Module Identifier MA13010 Module Title BASIC CALCULUS Academic Year 2001/2002 Co-ordinator Dr R 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