## MATH130 Fundamentals of Modern Mathematics 1

First Semester | Also available: Second Semester |

The aim of this paper is to develop confidence and capability in the concepts and techniques of modern mathematics and its applications.

On successful completion of the paper you will be able to:

- Realise how mathematical formalism is used for solving real-world problems.
- Use notation, algebra and the language of mathematics with increasing fluency and confidence.
- Apply mathematics within multiple disciplines.
- Work with mathematical expressions.

### Course Outline

#### Introducing the key concepts

We introduce and review the key ideas and problems which are central to the rest of the paper (and indeed to mathematics in general). Concepts discussed in this section include mathematical definitions of sets, functions, formulas.

We introduce vectors and Euclidean space, showing how different geometric objects (lines, planes, surfaces, subspaces, angles etc.) are depicted mathematically. Part of that will include a study of the properties of functions (mainly in one dimension, but also in general).

#### The mathematics of continuous change

We examine how to think mathematically about phenomena which change continuously in time, with examples drawn from mechanics, physiology, physics, mathematical biology, physical geography, surveying, and a wide range of other disciplines.

We introduce the idea of a limit, leading on to definitions for continuity and rates of change.

We develop the ideas of derivatives and integrals from scratch, exploring their applications (e.g. mechanics, statistics), geometry, and use in modelling.

We see how properties of a function can be expressed using derivatives. The connections between antiderivatives, area, and probability density are discussed, as are the different kinds of integral. We will discuss linear approximation of functions (based on Taylor's approximation formulae) and their applications, e.g. to simplify non-linear complex equations.

Throughout, we show how modern computing tools change the way we practice this kind of mathematics.

#### Matrices and geometry

We introduce matrices and matrix algebra, demonstrating links with geometry and data analysis.

We discuss fundamental matrix operations and algorithms and show how a single matrix equation can represent a whole system. The subspaces and geometry associated with matrices are explored, with applications in data analysis and statistical linear models.

#### Mathematics of higher dimensions

We look at how the concepts introduced extend to multiple dimensions, including high dimensional (partial) derivatives and integral and equations for least squares.

We show how the ideas in this paper link together and anticipate key results in MATH 140 and beyond.

### Assessment

Quizzes are 5-6 minute short tests which are conducted in the 24 hours following most lectures and review key points of the lecture that day. Only the top 80% of quiz marks will contribute to the student's grade. We have used quizzes for the last two years in MATH 160 and found them both popular and effective.

Assignments are handwritten answers to questions that explore in more detail the lecture material over the previous 2 weeks.

Tutorial problems examine the lecture material over the past 4 lectures and are completed in the tutorials. These are assessed on the basis of completion of the tutorial problems.

The midterm examination will test the material from the first half of the course and will allow students to familiarise themselves with the examination format used at the end of the paper.

There are no terms requirements for this paper.

### Final mark

Your final mark F in the paper will be calculated according to this formula:

**F = 0.45*E + 0.2*A + 0.1*T + 0.15*M + 0.1*Q**

where:

- E is the Exam mark
- A is the Assignments mark
- M is the Midterm mark
- T is the Tutorials mark
- Q is the Quizzes mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

**Academic integrity** means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.

**Academic misconduct** is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.

Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.

All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.

Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.

If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.

Types of academic misconduct are as follows:

**Plagiarism**

The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).

- Although not intended,
*unintentional plagiarism*is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided. *Intentional plagiarism*is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.

**Unauthorised Collaboration**

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..

**Impersonation**

Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.

**Falsiﬁcation**

Falsiﬁcation is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.

**Use of Unauthorised Materials**

Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.

**Assisting Others to Commit Academic Misconduct**

This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.