Module Identifier MA13010 Module Title BASIC CALCULUS Academic Year 2000/2001 Co-ordinator Dr R S Jones Semester Semester 2 Pre-Requisite A or AS level Mathematics or equivalent. Mutually Exclusive May not be taken at the same time as, or after any of MA10020, MA11010, MA11110, MA12610, MA13510. Course delivery Lecture 22 x 1 hour lectures Seminars / Tutorials 6 x 1hour example classes Assessment Exam 2-hour written examination 100% Resit assessment 2-hour written examination 100%

General description
This is a calculus course with the emphasis on methods, techniques and applications. The topics to be covered include differentiation, integration, Taylor and Mclaurin series, special functions, higher derivatives and partial differentiation.

Aims
To present the methods and techniques of the differential and the integral calculus so that they can be applied in a variety of contexts.

Learning outcomes
On completion of this module, a student should be able to:

• sketch graphs of elementary functions;
• solve inequalities by routine methods;
• explain the geometrical significance of the derivative of a function at a point as the slope of the tangent to a curve;
• differentiate elementary functions from first principles;
• differentiate using the function of a function rule, the product rule, and the quotient rule;
• differentiate parametrically and differentiate implicit functions;
• differentiate repeatedly including using Leibnitz' theorem;
• obtain the Taylor and Maclaurin expansions of a function;
• evaluate indeterminate limits using L'Hopital's rule;
• integrate using the method of substitution and integration by parts;
• apply the theory of integration to determine the area of regions in a plane and volumes of solids of revolution;
• locate stationary points and determine their nature;
• determine the first and second partial derivatives of functions of two variables.

Syllabus
1. FUNCTIONS: Curve sketching
2. INEQUALITIES: Simple inequalities
3. DIFFERENTIATION: Including differentiating from first principles. Function of a function rule, produce rule, quotient rule. Parametric differentiation, implicit differentiation
4. SPECIAL FUNCTIONS: Exponential, logarithmic, hyperbolic and trigonometric functions
5. HIGHER DERIVATIVES: Leibnitz' theorem
6. TAYLOR'S THEOREM: The mean-value theorem of the differential calculus and applications. Taylor and Maclaurin series. L'Hopital's rule
7. INTEGRATION: Integration techniques, integration by substitution and integration by parts
8. APPLICATIONS OF DIFFERENTIATION: Locate local maxima and minima of functions
9. APPLICATIONS OF INTEGRATION: Area under curve and volumes of solids of revolution
10. PARTIAL DIFFERENTIATION: First and second order partial derivatives of functions of two variables