Department of Mathematical Sciences



Further Calculus and Linear Algebra

The module follows on the module Mathematics 114 and covers further Calculus and Linear Algebra. In the Calculus section we begin with applications of integrals to the calculation of areas and volumes before using the definite integral to define the natural logarithm function. The exponential function is then defined as the inverse of the logarithmic function. Thereafter we learn more involved integration techniques and how to solve basic differential equations. The Calculus section of the course then finishes with a selection of topics: conic sections, polar coordinates, parametric equations and l’Hospital’s rule.

In Algebra we learn about systems of linear equations, matrices and determinants. This is followed by a study of vectors in 2 and 3 dimensions – the algebra of vectors and vector methods for working with lines and planes.

Module Information

  • 21539 144 (18) Mathematics 144
  • Academic year 1, Semester 2.
  • Lecture load: 5 lectures and 1 tutorial of 2 hours per week.
  • Module prerequisites: A Class Mark of at least 40% for Mathematics 114.
  • Language specification : A & E


Learning Material

There are two prescribed textbooks for the module. (The same textbooks are used in the prerequisite module Mathematics 114, and in second-year mathematics.)

  • J Stewart: CALCULUS  7th Edition, Thomson.
  • D Poole, LINEAR ALGEBRA  - A MODERN INTRODUCTION, 4th Edition, Cengage Learning.

Module Contents

Calculus (50% of the lectures)

  • Mean-Value Theorem for integrals, average values.
  • Areas and volumes by integration.
  • Inverse functions.
  • Natural logarithm and exponential function: definitions, arbitrary powers, derivatives, growth and decay.
  • Inverse trigonometric and hyperbolic functions
  • Integration techniques: Integration by parts, trigonometric integrals and substitutions, partial fractions and rationalising substitutions.
  • Differential equations: first order linear and separable equations.
  • Conic sections.
  • Polar coordinates: graphing, derivatives and integrals.
  • Parametric curves: derivatives, integrals and arc length.
  • L’Hospital’s rule.

Algebra (50% of the lectures)

  • Systems of linear equations.
  • Determinants: properties, evaluation by row reduction and co-factor expansion, inverse matrices and Cramer’s rule.
  • Vectors in 2 and 3 dimensions: Vector arithmetic, dot product, cross product, lines and planes.

Learning Opportunities

  • There is one double period 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 of the course.

Module Web-site

  • The module has a web site on WebCT,
  • Information about the module, tests and examinations is available on the site.
  • Solutions to tutorials and tests are placed on the site.


  • There is an Early Assessment Test and a Class Test for this course.
  • Your class mark (KP) is determined by your mark for the early assessment and midsemester test and by occasional tests written during tutorials.
  • If you are ill for the Early Assessment Test or Class Test and have a valid medical certificate, there is an oral test for this module.
  • You require a KP of at least 40% to be able to write the examination.
  • Two final exam papers will be written. The average mark obtained in these two exams determine you exam mark (EP).
  • To pass the module, you require a final mark (PP) of at least 50%, which is calculated by the following formula:  (0.4 x KP) + (0.6 x EP), where EP denotes your mark in the examination.

General Information

  • You will require a basic scientific calculator on occasions during the course.
  • Calculators are not allowed in tests.
  • If you miss a test or tutorial because of illness, you should provide Mrs Marais (room 1008B in the Mathematics Building) with a medical certificate when you return to classes.

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 doing many exercises.
  • Please consult your lecturer at any time if you are having any trouble with the course material.


This module together with Mathematics 114 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. Science students registered for the degree programmes in the following directions all take Mathematics 114: Actuarial Science, Physics/Chemistry, Mathematics, Geo-informatics, Applied Earth Science.


A student who has passed this module should:

  • Have an extended theoretical and practical grounding in integral calculus.
  • Know and understand the definitions, properties and essential applications of the natural logarithm and exponential functions.
  • Be able to use integration techniques to calculate areas, volumes and arc lengths and solve elementary differential equations.
  • Identify and sketch various conic sections and polar graphs.
  • Use techniques of calculus to study parametric and polar curves.
  • Be acquainted with l’Hospital’s rule.
  • Be able to solve systems of linear equations.
  • Be able to calculate matrix determinants, and use them in the solution of systems of linear equations.
  • Be able to manipulate vectors in 2- and 3-space, and use vector methods to describe lines and planes.
  • Be equipped to proceed with further mathematics studies in the second academic year.