Mathematics
Te Tari Pāngarau me te Tatauranga
Department of Mathematics & Statistics

## MATH104 Supplementary Calculus 2

 Second Semester Also available:  First Semester
9 points

### *** Not available in 2019! ***

This 9-point half-paper covers methods and applications of calculus, building on MATH 160/102. It consists of the calculus component of MATH 170. However, it is not by itself a sufficient foundation for second-year calculus (MATH 203, COMO 202) - for these papers, MATH 170 is required.

### Paper details

This paper 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

MATH 104 is taken only by students who need the calculus component (but not the algebra) of MATH 170. This situation may arise when a student has transferred from another university, or is majoring in another subject which requires a knowledge of calculus but not the same level of algebra.

### Prerequisites

MATH 160 or MATH 102 or high achievement (mostly Excellences and Merits) in NCEA Level 3 Mathematics with Calculus

### Main topics

• Review of trigonometry and basic 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

### Text

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:

• Calculus with Analytic Geometry by Howard Anton (Wiley)
• Calculus and Analytic Geometry by George Thomas and Ross Finney (Addison Wesley)

### Lecturer (Semester 2)

Dr Richard Norton (room 513)

### Lectures (Semester 2)

Tuesday, Thursday and alternate Fridays at 12 noon

### Tutorials

Attendance at tutorials is voluntary. An open system operates: tutorial classes run for up to 10 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 make up 20% of your final mark. The other 80% comes from a mix of your final exam mark and the assignment mark which is based solely on the ten marked weekly assignments.

### 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:

• achieve an overall mark of 40% on the 10 assignments

### Exam format

The 90-minute 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 about 15 questions.

Previous exams

### Final mark

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

F = 0.8max(E, (4E + A)/5) + 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.

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

### 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.

Further information

While we strive to keep details as accurate and up-to-date as possible, information given here should be regarded as provisional. Individual lecturers will confirm teaching and assessment methods.
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 applying a bit of calculus, it is possible to give quite accurate estimates of age, and hence to detect forgeries.
..., 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”.
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.