Module MAU11002: Mathematics, Statistics and Computation II
- Credit weighting (ECTS)
- 10 credits
- Semester/term taught
- Hilary term 2021-22
- 1 or 2 lecturers from the school of mathematics
- 1 lecturer from the department of statistic
- teaching assistants/demonstrators for tutorial groups and practicals
- Prof Anthony Brown & STATS
- Learning Outcomes
On successful completion of this module, students will be able to:
- use graphs of functions in the context of derivatives and integrals
- compute derivatives and equations of tangent lines for graphs of standard functions including rational functions, roots, trigonometric, exponential and logs and compositions of them;
- find indefinite and definite integrals including the use of substitution and integration by parts;
- solve simple maximisation/minimisation problems using the first derivative test and other applications including problems based on population dynamics and radioactive decay;
- select the correct method from those covered in the module to solve wordy calculus problems, including problems based on population dynamics and radioactive decay;
- algebraically manipulate matrices by addition and multiplication and use Leslie matrices to determine population growth;
- solve systems of linear equations by Gauss-Jordan elimination;
- calculate the determinant of a matrix and understand its connection to the existence of a matrix inverse; use Gauss-Jordan elimination to determine a matrix inverse;
- determine the eigenvalues and eigenvectors of a matrix
- determine the eigenvalues and eigenvectors of a matrix and link these quantities to population dynamics;
- Understand the basic ideas of descriptive statistics, types of variables and measures of central tendency and spread.
- Appreciate basic principles of counting to motivate an intuitive definition of probability and appreciate its axioms in a life science context.
- Understand common discrete and continuous distributions and how these naturally arise in life science examples.
- Understand how to take information from a data set in order to make inference about the population, appreciating the core ideas of sampling distributions, confidence intervals and the logic of hypothesis testing.
- Have a basic understanding of the statistical software R including importing and exporting data, basic manipulation, analysing and graphing (visualisation) of data, loading and installing package extensions, and how to use help files and on-line resources to solve error queries or to achieve more niche capabilities.
- Module Content
The module is divided into a maths and a statistics part, with maths further divided into calculus and linear algebra/discrete mathematics.
3 lectures plus one tutorial per week. The syllabus is largely based on the text book [Stewart-Day], and will cover most of Chapters 1-6 along with the beginning of Chapter 7 on differential equations:
- Functions and graphs. Lines, polynomials, rational functions, exponential and logarithmic functions, trigonometric functions and the unit circle.
- Limits, continuity, average rate of change, first principles definition of derivative, basic rules for differentiation
- Graphical interpretation of derivatives, optimization problems
- Exponential and log functions. Growth and decay applications. semilog and log-log plots.
- Integration (definite and indefinite). Techniques of substitution and integration by parts. Applications.
- Differential equations and initial value problems, solving first order linear equations. Applications in biology or ecology.
b) Linear algebra/discrete mathematics:
1 lecture and 1 tutorial per week. The syllabus will cover parts of chapter 1 on sequences, limits of sequences and difference equations and then chapter 8 of [Stewart-Day] on linear algebra.
The syllabus is approximately:
- Sequences, limits of sequences, difference equations, discrete time models
- Vectors and matrices , matrix algebra
- Inverse matrices, determinants
- Systems of difference equations, systems of linear equations, eigenvalues and eigenvectors. Leslie matrices, matrix models
There will be 1 lecture per week and 1 computer practical. The syllabus will cover much of chapters 11-13 of [Stewart-Day] and use [Bekerman-et-al] as main reference for R in the computer practicals.
The syllabus is approximately:
- Numerical and Graphical Descriptions of Data
- Relationships and linear regression
- Populations, Samples and Inference
- Probability, Conditional Probability and Bayes’ Rule
- Discrete and Continuous Random Variables
- The Sampling Distribution
- Confidence Intervals
- Hypothesis Testing
- [Stewart-Day] “Biocalculus: Calculus, Probability and Statistics for the Life Sciences”, James Stewart and Troy Davis, Cengage Learning (2016)
- [Beckerman-et-al] Getting Started with R: An Introduction for Biologists (2nd Ed). Beckerman, Childs and Petchy, Oxford University Press.
- Module Prerequisite
- None (except Leaving certificate minimum for entry)
- Assessment Detail
- 70 percent of the mark will come from the maths part of the module with 50 percent from a 3 hour end of semester exam and 20 percent based on continuous assessment (tutorials)
- 30 percent of the mark will come from the statistics component, consisting of group assessment (1-3 students working together on a data analysis project during the last weeks of teaching term)
Re-assessments if required will consist of 100% exam.
Methods of Teaching and Student Learning
11 weeks; 8 hours per week, including 5 lectures, 2 tutorials and 1 computer practical.
4 lectures + 2 tutorials per week will be covered by the school of maths;
1 lecture + 1 computer practical per week will be covered by the department of statistics