# Calculus 2

## Overview

• Credit value: 30 credits at Level 5
• Convenor: Robert Russell
• Assessment: two problem sets (25% each) and a three-hour examination (50%)

## Module description

In this module we develop the ideas and techniques of calculus introduced in Calculus 1 to functions of more than one variable. We will cover exact and numerical solutions of ordinary differential equations, as well as modelling problems using differential equations.

### Indicative syllabus

• Sequences and series: types of sequences, null sequences and their properties, convergent sequences and their properties, divergent sequences, sequences tending to infinity, types of series, tests for convergent and divergent series
• Functions: functions of one and two variables, limit of a function at a point, rules for evaluating limits, continuous functions
• Differentiation: definition of the derivative of a function, rules for evaluating derivatives, partial derivatives, tangent planes, the gradient of a function, directional derivatives, functions of three or more variables, higher derivatives, stationary points, finding and classifying stationary points, local and global extrema, Lagrange multipliers, the generalised chain rule, Taylor series
• Integration: rules for evaluating integrals, integrals with infinite limits, double integrals, splitting the integral, changing the order of integration, unbounded regions of integration, change of variables, polar coordinates
• Special functions: hyperbolic trigonometric functions and their properties, relationships between the hyperbolic and standard trigonometric functions, the inverse hyperbolic trigonometric functions, the derivatives of the hyperbolic trigonometric functions and their inverse, the gamma function and its properties, the beta function and its properties, the link between the gamma function and the beta functions, different forms of the beta function, using special functions to evaluate certain types of integral

## Learning objectives

By the end of this module, you will be able to:

• use techniques of calculus of more than one variable and methods of solution of ordinary differential equations
• understand the theory underpinning the techniques of calculus and produce proofs of some results in calculus
• use modelling oscillating systems, modelling problems in biology and modelling problems in finance and economics
• understand approximating functions using Taylor series
• find numerical solutions to differential equations
• estimate error in numerical solutions to differential equations
• use modelling problems using differential equations
• use a modern mathematical and/or statistical computer package with a programming facility, and demonstrate knowledge of other suitable packages.
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