Module Identifier MA10020 Module Title ALGEBRA AND CALCULUS Academic Year 2001/2002 Co-ordinator Dr T McDonough Semester Semester 1 Other staff Dr David Binding, Dr Jane Pearson, Dr R Jones Pre-Requisite A-level Mathematics or equivalent. Mutually Exclusive MA12110, MA12510, MA12610, MA13010, MA13510, MA13610 Course delivery Lecture 40 x 1 hour lectures Seminars / Tutorials 10 x 1 hour tutorials Workshop 4 x 1 hour workshops (including test) Assessment Continuous assessment 25% Exam 2 Hours (written examination) 75% Resit assessment 2 Hours (written examination) 100%

#### General description

This module covers the algebra and calculus which are fundamental to the development of mathematics.

#### Aims

To introduce students to the ideas of algebra through the study of complex numbers and polynomials; to establish a clear understanding of the ideas of limit and derivative; to develop technical facility in calculations involving limits and derivatives and to develop techniques for determining definite and indefinite integrals.

#### Learning outcomes

On completion of this module, a student should be able to:
• use the notation for sets and mappings;
• construct proofs using the Principle of Mathematical Induction;
• apply the Binomial Theorem for an integer exponent in various situations;
• obtain the sums of arithmetic and geometric series;
• manipulate complex numbers and use DeMoivre's Theorem;
• use the Division Algorithm for polynomials;
• derive identities involving the roots of a polynomial and its coefficients;
• sketch the graphs of simple functions;
• calculate the limits of real valued functions;
• determine whether given functions are continuous or not;
• explain the idea of derivative and compute derivatives from first principles;
• explain the notion of inverse function;
• derive the formulae for the derivative of products and quotients of functions;
• compute the derivative of functions;
• determine the local maxima and minima of functions and their points of inflexion;
• compute integrals by the methods of substitution and integration by parts;
• compute integrals of rational functions and trigonometric functions.

#### Syllabus

1. SETS AND MAPPINGS: Introduction to number systems and mappings.
2. FINITE SUMS: The Binomial Theorem, arithemetic and geometric series. The Principle of Mathematical Induction.
3. COMPLEX NUMBERS: Geometric interpretation. DeMoivre's Theorem.
4. POLYNOMIALS: The Division Algorithm and the Remainder Theorem. Symmetric functions. Relations between roots of a polynomial and its coefficients.
5. FUNCTIONS OF A REAL VARIABLE: Graphs of elementary functions (polynomial, trigonometric, exponential, logarithm, absolute value, integer part). Periodic functions, even and odd functions. Operations on functions: addition, multiplication, division, composition.
6. LIMITS AND CONTINUITY: Limit notation. Rules for manipulation of limits. Sandwich theorem for limits, applications. Definition of continuity at a point in terms of limits. Continuity of sum, product, quotient and composite of continuous functions. Intermediate - value theorem.
7. DIFFERENTIATION: Fermat's difference quotient (f(x)-f(a))/(x-a). Definition of the derivative of f(x) at a point. Geometric significance of the derivative. Differentiation from first principles of some elementary functions. Continuity of differentiable function; examples of continuous nondifferentiable functions. Rules for differentiation. Examples on differentiation, including logarithmic differentiation. Second-order derivatives.
8. INVERSE FUNCTIONS: Definition. Trigonometric and polynomial examples. Differentiation of elementary inverse functions.
9. LOCAL MAXIMA AND MINIMA, CURVE SKETCHING: Locating the critical points of a function. Using the first derivative test to determine local maxima and minima. Points of inflexion. Graphs of rational functions, vertical asymptotes, horizontal asymptotes.
10. INTEGRATION: The Fundamental theorem of the integral calculus. Linearity properties of integration. Indefinite integrals. Methods of integration: integration by substitution, integration by parts. Definition of log x as an integral. Properties of the log function from the properties of the integral. The exponential function as the inverse of the log function. The hyperbolic functions. Integration of rational functions, use of partial fractions.