ENGSCI 111 : Mathematical Modelling 1


2024 Summer School (1240) (15 POINTS)

Course Prescription

Introduction to mathematical modelling. Differentiation and integration (polynomials, trigonometric, exponential, logarithmic, and rational functions). Integration by parts, substitution and partial fractions. Differential equations and their solutions (including Euler's method). Vector and matrix algebra, transformations, solving systems of linear equations. Modelling using probability.

Course Overview

This course covers a range of material related to the mathematical modelling process.  Below is an overview of the material covered in each of the key sections of the course.

Mathematical Modelling
• Application of the mathematical modelling process, including the use of free body diagrams to formulate mathematical models from a word problem.
• Comprehend given proportionality relationships and then apply them to formulate mathematical models.
• Application of dimensional analysis to create and/or check the dimensional consistency of a model. 
• Application of function approximation to simplify a model for large or small values.
• Application of function optimisation by applying differential calculus. 
Modelling with Uncertainty
• Application of probability techniques including probability distributions, probability trees and Bayes theorem to compute the the probability of particular events/random variables given a situation described in words. 
Differentiation & Applications 
• Application of differentiation to solving relates rates of change problems, including the use of the chain rule to formulate the problem and implicit differentiation to find derivatives.
• Application of the formula for various types of series to determine polynomial approximations for functions, including the use of Maclaurin, Taylor and binomial series. 
• Application of finite difference formula for finding numerical derivatives using discrete tabulated data. 
• Comprehension of when polynomial approximations are valid (i.e. when they converge). 
• Application of integration techniques to compute integrals including the use of integration by substitution, parts and partial fractions. 
Ordinary Differential Equations
• Application of mathematical modelling to create Ordinary Differential Equations describing physical systems, identifying appropriate initial conditions where relevant. 
• Comprehension of different types of Ordinary Differential Equations, in order to classify them and identify the appropriate method of solution. 
• Application of appropriate types of solutions methods to relevant first order ODEs including the methods of direct integration, separation of variables, integration factor and Euler's method. 
• Application of boundary/intial conditions to solve spatially-varying/time-dependent systems described by an ODE, including finding and identifying steady state solutions. 
• Application of the method of exponential trial to solve second order, linear, homogeneous ODEs that have a characteristic equation that has distinct real roots. 
Vectors and Matrices
• Application of vector operations to solve 2/3D geometric problems described in words. 
• Application of matrix algebra in a physically meaningful context including geometric transformation and transitions matrices. 
• Application of linear algebra techniques to solving systems of linear equations.

Course Requirements

Restriction: ENGSCI 211, 213, 311, 313, 314

Capabilities Developed in this Course

Capability 3: Knowledge and Practice
Capability 4: Critical Thinking

Learning Outcomes

By the end of this course, students will be able to:
  1. Formulate Mathematical Models via the use of diagrams, assumptions, data, physical laws, dimensional analysis and approximation. (Capability 3.1 and 4.1)
  2. Use Modelling with Uncertainty to solve real world problems. (Capability 3.1 and 4.1)
  3. Calculate derivatives to solve problems involving rates of change and series. (Capability 3.1 and 4.1)
  4. Calculate definite and indefinite integrals, using the techniques of integrations by parts, substitution and partial fractions where needed. (Capability 3.1 and 4.1)
  5. Model real world problems with Ordinary Differential Equations and then solve them. (Capability 3.1 and 4.1)
  6. Use Vectors and Matrices to describe real world problems and then solve them. (Capability 3.1 and 4.1)


Assessment Type Percentage Classification
Assignments 30% Individual Coursework
Tests 20% Individual Coursework
Final Exam 50% Individual Coursework
Assessment Type Learning Outcome Addressed
1 2 3 4 5 6
Final Exam

Workload Expectations

This course is a standard 15 point course and students are expected to spend approximately 20 hours per week involved in each 15 point summer school course that they are enrolled in.

For this course each week you can expect 8 hours of lectures/tutorials and approximately 12 hours of self study. In any given week self study may include activities such as attending reviewing lecture recordings, working through practice problems, completing assignments and preparing for upcoming tests.

Delivery Mode

Campus Experience

Attendance is expected at scheduled activities including lectures and tutorials.

Lectures will be available as recordings. 
The course not include live online events.
Attendance on campus is required for the test.

Learning Resources

Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).

Please remember that the recording of any class on a personal device requires the permission of the instructor.

There are three course book manuals for this course: Volume 1, Volume 2 and Practice Problems. The summer school version of the course uses the same version of the course books as used in semester 1 of the previous year.
Hard copies of these course books can be purchased from UBIQ and pdf versions are available for free on Canvas.  

Health & Safety

Students are expected to adhere to the guidelines outlined in the Health and Safety section of the Engineering Undergraduate Handbook.

Student Feedback

At the end of every semester students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback and respond with summaries and actions.

Your feedback helps teachers to improve the course and its delivery for future students.

Class Representatives in each class can take feedback to the department and faculty staff-student consultative committees.

Students expressed concern that not enough time was devoted to vectors and matrices, topics that many found challenging as the concepts were new to them.  Summer school is by its nature very fast-paced but we will explore how we could devote a little more time to vectors and matrices.

Academic Integrity

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed for potential plagiarism or other forms of academic misconduct, using computerised detection mechanisms.

Class Representatives

Class representatives are students tasked with representing student issues to departments, faculties, and the wider university. If you have a complaint about this course, please contact your class rep who will know how to raise it in the right channels. See your departmental noticeboard for contact details for your class reps.

Inclusive Learning

All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.

Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website http://disability.auckland.ac.nz

Special Circumstances

If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.

If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page https://www.auckland.ac.nz/en/students/academic-information/exams-and-final-results/during-exams/aegrotat-and-compassionate-consideration.html.

This should be done as soon as possible and no later than seven days after the affected test or exam date.

Learning Continuity

In the event of an unexpected disruption, we undertake to maintain the continuity and standard of teaching and learning in all your courses throughout the year. If there are unexpected disruptions the University has contingency plans to ensure that access to your course continues and course assessment continues to meet the principles of the University’s assessment policy. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator/director, and if disruption occurs you should refer to the university website for information about how to proceed.

Student Charter and Responsibilities

The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter https://www.auckland.ac.nz/en/students/forms-policies-and-guidelines/student-policies-and-guidelines/student-charter.html.


Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.

In this course students may be asked to submit coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. In exceptional circumstances changes to elements of this course may be necessary at short notice. Students enrolled in this course will be informed of any such changes and the reasons for them, as soon as possible, through Canvas.

Published on 30/11/2023 08:08 a.m.