33190 Mathematical Modelling for Science
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Credit points: 6 cp
Result type: Grade and marks
Anti-requisite(s): 33130 Mathematical Modelling 1 AND 35101 Introduction to Linear Dynamical Systems AND 37131 Introduction to Linear Dynamical Systems
Recommended studies: two units of HSC Mathematics
Description
Mathematical modelling is essential in all branches of science. This subject develops the knowledge and skills necessary for problem-solving and mathematical modelling at an introductory level. Topics covered include: vectors and geometry; complex numbers; calculus and its relationship to science; differentiation and integration of functions; inverse, trigonometric and hyperbolic functions; the solution of differential equations with applications to exponential growth and decay and oscillating systems; Taylor series; and an introduction to linear algebra. The computer algebra system Mathematica is used for symbolic, graphical and numerical computations.
Subject objectives
Upon successful completion of this subject students should be able to:
| 1. | Identify, manipulate and graph functions including inverse trigonometric functions, hyperbolic functions and inverse functions. |
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| 2. | Differentiate functions, solve real-world problems involving derivatives, apply techniques of integration to perform integrals, and then use these to make predictions and understand aggregate quantities. |
| 3. | Define vectors and their operations, and use vectors in solving problems in three dimensions. |
| 4. | Define matrices and their operations, including multiplication and inverses and use matrix manipulations to solve problems of several variables. |
| 5. | Define complex numbers, use complex algebra and find roots and exponents of complex numbers. |
| 6. | Solve simple linear first and second order differential equations, and use these to study oscillations, simple harmonic motion and resonance. |
| 7. | Define and identify sequences and series, and use Taylor series of functions to simplify problems. |
| 8. | Communicate using mathematical reasoning in the solution to problems |
This subject also contributes specifically to the development of following course intended learning outcomes:
- An understanding of the nature, practice and application of the chosen science discipline. (1.0)
- Encompasses problem solving, critical thinking and analysis attributes and an understanding of the scientific method knowledge acquisition. (2.0)
- An understanding of the different forms of communication - writing, reading, speaking, listening -, including visual and graphical, within science and beyond and the ability to apply these appropriately and effectively for different audiences. (6.0)
Contribution to the development of graduate attributes
This subject contributes to the development of the following graduate attributes:
Graduate Attribute 1 - Disciplinary knowledge and its appropriate application
A broad introduction to the most important and widely used concepts in mathematics is provided throughout this subject. Students will explore a range of fundamental topics, with the core disciplinary knowlegde presented in lectures used to underpin the development of problem solving strategies in the tutorials.
Graduate Attribute 2 - An Enquiry-oriented approach
The application of problem solving strategies to real-world problems is a focus of the subject. Numerous approaches to solving problems will be developed throughout the subject, and students will be required to select the most appropriate approach for the problems presented. This is a core skill in the context of mathematical modelling.
Graduate Attribute 6 - Communication skills
The ability to communicate mathematical reasoning using correct terminology is a each skill. In this subject, students will develop the ability to present a problem solving strategies in a logical format, with a focus on the use of appropriate terminology. This skill will be introduced during lectures and reinforced through practice questions during tutorial sessions.
Teaching and learning strategies
Lectures: three hours/week
Tutorials: one hour per week
Mastery Tests: four per semester
New material will be initially presented in lectures. The lecture slides will be placed on UTSOnline before the lecture so that students can prepare and follow the worked examples in class. The slides and examples presented in class will then be available following the lecture for further study and to consolidate the learning of new material and concepts. Students are then expected to use these to work through the "Basic Skills" section of the Problem Sets prior to the tutorials. During the tutorials the Basic Skills work will be checked, providing valuable feedback on progress. If completed correctly, the students will use the skills gained to solve more complex, real-world problems which will be encountered in the exam. The tutorial sessions provide an opportunity for collaborative learning, with students encouraged to work in groups to refine and improve their problem solving strategies.
The Basic Skills will be formally assessed using a series of four Mastery Tests. There will be four tests spaced throughout the semester, and each student will have three chances to pass each test. Sample questions and simulated tests will be available online for the students to practise prior to each of the tests - many of these questions are linked to video tutorials and to the relevant section in the textbook. Students can practise these sample questions as many times as they like, receiving feedback along the way.
Content
Topics to be presented throughout semester will include:
- Vectors and scalars, and their relation to geometry.
- Complex numbers.
- Functions and derivatives, and their relationship to measurement and the interpretation of physical results.
- Differentiability.
- Differential equations arising from physical problems.
- Solution by series.
- Oscillatory motion.
- Trigonometric functions and inverse trigonometric functions. Integrals and logarithms, inverse functions.
- Methods of integration.
- Introduction to matrices and linear algebra.
The computer algebra system Mathematica will be used in the subject as an aid to computation, graph plotting and visualization.
Assessment
Assessment task 1: Final examination
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1. Disciplinary knowledge and its appropriate application 2. Enquiry-oriented approach 6. Communication skills Final exam involves Problem posing and solving – ability to identify, assess and formulate problems relevant to one’s academic discipline and apply appropriate approaches and methods of problem solving. |
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| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3, 4, 5, 6, 7 and 8 This assessment task contributes to the development of course intended learning outcome(s): 1.0, 2.0 and 6.0 |
| Weight: | 37.5% |
| Criteria: | Students will be assessed on:
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Assessment task 2: Mastery Tests
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1. disciplinary knowledge and its appropriate application Mastery Tests targets Problem posing and solving – ability to identify, assess and formulate problems relevant to one’s academic discipline and apply appropriate approaches and methods of problem solving. |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3, 4 and 6 This assessment task contributes to the development of course intended learning outcome(s): 1.0 |
| Weight: | 62.5% |
| Length: | 45 minutes |
| Criteria: | Students will be assessed on:
See 'Further Information' for more detail on requirements. |
Minimum requirements
Students must receive a minimum of 50% of the overall mark to pass the subject.
Students should note that in order to sit the final exam they must achieve a minimum of 80% for each and every Mastery test.
Required texts
J. Stewart: Calculus - concepts and contexts, Metric International 4th Edition, Cengage
References
- G. H. Smith and G. J. McLelland (2003). On the shoulders of giants: A course in single variable calculus. Sydney, UNSW Press.
- C. H. Edwards and D. E. Penney, Calculus with Analytic Geometry, 3rd or 4th Editions. Prentice Hall.
- S.L. Salas and E. Hille, Calculus: one and several variables, 7th edition, John Wiley and Sons, 1995
- J. Callahan and K. Hoffman, Calculus in context, W. H. Freeman and Company, 1995
