1. To introduce the basic ideas of differential and integral calculus, to develop the basic skills required to work with them and to apply these skills to a range of problems.

2. To introduce some of the fundamental concepts and techniques of real analysis, including limits and continuity.

3. To introduce the notions of sequences and series and of their convergence.

(LO1) Apply key definitions and theorems that underpin the definition of real numbers and the convergence of sequences.

(LO2) Apply key definitions and properties that are relevant for simple elementary functions.

(LO3) Apply standard notation, definitions and theorems of real analysis.

(LO4) Differentiate and integrate a wide range of functions.

(LO5) Apply definitions and theorems that underpin the properties of continuous functions.

(LO6) Sketch graphs and solve problems involving optimisation and mensuration

(LO7) Determine the convergence/divergence of a series.

(LO8) Construct proofs of mathematical results in real analysis.

(S1) Numeracy

Properties of subsets of the real numbers including their boundedness, supremum and infimum.

Algebraic and trigonometric functions.  
Absolute values and inequalities.  
Inverse functions. 
Sequences.
Limit of a sequence.
Continuity of functions (via sequences).
Derivative.
Differentiation of sums, products and quotients.  
Implicit differentiation.  
Critical points and extrema.
Optimisation.
L'Hôpital's Theorem.
Indefinite integrals, definite integrals and the Fundamental Theorem of Calculus.
The exponential and logarithm functions, hyperbolic functions.
Techniques of integration.
Series.
Convergence of a series.  
Tests for convergence.  
Alternating series and absolute convergence.