Te Tari Pāngarau me te Tatauranga
Department of Mathematics & Statistics

MATH104 Supplementary Calculus 2

Second SemesterAlso available:  First Semester
9 points

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

Beginning with a brief review, the paper extends some of the topics covered in MATH 160, and introduces others that are new. It includes a section on useful functions, methods of solving differential equations (and examples of their many uses), and further integration techniques and applications. Knitting the whole section together are power series.

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.


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


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 John Shanks (room 513)

Lectures (Semester 2)

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


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.

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 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


  • 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 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.

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.