Module MA1M01: Mathematical Methods

Credit weighting (ECTS)
10 credits
Semester/term taught
Michaelmas term 2012-13
Contact Hours
11 weeks, 5 lectures, 2 tutorials and 1 computer lab per week
Lecturers
Prof. Nicolas Garron, Prof. Sinéad Ryan
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 stadard 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 and link these quantities to population dynamics;
• state and apply the laws of probability;
• determine the results of binomial experiments with discrete random variables;
• calculate probabilities using probability density functions for continuous random variables.
Module Content

Calculus for Life Scientists

This part will be lectured by Prof. Nicolas Garron and there will be 3 lectures plus one tutorial per week. The syllabus is largely based on [Bittinger-G-N].

The calculus part of the syllabus is approximately Chapter 1-5 along with a little of Chapter 8 on differential equations (sections 8.1 and 8.2) from [Bittinger-G-N].

• Functions and graphs. Lines, polynomials, rational functions, trigonometric functions and the unit circle.
• Differentiation. Limits, continuity, average rate of change, first principles definition, basic rules for differentiation.
• Graphical interpretation of derivatives, max/min.
• Exponential and log functions. Growth and decay applications.
• Integration (definite and indefinite). Techniques of substitution and integration by parts. Applications.
• Differential equations and initial value problems, solving first order linear equations. Some application in biology or ecology.

Discrete Mathematics for Life Scientists

Prof. Sinéad Ryan will be the lectuer for this part. There will be 2 lectures per week, one tutorial and, for several of the weeks, a computer practical.

The syllabus is approximately:

• Linear algebra. Matrices, solving systems of linear equations, inverse matrices, determinants, eigenvalues and eigenvectors, solving difference equations. Population growth. (Chapter 6 of [Bittinger-G-N].)

• Spreadsheets. Basic concept of programming formulae in a spreadsheet such as Excel (absolute and relative cell references, some typical built in functions like sum, count, if). Formula for least squares fit of a line to points in the plane (without justification?). Graphs. Use of log scales.

• Probability. Basic concepts of probability. The binomial distribution, expectation and standard deviation for discrete random variables. Continuous random variables, probability density functions, expectation and standard deviation of continuous random variables. (Sections 10.1, 10.3, 10.4 of [Bittinger-G-N].)

Textbook:

[Bittinger-G-N] Calculus for the Life Sciences. Marvin Bittinger, Neal Brand, John Quintanilla. Pearson Dec 2005
Module Prerequisite
None (except Leaving certificate minimum for entry).
Assessment Detail
This module will be examined in a 3 hour examination in Trinity term. The exam will count 75% of the final grade with the remaining 25% for continuous assessment. For students who are required to sit supplementals, the same weighting applies.