## MATH160 Mathematics 1

Summer School | Also available: First Semester Second Semester |

### Introduction

Algebra and Calculus form the basic tools used to produce most mathematical frameworks for modelling quantifiable phenomena. For example, to model the movement of an object through space we need first to create an algebraic structure in which to specify where our object is, and then we can study how that position changes with time (i.e. its movement) using calculus.

Many other problems arising in areas such as Economics or Chemistry, can be examined in a mathematical way using the same basic ideas. For example, we may need to minimize a manufacturing cost, or the time for a chemical reaction to take place, or the effects of river pollution; in each case the techniques used for the minimization are based on a mixture of algebra and calculus theories.

This course aims to develop skills with these tools both for use in other subjects and in preparation for further study of Mathematics.

### Paper details

This paper is the natural continuation of Year 13 Mathematics, and is divided between algebra and calculus.

After a review of basic trigonometry, the algebra half focusses on three-dimensional vectors and their many uses (such as in geometry, computer graphics, surveying and even calculus). The vector representation of lines, planes and projections leads naturally to the discussion of linear systems of equations. The basic properties of matrices are studied together with some applications. Complex numbers and polynomials complete this section of the course.

In the calculus half you will study the ideas and methods of differentiation and integration, using an approach that is intuitive and avoids excess formality. Applications will include optimization, related rates, the use of differentials, finding areas, the Taylor series, solving simple differential equations, and an introduction to partial derivatives.

### Potential students

Math 160 is intended both for those with a main interest in studying Mathematics and/or Statistics, and those whose interest in Mathematics is mainly to support other areas of study. These might include the physical, health and biological sciences, computer and information science, engineering, surveying, architecture, economics and finance, and philosophy of science. An understanding of basic algebraic and differential and integral techniques is of benefit to all students exposed to the analysis of processes, whether involving one or several variables.

The paper is suitable for students who have passed at least 12 credits in NCEA level 3 Calculus (or equivalent) or have passed at least 18 credits in NCEA level 3 Statistics. Weaker students should seek advice and might first consider taking Math 151.

### Prerequisites

None

### Main topics

Algebra:

- Vectors; linear and planar geometry and applications
- Solving linear systems
- Matrices and applications
- Complex numbers
- Polynomials and their roots.

Calculus:

- Functions
- Introduction to calculus
- Techniques of differentiation and integration

### Texts

**Algebra**: Course materials will be available on the resource page. The book *MATH 160 Algebra Outline Notes* is available for purchase from the Print Shop.

**Calculus**: Course materials will be available on the resource page. For further reading we recommend the text: *Calculus* by James Stewart (Truncated edition), available from the University Book Shop; if you are planning on taking MATH 170, you should consider getting the full *Calculus, metric edition 8*.

### Useful references

Several standard texts are suitable for reference. For example:

*Elementary Vector Algebra*by A.M. MacBeath*Algebra, Geometry and Trigonometry*by M.V. Sweet*Calculus with Analytic Geometry*by Howard Anton (Wiley)*Calculus*by James Stewart (Full edition.)

### Lecturers (Summer School)

- Algebra: Ilija Tolich, email: itolich@maths.otago.ac.nz
- Calculus: Johannes Mosig, email: jmosig@maths.otago.ac.nz

### Lectures (Summer School)

- Algebra: Mon, Tue, Wed, Thu 10-11am
- Calculus: Mon, Tue, Wed, Thu 2-3pm

### Tutorials

11–12 and 3–4 Monday to Thursday

Attendance at tutorials is voluntary. An open tutorial system operates and students may attend as many as they need to and are able to.

### Internal Assessment

There are ten marked assignments which make up your assignment mark (A).

Five computer Skills Tests in each of Algebra and Calculus together make up 20% (T) of your final mark.

You can check your marks by clicking on the Resources link at the top of this page.

### Terms Requirement

**You have to fulfil the terms requirement in order to be allowed to sit the final exam.**

In this paper, to pass “terms” you need to:

- gain at least 5/10 in each of the first four Skills Tests
- achieve an overall mark of 40% on the 10 assignments

### Exam format

The 3-hour final exam is answered in spaces provided on the question booklet. All questions should be attempted and the number of marks available for each question is indicated on the paper. There are usually from 15 to 20 questions for each of Algebra and Calculus. You may allocate your time between the two sections as you wish.

### Final mark

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

**F = 0.8max(E, 0.8E + 0.2A) + 0.2T**

where:

- E is the Exam mark
- A is the Assignments mark
- T is the Tests mark

and all quantities are expressed as percentages.

This means your internal assessment can boost your exam mark if that helps you. Notice how important the tests are — to gain terms and for their contribution to the final mark.

### 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.

### Sample problem

The aircraft’s flightpath goes through coordinates (1,2,0) and (23,-19,3). The top of the hill is at (18,-13,2).How close does the aircraft get to the top of the hill? Vectors make this an easy calculation.

### J Willard Gibbs...

..., 1839-1903, was a pioneer in vector analysis. His family lived in Connecticut and Gibbs became Professor of Mathematical Physics at Yale in 1871 — rather surprisingly before he had published any work! He made major contributions to thermodynamics, the electromagnetic theory of light and statistical mechanics.### Sample problem

In a certain city, commuters go to work by car or bus. A study shows that from each year to the next year 20% of car users change to travelling by bus, while 15% of bus users change to travelling by car. What percentage of commuters travel by car, once things have settled down?

### Suspension bridges

The main cable of a suspension bridge naturally forms a curve called a catenary. When it is loaded with the horizontal road structure it deforms into a parabola. Unless very carefully designed, suspension bridges are susceptible to collapse from high winds or earthquakes.### Sample problem

Let $N$ be the number of individuals in a population. One model for studying $N$ says that the rate of increase of $N$ depends on both $N$ itself (since the more individuals there are the more offspring will be produced) and on some residual amount $M-N$ (since there will be competition for resources like food); so we have $$\frac{dN}{dt}=N(M-N)$$ for some constant $M$.Will the population die out or reach some maximum value?