## MATH170 Mathematics 2

Second Semester | Also available: First Semester |

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.

MATH 170 is the natural continuation of MATH 160 and provides the basis for progression to 200-level Mathematics as well as a good mathematical background to support other subjects.

### Paper details

Like MATH 160, MATH 170 is divided between algebra and calculus, and focuses on both ideas and methods.

The algebra component first expands on the material on vectors and matrices begun in MATH 160. (Note, however, that sufficient background is provided on these topics to enable MATH 170 to be taken directly from school.) The course goes on to consider the determinant (a number associated with a matrix) and linear transformations (special mappings associated with computer graphics). The final section considers numbers and factorization, induction (used to prove sequential statements involving integers) and various counting techniques.

The calculus component extends some of the topics covered in MATH 160, and introduces others that are new. It starts with sequences (an ordered list of numbers, possibly infinite) and series (the sum of all the numbers in a sequences). The course then introduces special functions such as the natural logarithm, hyperbolic functions, and inverse trigonometric and hyperbolic functions. After further methods of integration and applications of integration to arclength and volumes, the course concludes with the study of differential equations (and examples of their many uses).

### Potential students

This paper should appeal to a wide variety of students including Mathematics and Statistics majors, those studying Computer Science, Physics, Chemistry, Surveying or any discipline with a quantitative component requiring competent manipulation of mathematical formulae and interpretation of mathematical representations of systems.

### Prerequisites

MATH 160 or high achievement (mostly Excellences and Merits) in NCEA Level 3 Calculus

### Main topics

**Algebra**

- Algebra and geometry of 3 dimensional vectors
- Manipulation of matrices and matrix equations
- Introduction to linear transformations
- Eigenvalues and eigenvectors
- Discrete mathematics, including mathematical induction, Diophantine equations and basic counting techniques

**Calculus**

- Sequences, series and Taylor series
- Natural log, exponential, hyperbolic, inverse trigonometric and hyperbolic functions
- Methods of integration
- Arc length; volumes and surfaces of revolution
- Solving differential equations

### Texts

**Algebra**

Course materials will be available on the resource page. A pdf copy of the book *MATH 170 Algebra Outline Notes * will be available on the resource page. A printed copy can be purchased from the Print Shop.

**Calculus**

Course materials will be available on the resource page. The book *MATH 170 Calculus Outline Notes* is available for purchase from the Print Shop.

Recommended text: *Calculus* by James Stewart, metric version, 8th edition (available from the University Book Shop). Older editions of this textbook are perfectly good.

### 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*Elementary Linear Algebra (Applications version)*by H. Anton and C. Rorres (7th edition)*Introductory Linear Algebra (with applications)*by B. Kolman (6th edition)*Calculus with Analytic Geometry*by Howard Anton (Wiley)*Calculus and Analytic Geometry*by George Thomas and Ross Finney (Addison Wesley)

### Lecturers (Semester 2)

Algebra: Prof Mike Hendy (room 517)

Calculus: Dr John Shanks (room 513)

### Lectures

Algebra: Monday, Wednesday and alternate Fridays (from July 14) at 12 noon

Calculus: Tuesday, Thursday and alternate Fridays (from July 21) at 12 noon

### Tutorials

Attendance at tutorials is voluntary. An open system operates: tutorial classes run for up to 8 hours per week (depending on demand), and students may attend as many as they need to and are able to.

### Internal Assessment

Five computer Skills Tests (in each of Algebra and Calculus) make up 20% (T) of your final mark. The other 80% comes from a mix of your exam mark (E) and the assignment mark (A) which is based solely on the ten marked weekly assignments.

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 6/10 in each of the first four Skills Tests before the end of the 12th week
- 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 10 to 15 questions for each of Algebra and Calculus. You may allocate your time between the two sections as you wish.

### Final mark

If:

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

all expressed out of 100, then your final mark F in this paper will be calculated according to this formula:

**F = 0.8max(E, (4E + A)/5) + 0.2T**

So your internal assessment counts at 1/5 weighting if that helps you.

### 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.

### Linear transformations

Multiplication by a 2x2 matrix can be used to transform points in the x-y plane — such

**transformations**are called

**linear**since they map straight lines to straight lines.

Apply each of these matrices to the image and see what happens to the kiwi:

Which of these do you think is reversible?

### Sample problem

After an evening’s competition the six couples from the “Twirling Dervishes” Ballroom Dancing team headed off to the local pub for a drink. A fairly simple tavern, their local offers 6 different beers and 4 types of wine. Having crammed themselves around a table their captain, Fred, volunteered to get in the first round. Armed with the numbers of each type of drink required he set off for the bar. Being a bit of a wag, Fred decided to perform a pirouette on the way and became dizzy. When he reached the bar he had completely forgotten the order. Too embarrased to go back and ask for the order again, he just made up a selection of twelve drinks from those on offer. What is the probability that he got the order right?

### Arthur Cayley...

... 1821-1895, published over 900 papers and notes covering nearly every aspect of modern mathematics. The most important of his work was in developing the algebra of matrices, work in non-euclidean geometry and n-dimensional geometry.### Sample problem

How can we tell the age of paintings? Some pigments contain a mineral which has two unstable isotopes occurring naturally. These isotopes decay at different rates, and one of them is a biproduct of the other’s decay process. By knowing their original relative proportions in the pigment and solving a differential equation, it is possible to give quite accurate estimates of age, and hence to detect forgeries.### Brook Taylor...

... 1685-1731, contributed greatly to 18th century mathematics — much more so than the single result that bears his name would suggest. He was a champion of Newton’s approach to calculus and produced many important works developing that area of mathematics. He invented the “calculus of finite differences” and “integration by parts”, as well as discovering “Taylor’s series”.### Sample problem

The interaction of two species, one a predator and the other its prey, can be studied using linked first order differential equtions. By varying the initial conditions (essentially the numbers of each species) different scenarios can be considered, the critical one being the circumstances under which the prey becomes extinct. We will solve the equations in class and also graph the changing populations using computer software.