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Course Overview

Analysis provides the rigourous, foundational underpinnings of calculus. This course builds on the foundations in Analysis I, and explores the notions of Riemann integrability, Cauchy sequences, sequences of functions, and power series.

The techniques of careful rigourous argument seen in Analysis I will be further developed. Such techniques will be applied to solve problems that would otherwise be inaccessible. As in Analysis I, the emphasis of this course is on valid mathematical proofs and correct reasoning.

Course Details

Study Type Level Undergraduate 2 Second Sub Session 15 credits (7.5 ECTS credits) Old Aberdeen No Dr Alexey Sevastyanov

Qualification Prerequisites

• Programme Level 2

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Course Description

- Differentiation of functions of one variable: basic definitions and properties, chain rule, basic results on differentiable functions, Rolle's Theorem, Mean Value Theorem.

- Riemann integrability: Riemann sums, basic properties, the Fundamental Theorem of Calculus, improper integrals - Sequences of functions: pointwise convergence, uniform convergence, properties of limits of functions, series of functions

- Power series: convergence, continuity, differentiability, integrability, Taylor series

Syllabus

• Differentiation: Definitions and properties, Standard rules for differentiation, Extrema, Mean value theorem, Monotonicity and Convexity.
• Riemann integrability and the Riemann integrals.
• Integrability of continuous functions; characterisations of integrability; properties of the integral.
• Sequences and series of functions:
• Pointwise and uniform convergence; examples of pointwise converging sequences with bad  behaviour (regarding continuity, differentiation, integration);
• Theorems about good behaviour under uniform convergence; Weierstrass' M-test;
• Dominated Convergence Theorem; pointwise limits of continuous functions.
• Taylor series. Computing radius of convergence; uniform convergence of power series; Lagrange's form of the remainder.

Course Aims

To further develop understanding of the concepts, techniques, and tools of calculus. Calculus is the mathematical study of variation. This course emphasises differential and integral calculus, sequences and series of functions.

Learning Objectives

By the end of this course the student should:

• be able to state the main definitions and theorems of the course;
• be able to prove most results from the course;
• be familiar with the concept of differentiability and understand theorems about differentiable functions;
• understand Riemann integration and theorems about the Riemann integral;
• be able to apply techniques for showing integrability or non-integrability of functions;
• be able to distinguish between pointwise and uniform convergence of sequences of functions;
• be able to compute Taylor series, compute the interval of convergence of power series, and use Taylor's theorem to estimate functions by polynomials.

Contact Teaching Time

Information on contact teaching time is available from the course guide.

Summative Assessments

1st attempt - 1 two-hour written examination (80%); in-course assessment (20%).

Resit – 1 two-hour written examination paper. Maximum of written exam (100%) or written exam (80%) with carried forward in-course assessment (20%).

Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

Feedback

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact Course Coordinator for feedback on the final examination.

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