Unit rationale, description and aim
Along with Geometry and Algebra, Analysis is one of the truly fundamental and essential branches of mathematics. Real analysis is a sub-branch of Analysis. It has its origins in Greek mathematics but it really began to take shape (in a rather geometric way) around 1700 with the development of the differential and integral calculus. In the 19th century, mathematicians sought to disengage analysis from its geometric ties and to formulate its basic concepts and techniques in a rigorous and logical manner. This program continued will into the early decades of the 20th century when, in line with the developments in set theory, real analysis began to connect itself with what has since come to be known as point-set topology. The outcome of this long but fruitful intellectual struggle was finally to have placed the differential and integral calculus of 1700 on a secure logical and rigorous foundation. This unit offers an introduction to this logical and rigorous approach to functions of a real variable.
Learning outcomes
To successfully complete this unit you will be able to demonstrate you have achieved the learning outcomes (LO) detailed in the below table.
Each outcome is informed by a number of graduate capabilities (GC) to ensure your work in this, and every unit, is part of a larger goal of graduating from ACU with the attributes of insight, empathy, imagination and impact.
Explore the graduate capabilities.
Prove simple results involving the limits of seque...
Learning Outcome 01
Solve simple theoretical problems using the more c...
Learning Outcome 02
Understand basic notions of point-set topology. Us...
Learning Outcome 03
Content
Topics will include:
- Sequences of real numbers, limits and convergence of sequences
- Series and tests for convergence of series
- Functions and limits of functions
- Continuous functions
- Properties of continuous functions including the intermediate value theorem
- Differentiation, differentiability, the mean value theorem
- Power series
- Open/closed (and other) sets in the real numbers
- Supremums, infimums, and the completeness of the real numbers
- Introduction to point set topology including properties of open sets, inverse images of sets, and continuity via open sets
Assessment strategy and rationale
To successfully complete an undergraduate Mathematics sequence, students need an understanding of a variety of basic Mathematical topics and an ability to apply that understanding to a variety of problems. To succeed at problem solving in Mathematics, students must have these skills at their fingertips and be able to recall them and choose an appropriate approach under some pressure. The assessment strategy chosen, while traditional, tests and supports student learning by providing opportunities to develop and test their problem-solving skills through the unit.
At specified times, through the semester, students will sit two supervised components to ensure that students have fully integrated the learning and can independently bring a variety of strategies to solve a variety of problems. The take home assignment will allow students to develop other skills including traditional and technological problem-solving methods.
Typing of Mathematical notation either requires a significant investment of time, or knowledge of advanced Mathematical typesetting software. Neither of those skills are suitable for an undergraduate course in Mathematics. Consequently, the lecturer may request that assignments and tests are handwritten.
Overview of assessments
Assessment Task 1: Take Home Assignment(s) Conti...
Assessment Task 1: Take Home Assignment(s)
Continuous assessment – A single task which is submitted in 1, 2 or 3 parts across the semester as stipulated by the lecturer.
30%
Assessment Task 2: Test Content and application ...
Assessment Task 2: Test
Content and application test 1. This test will be conducted under supervision and on campus.
35%
Assessment Task 3: Test Content and application ...
Assessment Task 3: Test
Content and application test 2. This test will be conducted under supervision and on campus.
35%
Learning and teaching strategy and rationale
As is common in Mathematics a variety of Active Learning approaches promote the best acquisition of skills and understanding. Lectures will typically be structured to provide explanations of the material to be covered, along with examples of the applications of that material, as well as formal and informal tasks for students that will reinforce that learning. Splitting the taught content of the lectures into short topics will ensure increased student engagement with the material, as well as providing opportunities for students to consolidate their learning before new material is covered.
This allows students to learn the skills and then build understanding, competence and confidence through tutorials that ideally occur face to face. Face-to-face learning supports immediate feedback, peer interaction, and collaborative problem-solving—essential elements in developing mathematical thinking—while also enabling tutors to identify and address misconceptions early.
Continuous formative learning opportunities allow for the early detection of problems a student might be having and ensures that appropriate guidance can be given early enough to support later learning in the unit. Students will be expected to submit written responses to questions involving core mathematical concepts, methods, and problem-solving strategies. In all cases this should be supported using available online technology.