Mathematical Studies
MA161: Mathematical Studies (15 ECTS)
Mathematical Studies MA161 is an introduction to university mathematics aimed at students studying the physical and life sciences. The majority of MA161 students will have studied ordinary level mathematics at Leaving Certificate, though some will have studied the subject at higher level. Although students are unable to proceed from this module to a Mathematics BSc, the module does lead on to a range of second and third year Mathematical Studies modules relevant to the physical and life sciences. (By taking a total of 60 ECTS of appropriate Mathematical Studies modules during their undergraduate studies, students can satisfy the Teaching Council's subject requirements for mathematics teachers while majoring in another Science subject.)
Taught in Semester(s) 1 and 2. Examined in Semester(s) 1 and 2.
Workload: 300 hours (96 Lecture hours, 24 Tutorial hours, 180 Self study hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
- perform algebraic operations with $2\times2$ matrices, and translate some geometric problems into the language of $2\times2$ matrices. You will be able to identify some commonly occurring linear scientific phenomena.
- calculate the eigenvalues and eigenvectors of a $2\times2$ matrix and use these calculations to solve some recurrence problems occurring in science.
- solve a system of $n$ equations in $n$ unknowns (for low values of $n$) and, in particular, decide when such a system has no solution, a unique solution or infinitely many solutions.
- sketch the graph of a number of basic functions; calculate the limit of a function at a point or at infinity; decide whether a given function has an inverse and, if it does, calculate it; use the Intermediate Value Theorem to find roots of equations; apply the material learned to a variety of problems coming from physics and earth sciences.
- use the definition of derivative to compute the derivative of simple functions; apply different techniques of differentiation to calculate derivatives; apply the Mean Value Theorem to finding roots of equations; find maxima/minima/inflection points, and use these to sketch graphs of functions; apply differentiation techniques to solve optimisation problems.
- perform calculations with logarithms and the exponential function; use anti-derivatives to solve some basic problems in biology, chemistry and physics.
- perform basic arithmetic operations with complex numbers, and factorize polynomial as a product of linear factors.
- quantify the likelihood of some simple events, and calculate the expected value of some simple random variables.
- describe data using the notions of median, mode, percentile, mean, standard deviation; make inferences based on the estimated mean and standard deviation of a population.
- explain the connection between differential and integral calculus using the Fundamental Theorem of Calculus, and you will be able to apply this connection to some practical scientific problems.
- evaluate definite and indefinite integrals using a variety of techniques.
- solve separable differential equations and apply this skill to study population models in biology and physics.
Indicative Content
The module covers 12 topics.
Algebra and geometry of $2\times2$ matrices: Addition, multiplication, determinants and inverses of $2\times2$ matrices. Linear transformations of the plane and the geometric interpretation of these concepts. Examples of linear phenomena in science.
Eigenvalues and eigenvectors of $2\times2$ matrices: Definition of eigenvalues and eigenvectors; geometric interpretation; stochastic matrices; applications to practical recurrence problems in biology, chemistry and physics; Principle of Induction.
Systems of equations and matrices: Systems of n equations in n unknowns. Multiplication, determinants, adjoints, inverses of $n\times n$ matrices. Applications to practical problems in biology, chemistry and physics.
Continuous functions: Basic functions and their graphs; inverse functions; limits; Intermediate Value Theorem; roots of equations;
Rates of change and optimization: Definition of derivative and its physical interpretation; techniques of differentiation; differentiability implies continuity; Mean Value Theorem; roots of equations.; detecting maxima/minima; monotonicity, concavity; application to graph sketching; optimisation problems.
Anti-derivatives: Exponentials, logarithms and pH calculations; anti-derivatives; real-world problems involving anti-derivatives.
Complex numbers: Cartesian and polar coordinates; geometric interpretation using Argand diagrams; roots of unity; roots of polynomials; complex conjugates.
Probability: Probability of events; conditional probability and independence of events; Bayes' Theorem; expected values.
Descriptive statistics: Histograms; mode, median, mean, quartile; standard deviation. Population, samples and estimators; applications to practical problems in biology, chemistry and physics.
Integral Calculus: Definite integrals and the Fundamental Theorem of Calculus; applications of integration to real-world problems.
Techniques of integration: A range of techniques for calculating definite and indefinite integrals; further applications to real-world problems.
Differential equations: separable differential equations; logistic equation; applications to radioactive decay and biological population models.
Module Resources
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