# 3H: Analysis of Differentiation and Integration MATHS4073

**Academic Session:**2019-20**School:**School of Mathematics and Statistics**Credits:**10**Level:**Level 4 (SCQF level 10)**Typically Offered:**Semester 1**Available to Visiting Students:**Yes**Available to Erasmus Students:**Yes

#### Short Description

This course aims to provide a rigorous foundation for calculus. It will examine differentiability (mainly on the real line), power series and integration of continuous functions on closed and bounded sets.

#### Timetable

17 x 1hr lectures and 6 x 1hr tutorials in a semester.

#### Requirements of Entry

None

#### Excluded Courses

Analysis I (86MG)

Analysis II (86PJ)

#### Assessment

90% Examination, 10% Coursework.

**Main Assessment In:** April/May

**Are reassessment opportunities available for all summative assessments?** Not applicable

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below.

#### Course Aims

This course aims to provide a rigorous foundation for calculus. It will examine differentiability (mainly on the real line), power series and integration of continuous functions on closed and bounded sets. The focus of the course will be on how to prove standard intuitive facts which are often taken for granted and we will see how to define mathematical functions such as sin, cos, exp and log. General problem solving and proving skills will be developed in the context of real analysis.

#### Intended Learning Outcomes of Course

By the end of this course students will be able to:

■ determine, with proof, whether a real or complex function is differentiable and compute (with justification) its derivative; explain, with justification, the connection between differentiability and other properties of functions such as continuity;

■ prove standard properties of real differentiable functions including Rolle's theorem, the mean value theorem; give a rigorous explanation of the connection between the derivative and monotonicity of a real function; use properties of the derivative to solve problems relating to real differentiable functions;

■ compute the radius of convergence, and determine the set of convergence of suitable real and complex power series of the type discussed in lectures; discuss the validity of differentiating power series term by term; define exponential and trigonometric functions using power series and rigorously establish their properties;

■ state and prove Taylor's theorem for real functions using both the Lagrange and Cauchy forms of the remainder; estimate the remainder of appropriate functions in order to establish the validity of the Taylor-MacLaurin series;

■ explain how to define the integral of a continuous function on a closed and bounded interval, establish properties of these integrals including the fundamental theorem of calculus, use properties of the integral to solve appropriate problems.

#### Minimum Requirement for Award of Credits

None