## Trinity College Dublin, The University of Dublin

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# Module MAU11001: Mathematics, Statistics and Computation

Credit weighting (ECTS)
10 credits
Semester/term taught
Michaelmas term 2019-20

Methods of Teaching and Student Learning
11 weeks; 8 hours per week, including 5 lectures, 2 tutorials and 1 computer practical.

• 1 or 2 lecturers from the school of mathematics
• 1 lecturer from the department of statistic
• teaching assistants/demonstrators for tutorial groups and practicals

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

Lecturers
Prof Anthony Brown, Prof Alberto Ramos & 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;
• algebracially 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.

Mathematics:
a)  Calculus:
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

Statistics:
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

Textbook:

•  [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 with 50 percent from a 2 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.