## 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 endeavours at the University of Otago are built upon an essential commitment to academic integrity.**

The two most common forms of academic misconduct are *plagiarism* and *unauthorised collaboration*.

#### Academic misconduct: **Plagiarism**

Plagiarism is defined as:

- Copying or paraphrasing another person’s work and presenting it as your own.
- Being party to someone else’s plagiarism by letting them copy your work or helping them to copy the work of someone else without acknowledgement.
- Using your own work in another situation, such as for the assessment of a different paper or program, without indicating the source.
- Plagiarism can be unintentional or intentional. Even if it is unintentional, it is still considered to be plagiarism.

All students have a responsibility to be aware of acceptable academic practice in relation to the use of material prepared by others and are expected to take all steps reasonably necessary to ensure no breach of acceptable academic practice occurs. You should also be aware that plagiarism is easy to detect and the University has policies in place to deal with it.

#### Academic misconduct: **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 student’s answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer.

### 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?