Calculus 1

• Credit value: 30 credits at Level 4
• Convenor: Dan Mcveagh
• Assessment: three problem sets (10% each) and a three-hour examination (70%)

In this module we introduce you to the methods of calculus and some of its applications, together with essential algebraic methods required throughout the programme. You will also learn about methods of approximation, and their limitations, and develop your knowledge of a mathematical computer package which you will then use to investigate and solve problems covered in the module.

Indicative syllabus

• Algebraic methods: polynomials, the Factor Theorem, polynomial equations, exact solutions to linear and quadratic equations, completing the squares, the Binomial Theorem, partial fractions, inequalities, arithmetic of complex numbers, complex numbers as the roots of polynomials.
• Coordinate geometry: straight lines, finding the equation of a straight lines, perpendicular lines, circles, tangent to a point on a circle, equation of a circle, finding the centre and radius of a given circle.
• Real functions: the properties and graphs of exponential, logarithmic and trigonometric functions, inverses, trigonometric identities
• Sequences and series: definitions, intuitive idea of a limit of a sequence, sigma notation, sum of i, i2 and i3 (i=1..n), arithmetic and geometric progressions.
• Differentiation: derivatives of standard functions, the chain rule, the product rule, the quotient rule,  the inverse function rule, implicit differentiation, logarithmic differentiation.
• Integration: integrals of standard functions, definite integration and the area under a curve, integration by substitution, integration by parts, integration of rational functions.
• Methods of approximation: the bisection method, the Newton-Raphson method, the Trapezium rule, Simpson’s rule, Maclaurin and Taylor approximations, power series of standard functions
• Applications of calculus: tangents, stationary points, maxima, minima and points of inflexion, curve sketching, rates of change, motion in a straight line, arc length, volumes of revolution, first order ODEs: variables separable and integrating factors.

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

• solve polynomial equations and simple inequalities
• express a rational function in partial fractions
• understand the methods of differentiation and integration, and differentiate and integrate functions of one variable
• understand arithmetic and geometric progressions
• understand the basic notation and terminology of calculus
• understand the Binomial Theorem
• understand the properties of standard functions of one variable
• use simple numerical methods for solving equations and for evaluating definite integrals
• express a function of one variable as a power series, and use the power series as an approximation for the function
• understand the importance of convergence to the solution of a problem when using a numerical method, and the fact that, in some cases, such a method may fail to produce a valid solution
• use a mathematical computer package to investigate and find solutions to the problems considered in the module.