## MATH151 General Mathematics

Summer School | Also available: First Semester |

Mathematics occurs in almost every field of study and certainly in every quantitative discipline. Getting on top of even basic mathematical techniques is an important step to being able to understand the analytical processes in those fields — processes that deal with, for example, chemical reactions, financial models, population interactions between species, and the stresses in the structural members of a bridge.

### Paper details

This is an ideal paper for those who need or want to take at least a service paper in mathematical methods and do not yet have a background in mathematics sufficiently strong to join the MATH 160 class. Emphasis is placed on understanding via examples, and you will use the methods taught to study a variety of practical problems. In the process your manipulation skills in algebra and calculus will improve, and you will gain insight into the usefulness of the techniques. It will also provide you with an appreciation of the value and power of Mathematics and the motivation to progress to further MATH papers.

In particular you will cover such topics as linear and quadratic models, linear programming, functional notation, differentiation, rates of change, graphing of functions, optimization problems, exponentials and logarithms, compound interest, exponential growth and decay, simple integration.

### Potential students

This course is intended for students whose mathematical background is insufficient to embark on MATH 160 but who want to improve their maths skills, either to assist in their studies of other subjects or to prepare themselves to take MATH 160.

### Prerequisites

NCEA Level 2 Mathematics

### Main topics

- Basic algebraic manipulation
- Equations of lines
- Systems of linear equations
- Quadratics
- Arithmetic progressions
- Compound interest
- Linear programming
- Rates of change
- Differentiation
- Graphing functions
- Optimization problems
- Simple integration
- Finding areas
- Exponential, log and trig functions
- Differential equations

### Required text

None. Course materials will be available on the resource page. A book of complete notes for MATH 151 is available for purchase at the Print Shop.

### Useful references

Reviewing of texts from year 12 or year 13 mathematics may be useful from time to time. There are also two books available on close reserve at the Science Library, Foundation Maths and Maths for Higher Education.

### Lecturer

Frances Baeumer, room 213

### Lectures (Summer School)

Lectures: 10:00 to 11:50, Monday, Tuesday, Wednesday and Thursday

### Tutorial (Summer School)

13:00 to 14:50 Wednesday and Thursday

### Internal Assessment (Summer School)

There are three computer Skills Tests which count 20% (T) of your final mark. Your assignment mark A counts 15% and is based on ten on-line assignments. The remaining 65% of the final mark comes from the examination.

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 a mark of 5 or better on the first three Skills Tests.**

### Final mark

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

**F = 0.65E + 0.15A + 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.

### 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 Bright’nWarm Greenhouse people are installing a new lighting system. To provide light and warmth to their plants through overhead lighting they must decide on how many incandescent bulbs and fluorescent tubes to use to achieve the effect they want. Each bulb gives off 4 units of heat and 6 units of light while each tube gives off 2 units of heat and 9 units of light. The bulbs cost \$3.00 each and the tubes cost \$4.00 each. The system must provide at least 20 units of heat and at least 54 units of light. How many bulbs and tubes should they use to minimize the cost?### Babylonian Maths

Much of the basic mathematics we learn today can be traced back thousands of years. In Babylonian times (2000 to 1500 B.C.) **quadratic equations** could be solved, both by substituting into a general formula and by completing the square. Some cubic equations and biquadratic equations were also discussed. Tablets have been unearthed listing hundreds of unsolved problems involving simultaneous equations.

Their algebraic approach to solving geometric problems pre-dates by many centuries the advent of what we now call *algebraic geometry*. From an 1800 B.C. tablet: *An area X, consisting of the sum of two squares, is 1000. The side of one square is 10 less than 2/3 the side of the other square. What are the sides of the squares?*

Can you solve this problem?

### Sample problem

A large ship manoeuvring towards its berth travels at a speed (in km/hr) of $t^3-3t^2+4$ (starting from time $t$=0). How long does it take to come to a stop? How far does it travel over that period of time?