Mathematical Studies
MA160: Mathematics (10 ECTS)
Taught in Semester(s) I+II. Examined in Semester(s) I+II.
Workload: 96 hours (72 Lecture hours, 24 Tutorial hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
- use modular arithmetic and Euler's Phi function to detect errors in ISBNs, encipher messages using 1-dimensional affine and RSA cryptosystems, attack 1-dimensional affine cryptosystems, calculate with Chinese remainders. The student will also be able to present a proof of Fermat's little theorem;
- use matrices to solve resource allocation problems, encipher messages using higher dimensional affine cryptosystems, attack higher dimensional affine cryptosystems, solve some geometric problems;
- use eigenvalues, eigenvectors and the Principle of Induction to solve practical and theoretical problems about recurrence. The student will also be able to state the Hamilton-Cayley theorem and prove it in the $2\times2$ case;
- 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. You will be able to apply the material learned to a variety of problems coming from physics and earth sciences;
- use the definition of a 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;
- be able to perform calculations with logarithms and the exponential function. You will be able to use anti-derivatives to solve some basic problems in biology, chemistry and physics;
- be able to perform basic arithmetic operations with complex numbers, and factorize polynomials as a product of linear factors;
- be able to quantify the likelihood of some simple events, and calculate the expected value of some simple random variables;
- be able to describe data using the notions of median, mode, percentile, mean, standard deviation; you will be able to make inferences based on the estimated mean and standard deviation of a population;
- be able to 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;
- be able to evaluate definite and indefinite integrals using a variety of techniques;
- be able to solve separable differential equations and apply this skill to study population models in biology and physics.
Indicative Content
- Modular arithmetic, Euclidean algorithm, applications to ISBNs and cryptography Euler's Phi function, Fermat's little theorem (and its proof), application to public key cryptography, Chinese Remainder Theorem.
- Matrix addition, multiplication, inversion, systems of equations, applications to resource allocation problems; linear transformations, applications to cryptography; Cross products, applications to geometry.
- Calculation of eigenvalues, eigenvectors and matrix powers for $2\times2$ matrices, Hamilton-Cayley theorem (with proof for $2\times2$ matrices); proof by induction. Fibonacci sequence, golden ratio, applications to practical recurrence problems.
- 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.
- Exponentials, logarithms and pH calculations; anti-derivatives; real-world problems involving anti-derivatives.
- Cartesian and polar coordinates; geometric interpretation using Argand diagrams; roots of unity; roots of polynomials; complex conjugates.
- Probability of events; conditional probability and independence of events; Bayes' Theorem; expected values.
- Histograms; mode, median, mean, quartile; standard deviation. Population, samples and estimators; applications to practical problems in biology, chemistry and physics.
- Definite integrals and the Fundamental Theorem of Calculus; applications of integration to real-world problems.
- A range of techniques for calculating definite and indefinite integrals; further applications to real-world problems.
- Separable differential equations; logistic equation; applications to radioactive decay and biological population models.
Module Resources
Calculus James Stewart Brooks Cole
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