MAU11S02 Mathematics for scientists II

Module Code	MAU11S02
Module Title	Mathematics for scientists II
Semester taught	Semester 2
ECTS Credits	10
Module Lecturers	Prof. Miriam Logan Prof. Colm Ó Dúnlaing
Module Prerequisites	MAU11S01 Mathematics for scientists I

11 weeks of teaching with 6 lectures and 2 tutorials per week.

On successful completion of this module, students will be able to

Use standard techniques to compute definite integrals.
Use integrals to compute volumes, areas and lengths.
Evaluate improper integrals.
Formulate and solve first-order differential equations.
Determine whether a given sequence converges or not.
Test a given series for convergence.
Approximate a given function by polynomials using Taylor and Maclaurin series.
Compute determinants using either cofactor expansion or upper triangular forms.
Use Cramer's rule to solve linear equations.
Use the adjoint matrix to invert matrices.
Construct bases for the row space, column space and nullspace of a matrix.
Construct orthonormal bases in three dimensions.
Calculate the matrices of various linear maps.
Compute linear and quadratic curves matching data using the least squared error criterion.
Calculate eigenvalues and eigenvectors for 2x2 matrices, with applications to differential equations.
Derive probability distributions in some simple cases.
Solve problems involving the binomial distribution.
Calculate percentage points for continuous distributions such as the normal, chi-squared, and student's t-distribution.
Compute confidence intervals for the mean and standard deviation.

Applications of integrals: area between curves, volume of a solid, length of a plane curve, area of a surface of revolution.
Techniques of integration: integration by parts, trigonometric substitutions, numerical integration, improper integrals.
Differential equations: separable, first-order linear, Euler method.
Infinite series: convergence of sequences, sums of infinite series, tests for convergence, absolute convergence, Taylor series.
Parametric curves and polar coordinates.
Determinants, Cramer's rule, adjoint matrix formula for inverse.
Row space, column space and nullspace of a matrix.
Orthonormal bases in three dimensions.
Least squared error linear and quadratic estimates.
Eigenvalues and eigenvectors for 2x2 matrices, systems of linear differential equations.
Probability distributions: binomial, chi-squared, normal, Poisson, uniform.
Central limit theorem.
Hypothesis testing, confidence intervals for the mean and standard deviation.