Mathematics 1

Objectives

The objective is to develop mathematical knowledge and skills in subject areas including pre-calculus, calculus, matrices, statistics and probability, as preparation for a wide variety of university courses.

Course outline

Semester 1

• Review of algebra and trigonometry.
• Functions, domain, range, graphs, transformations of graphs.
• Composition of functions, inverse functions (including logarithms and exponential functions, and inverse trigonometric functions).
• Limits, continuity, differentiability, differentiation from first principles.
• Differentiation formulae, including the product rule, quotient rule, chain rule.
• Applications of derivatives, including rates of change, equations of tangents and normals, stationary points, curve-sketching, applied maximum/minimum problems.
• Anti-differentiation, fundamental theorem of calculus, evaluation of areas.

Semester 2

• Indefinite integrals and further applications of integration.
• Limits at infinity and improper integrals.
• Linear approximations.
• Matrix methods for solving systems of linear equations, including the triangular matrix method.
• Statistics, including tabulation and graphical representation of data, mean, median, mode, standard deviation, variance, quartiles.
• Permutations and combinations.
• Introduction to probability, including addition rule, multiplication rule, independent events, tree diagrams.
• Markov chains.
• Probability distributions, expected value, standard deviation.
• Binomial, hypergeometric and geometric distributions, as examples of discrete probability distributions.
• Continuous probability density functions, mean, median, variance, and applications.
• Normal distribution.

Assessment

Semester 1 Exam 30%

Class work in Semester 2 5%

Semester 2 Exam 65%

Required Texts

Maths 1 Lecture Notes (available from the Maths 1 lecturer-in-charge at Trinity College).