## MATH102 Supplementary Calculus 1

Summer School | Also available: First Semester Second Semester |

This 9-point half-paper covers methods and applications of calculus. It consists of the Calculus component of MATH 160. Note that to go on to MATH 170, you will need both MATH 101 and MATH 102 - or their equivalent, MATH 160.

### Paper details

This Calculus paper is the natural continuation of Year 13 Calculus.

Here 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 102 is taken only by students who need the calculus component (but not the algebra) of MATH 160. This situation may arise when a student has transferred from another university, or is looking for a 9-point paper, or has previously taken MATH 101, the Algebra half of MATH 160.

### Prerequisites

Formally MATH 102, like Math160, has no prerequisites other than "sufficient achievement in NCEA Level 3 Calculus". However, we strongly suggest that if you have not passed the externally assessed NCEA Calculus papers "Apply differentiation methods in solving problems" (AS91578) and "Apply integration methods in solving problems" (AS91579), then you should consider taking MATH 151 before attempting MATH 102.

### Main topics

- Functions
- Introduction to calculus
- Techniques of differentiation and integration

### Required texts

Required 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:

*Calculus with Analytic Geometry* by Howard Anton (Wiley)

*Calculus* by James Stewart (Full edition.)

### Lecturers (Summer School)

Johannes Mosig, email: jmosig@maths.otago.ac.nz

### Lectures (Summer School)

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 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 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 from 15 to 20 questions.

### 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.

So your internal assessment can boost your exam mark if that helps you. Note that the test component definitely counts and so the tests should be regarded as compulsory.

### 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.

### 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?