Module Descriptors

Mathematics

MA190: 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:

  1. 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;
  2. use matrices to solve resource allocation problems, encipher messages using higher dimensional affine cryptosystems, attack higher dimensional affine cryptosystems, solve some geometric problems;
  3. 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 2x2 case;
  4. 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;
  5. apply the material learned to a variety of problems coming from Physics and Earth Sciences;
  6. 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 arising from Business and Economics;
  7. find the general solution of a number of basic separable differentiable equations;
  8. solve basic word problems;
  9. distinguish between finite, countably infinite and uncountable sets of real numbers, explain these distinctions and provide examples to support these explanations;
  10. explain the meanings of the terms supremum and infimum, analyze boundedness properties of given sets and provide new examples of sets with prescribed properties;
  11. explain the concept of convergence and its importance in mathematics, and discuss and relate various properties of sequences of real numbers;
  12. determine with proof whether a given sequence of real numbers is convergent, and provide examples of sequences with certain specified properties;
  13. explain the connection between differential and integral calculus using the Fundamental Theorem of Calculus;
  14. evaluate definite and indefinite integrals using a variety of techniques;
  15. communicate ideas in a precise and clear manner using the specialized language of written mathematics.

Indicative Content

  1. 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.
  2. Matrix addition, multiplication, inversion, systems of equations, applications to resource allocation problems; linear transformations, applications to cryptography; Cross products, applications to geometry.
  3. Calculation of eigenvalues, eigenvectors and matrix powers for 2x2 matrices, Hamilton-Cayley theorem (with proof for 2x2 matrices); proof by induction; Fibonacci sequence, golden ratio, applications to practical recurrence problems.
  4. Basic functions and their graphs; inverse functions; limits; the intermediate value theorem; roots of equations.
  5. Definition of derivative and its physical interpretation. Techniques of differentiation. Differentiability implies continuity (with proof). The Mean Value Theorem; roots of equations.
  6. Detecting maxima/minima, monotonicity, concavity; application to graph sketching.
  7. Optimisation word problems.
  8. Exponentials and logarithms. Anti-derivatives and separable differential equations. World problems involving differential equations: radioactive decay, population models.
  9. Bounded and unbounded sets. Finite and infinite sets. Different kinds of infinities. The order relation on the real numbers. Suprema and infima. The completeness property of the real numbers. Sequences of real numbers:convergence and divergence.
  10. What is a sequence? Convergent and divergent sequences. Boundedness and monotonicity. The Mean Value Theorem and some applications.
  11. Definite integrals and the Fundamental Theorem of Calculus. Techniques of Integration.

Module Resources

Calculus James Stewart Brooks Cole

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