Mathematics

MATHEMATICS 114

Calculus

The module covers introductory Calculus. A key concept in Calculus is that of a limit; thus we begin with a study of functions and limits, providing a foundation for the work that follows. With our knowledge of limits we define the derivative of a function and discover rules and techniques for differentiating. (Derivatives capture mathematically the idea of instantaneous change.) We then apply these skills to help us understand functions, solve optimisation problems and sketch curves. Towards the end of the semester we begin the study of integration, motivated by the problem of finding the area under a graph, finishing the module with basic integration techniques and the Fundamental Theorem of Calculus. (This theorem uncovers the relationship between derivatives and integrals.)

Module Information

• 21539 114 (16) Mathematics 114
• Academic year 1, Semester 1.
• Lecture load: 5 lectures and 1 tutorial of 2 hours per week.
• Module prerequisites:  6 for Grade 12 Mathematics
• Language specification : A & E

Lecturers

• Prof F Breuer:  Office 1008C, Industrial Psychology/Mathematics Building (e-mail: fbreuer@sun.ac.za)
• Prof J de Villers: Office 3007 , Industrial Psychology/Mathematics Building (email: jmdv@sun.ac.za)
• Dr G Boxall: Office 1009C, Industrial Psychology/Mathematics Building (e-mail: gboxall@sun.ac.za)
• Dr K-T Howell:  Office 1009D, Industrial Psychology/Mathematics Building (e-mail: kthowell@sun.ac.za)

Learning Material

There is one prescribed textbooks for the module. (The same textbook is used in the subsequent module Mathematics 144, and in second-year mathematics.)

• J Stewart: CALCULUS  7th Edition, Thomson.

Module Contents

• Introduction: Numbers, functions, trigonometric functions, mathematical proof, the binomial theorem.
• Limits and continuity: limit theorems, continuity.
• Differentiation: Definition of the derivative, differentiation rules, implicit differentiation, Newton-Raphson approximations.
• Applications of the derivative: The mean value theorem, extreme values, optimisation, curve sketching.
• Integration: Definite integrals, anti-derivatives and the fundamental theorem of calculus, u-substitutions.

Learning Opportunities

• All the module material is covered in lectures.
• The module makes use of a differentiated tutorial system with two groups.
• Students are divided according to their support needs.
• There is one tutorial each week. In tutorials students work on a set of problems under the supervision of lecturers and senior students.
• Tutorial attendance is compulsory, even for repeaters.
• Tutorial tests are written every second week during tutorials.

Module Web-site

• The module has a web site on SunLearn.
• Information about the module, tests and examinations is available on the site.
• Solutions to tutorials and tests are placed on the site.
• This is the official means of communication for the module.

Assessment

• There will be a number of tests written during the module:
• Early Assessment Test (AT) written on 5 March.
• Class Test (CT) written on 21 April.
• Tutorial tests (TT) written every second week during tutorials.
• The Class Mark (CM) is calculates as follows: CM =  (30 x TT  +  20 x AT + 50 x CT)/100.
• You require a Class Mark of at least 40% to be able to write the examination.
• There are two final examination papers written at the end of the module, the average of which determines your Exam mark (EP).
• Your Final Mark (PP) is calculated by the formula: PP = (40 x CM +  60 x EP)/100.
• You require a Final Mark of at least 50% to pass the module.
• If you miss a test due to illness you need to give Mrs Marais (room 3003 in the Mathematics Building) a valid medical certificate otherwise you will be given 0 for the test.
• Please note that lecturers may follow up medical certificates and verify that they are authentic.
• If a student misses the class test or too many tests due to illness the lecturers have the option of giving such a student an oral exam in order to calculate their class mark.
• Students who do not attend tutorials may be awarded a class mark of less than 40% for this module, regardless of their other marks.
• There is no sick test for this module. If you are ill and provide a valid medical certificate, there will be an oral test in the place of the Class Test.

General Information

• You will require a basic scientific calculator on occasions during the course.
• Calculators may not be used in tests.

Study Guidelines

• To succeed in this module it is essential that you work regularly on your own, practising the techniques and applying the results taught in lectures.
• Lectures follow the textbooks very closely. Your lecturer will explain the theory in the textbook and work through extra examples. If you read the textbook before you come to class, you will learn a lot more from the lectures.
• After class you should make time to read your notes and consolidate what was taught in the lecture by working on exercises in the textbook.
• Please consult your lecturer at any time if you are having any trouble with the course material.

Rationale

This module together with Mathematics 144 forms the cornerstone for further study in Mathematics. The module is required by science and commerce students who require a thorough mathematical grounding for further study in mathematics and for their other subjects.

Outcomes

A student who has passed this module should:

• Have a solid theoretical and practical grounding in differential and integral calculus.
• Understand the concepts of function, limit, derivative and definite and indefinite integral.
• Know rules of differentiation and be able to differentiate algebraic and trigonometric functions as well as perform implicit differentiation.
• Be able to use differentiation techniques to solve optimisation problems and sketch graphs of functions.
• Be able to integrate basic algebraic and trigonometric functions.
• Understand methods of proof and reasoning, including mathematical induction, that are used to establish key results in the development of calculus.