Self assessment task on Units 1–3 contained in Addendum A? (1) 2. Did you find the self assessment task useful? (1) 3. Explain why (how) you found the self assessment task useful; or why you did not find it useful. (Maximum of 150 words.) (3) [5] Section B When you feel confident that you have a sound knowledge and understanding of Units 1–3, you can study Units 4–10 and then attempt this section of the assignment. When you answer a question you should once again study all the sections mentioned at the beginning of the question. Before attempting to answer a question, note whether it involves a function of one variable or a function of more than one variable. Make sure that you understand the essential difference between dealing with a limit of a function of one variable and a function of two variables. (Study Figures 4.3 and 4.4 in this regard.) Note that Units 5 and 6 deal with functions of one variable and that Units7 and 8 deal with functions of more than one variable. Before attempting a problem, it is important to note what type of function(s) the problem deals with. Take note of the notations used for the various types of derivatives and make sure that you use these correctly. Note, for instance, that the notation f 0 is not used when f is a function of more than one variable. (The reason for this will become clear when you study Sections 7.1 and 7.2.) The distance concept, which was introduced in Unit 2, plays a vital role in limits of functions. Please revise Sections 2.5 and 2.6 before you attempt this assignment. 24 Downloaded by Thomas Mboya () lOMoARcPSD| MAT2615/101 1. (Section 4.3) Consider the R2 R function f defined by f (x; y) = x2 p x2 + y 2 : Prove from first principles that lim (x;y)!(0;0) f (x; y) = 0: Hint: Note that x2 x2 + y 2 . [5] 2. (Sections 4.4 and 4.6) Consider the R2 R2 function F given by F (x; y) = y sin x x ; xy sin y : Use limit rules to determine lim (x;y)!( ; 2 ) F (x; y) : [4] 3. (Section 4.5) Consider the R2 R function f defined by f (x; y) = 2xy2 x2 + y 4 . (a) Write down the largest possible domain Df of f: (2) (b) Show that lim (x;y)!(0;0) f (x; y) does not exist, by considering limits along relevant curves.(5) Hints: Study Theorem 4.5.2, Remarks 4.5.3 and Remarks 4.5.5 carefully. Take care to use appropriate notation and to present your proof correctly. (Study Examples 4.5.4 in this regard.) [7] 4. (Sections 4.4 and 4.7) Consider the R2 R function f given by f (x; y) = 8 < : (x 2 y2 ) x + y if y 6= x 0 if (x; y) = (0; 0) or (x; y) = (1; 1) : (a) Write down the largest possible domain Df of f . (1) (b) Determine lim (x;y)!(0;0) f (x; y) and lim (x;y)!(1; 1) f (x; y): (3) 25 Downloaded by Thomas Mboya () lOMoARcPSD| (c) Write down the values of f (0; 0)and f (1; 1). (1) (d) Is f continuous at (x; y) = (0; 0)? (1) (e) Is f continuous at (x; y) = (1; 1)? (1) (f) Is f a continuous function? (1) Give reasons for your answers to (d), (e) and (f). Hint: Study Remark 4.3.2(3), Definition 4.7.1 and Remarks 4.7.2. [8] 5. (Sections 7.2, 7.4 and 7.7.) Let f be the R2 R function defined by f (x; y) = 2x 2 + 2xy 3: (a) Determine grad f(3; 2) by using the Rules of Differentiation forR R functions. (Read Remark 7.2.4.) (2) (b) Determine the rate of increase inf at the point(3; 2) in the direction of the vector(3; 1): (Study Definition 7.7.1 and Remark 7.7.2(1) carefully. Then use Theorem 7.7.3.) (4) (c) What is the rate of increase in f at (3; 2) in the direction of the positive Y-axis? (2) (d) Determine the maximum rate of increase in f at (3; 2): (Read Corollaries 7.7.4 and 7.7.5.) (2) [10] 6. (Sections 1.3 and 7.6) Let f be the R3 R function defined by f (x; y; z) = p (x 2 + y 2 ) z and let r be the R R3 function defined by r(t) = (t; t; ln t) : (a) Determine the composite function f r. (1) (b) Write down the domains of f , r and f r. (Use set-builder notation.) (3) (c) Determine the derivative function (f r) 0 by (i) differentiating the expression obtained in (a). (2) (ii) using the General Chain Rule (Theorem 7.6.1). (4) [10] 7. (Sections 3.2, 7.5 and 7.8) Consider the R2 R function f defined by f (x; y) = 1 + x 2 + y 2: Let Cbe the contour curve of f through the point (1; 1), let L be the tangent to Cat (1; 1)and let V be the tangent plane to the graph of f at (1; 1; 3): 26 Downloaded by Thomas Mboya () lOMoARcPSD| MAT2615/101 (a) Find the equation of the curve C: (2) (b) Find a vector in R2 that is perpendicular to Cat (1; 1): (2) (c) Find the Cartesian equation of the line L: (3) (d) Find a vector in R3 that is perpendicular to the graph of f at the point (1; 1; 3): (3) (e) Find the Cartesian equation of the plane V: (3) (f) Draw a sketch to visualize the graph of f , together with appropriate sections of the line L and the plane V. Also show the vectors that you obtained in (b) and (d) on your sketch. (5) Hints: Study Definitions 3.2.5 and 3.2.9. Note that the level of Cis given by f (1; 1): By a vector perpendicular to a curve at a given point, we mean a vector perpendicular to the tangent to the curve at that point. Use Theorem 7.8.1 to find a vector perpendicular to C at the point (1; 1). Study Remark 2.12.2(1) and use Definition 2.12.1 to find the Cartesian equation of L. Or, equivalently, use Definition 7.8.6. (Note that, in the case n = 2 , the formula in Definition 7.8.6 gives a Cartesian equation for a tangent to a contour curve.) By a vector perpendicular to a surface at a given point, we mean a vector perpendicular to the tangent plane to the surface at that point. Define anR3 R function g such that the graph of f is a contour surface of g; and then use Theorem 7.8.3 to find a vector perpendicular to V at the point (1; 1; 3): Use Definition 2.12.1 or Definition 7.8.6 (with g in the place of f ) to find the equation of V; or use Definition 7.5.4 (Read Remark 7.5.5(2).) [18] 8. (Sections 7.10, 8.2, 8.3 and 8.4) Consider the 3-dimensional vector fieldF defined by F (x; y; z) = 18x 2y 3 + 2x; 18x 3y 2 + 4yz3 + 2y; 6y 2z 2 4z : (a) Write down the Jacobian matrix JF (x; y; z) : (2) (b) Determine the divergence div F (x; y; z): (2) (c) Determine curl F (x; y; z): (2) (d) Give reasons why F has a potential function. (Refer to the relevant definition and theorems in the study guide.) (2) (e) Find a potential function for F , using the method of Example 7.9.1 Note, however, that the example concerns a 2-dimensional vector field, so you will have to adapt the method to be suitable for a 3-dimensional vector field. Pay special attention to the notation that you use for derivatives of functions of more than one variable. (6) [14] 27 Downloaded by Thomas Mboya () lOMoARcPSD| 9. (Unit 9) Consider the R2 R function f defined by f (x; y) = sin x cos y: Find the second order Taylor Polynomial off about the point 4 ; 4 : Leave your answer in the form of a polynomial in x 4 and y 4 : (This form is convenient for evaluating function values at points near 4 ; 4 ): [6] 10. (Section 10.2) Consider the R2 R function f defined by f (x; y) = cos x sin y; 2 < x < 2 ; 0 < y < 2 Find the critical points of f and determine their nature. Hint: There are two critical points. Determine whether they are saddle points or local minima or maxima. Also, remember to check for global extrema. [10] 11. (Section 10.3) You have to design a cylindricalcan that is open at the top (but closed at the bottom). Its volume has to be 1m3: Determine the height of the cylinder and the radius of its base circle to make the total surface area as small as possible. Hints: Determine the surface area of a cylinder with radius rm and height hm, which has only one base circle. Read the question carefully. Denote the quantity that you wish to minimize by f (r; h): Note that there is a constraint on the volume. Formulate the problem as a constrained optimization problem and then apply the Method of Lagrange to solve it. [10] 28 Downloaded by Thomas Mboya () lOMoARcPSD| MAT2615/101 B.2 ASSIGNMENT 02 ONLY FOR SEMESTER 1 STUDENTS ASSIGNMENT 02 Units 12-19: Integration FIXED CLOSING DATE: 4 April 2014 UNIQUE ASSIGNMENT NUMBER: The answers to only some questions will be marked. 1. (Sections 13.1 – 13.5) Let R be the region in R2 below the line y = x + 2 and above the parabola y = x 2: (a) Sketch the region R: Find the points of intersection of the line and the parabola and indicate them on your sketch. (3) (b) Is R a Type 1 region? If yes, describe it as a Type 1 region. If not, describe it as a union of Type 1 regions. Use set builder notation. Hints: Read the description of a Type 1 Region on p. 18 of Guide 3 and study Fig. 13.9 carefully. Shade the region R by means of vertical lines and highlight the curves which form lower and upper boundaries of R: Write the equations of these curves in the form y = g (x) : (3) (c) Is R a Type 2 region? If yes, describe it as a Type 2 region. If not, describe it as a union of Type 2 regions. Use set builder notation. Hints: Read the description of a Type 2 Region on p. 20 and study Fig 13.11 carefully. Shade the region R by means of horizontal lines and highlight the curves which form left and right boundaries of R: These curves are not smooth, since there is a sudden change in each of them where the parabola meets the line. Write the equation of each section of these curves in the form x= h (y) : (4) (d) Calculate the integral ZZ R xy dA; (i) by using the order of integration dy dx: (ii) by using the order of integration dx dy: If your answers to (i) and (ii) do not agree, then please try to find your mistake and correct it. If you find it difficult to evaluate these integrals, you should revise the techniques of integration taught in First Year Calculus, especially the techniques of integration by parts, and integration by substitution. (8) [18] 29 Downloaded by Thomas Mboya () lOMoARcPSD| 2. (Sections 3.2, 13.3, 13.7, 14.3, 14.6, 14.7) Consider the integral Z 1 1 Z p 1 x2 p 1 x 2 4 x 2 y 2 dy dx: (a) Sketch the region of integration. (3) (b) Give a geometric interpretation of the above integral by using a3-dimensional sketch. Hint: Study the geometric interpretation of a double integral in Section 13.3. (4) (c) Transform the above integral to a triple integral with cylindrical coordinates. (Do not evaluate the integral.) Hint: Study the geometric interpretation of a triple integral in Section 14.3. Also study cylindrical coordinates in Section 14.6 and examples on calculating volumes by using triple integrals in Section 14.7. (3) [10] 3. (Sections 3.2, 14.6) Let D be the 3-dimensional region inside the sphere x2 + y 2 + z 2 = 4 above the cone z = p 4x 2 + 4y 2: Evaluate the integral ZZZ D z dV by using spherical coordinates. Hints: Sketch the region D: Study the steps described in Example 14.6.7. Remember that represents the distance from the origin. All points on the given sphere lie at the same distance from the origin. Remember that is measured downward from the positiveZ–axis. [8] 4. (Units 15, 16; Section 7.9) Leta = ( 1; 0)and b = (2; 3) : Let L be the directed line segment in R2 with initial point a and end point b; and let C be the path from a to b along the curve y = x 2 1: (a) Sketch the directed curves L and C: (3) (b) Find a parametric equation for the line segment L (See pp. 68-70 of Guide 1.) (2) (c) Find a parametric equation for the line segment L: (2) 30 Downloaded by Thomas Mboya () lOMoARcPSD| MAT2615/101 (d) (Sections 15.1 and 15.2) Evaluate the following line integrals with respect to arc length Z L x ds; Z L x ds; Z C x ds: Comment on your answers. (7) (e) Evaluate the following line integrals: Z L 2xy dx + x 2dy; Z L 2xy dx + x 2dy; Z C 2xy dx + x 2dy; by (i) transforming each integral to an ordinary single integral (see Section 15.3). (ii) using the Fundamental Theorem of Line Integrals (Theorem 16.5.2) (12) Hint for (ii) Let F (x; y) = (2xy; x 2 ) : First show thatF has a potential functionf (see Remark 8.5.3 and Theorem 8.5.6). Then apply Theorem 16.5.2. [26] 5. (Sections 13.5, 16.3, 16.4, 19.1) Let D = C [ ( L) ; where C and L are the curves from Question 4. Use Green’s Theorem (Theorem 19.1.1) to calculate the work done by the force field F (x; y) = y 2 2xy; y 2 + 2xy on an object that moves once round the closed curveD (in the anticlockwise direction). Hints: Set up a line integral to represent the work done by applying equations (16.2) and (16.3). Check whether all the conditions of Green’s Theorem are satisfied before applying it. When you have applied Green’s Theorem, you have an ordinary double integral over R; which is a Type 1 region (see Section 13.5). [10] 6. (Unit 17) Consider the surface S in R3 defined by S = n (x; y; z) j z = p 9 x2 y2; 1 z 2 o : (a) Sketch the surface S, together with its projection onto the XY -plane. (4) (b) Calculate the area of S by using a surface integral. (7) [11] 31 Downloaded by Thomas Mboya () lOMoARcPSD| 7. (Sections 16.4, 19.2) Consider the surface S = (x; y; z) j z = 3 x 2 y 2 ; z 2 oriented upward. Evaluate the flux integral ZZ S (curl F ) ndS where F (x; y; z) = ( y; x; 1) ;by using Stokes’ Theorem. Hints: First sketch S and indicate the oriented boundary C of S: (Use the Right Hand Rule to obtain the orientation of C:) Make sure that the conditions of Stokes’ Theorem are satisfied before you apply it. Evaluate the line integral that comes from Stokes’ Theorem by first parametrizingC and then using the method of Section 16.4. [10] 8. (Sections 18.3, 19.3) Consider the surfaceS = S1 [ S 2 ; where S1 = n (x; y; z) j z = p 1 x2 y2 o and S2 = (x; y; z) j z = 1 x 2 y 2; z 0 : Calculate the outward flux of a fluid with velocity field F (x; y; z) = (xz; yz; z) by using formulas (18.2) and (18.3) and Gauss’ Theorem