First Year Analysis

(parts of 1A11 and 1A22)

This web site contains information and materials for the Analysis sections of 1A11 and 1A22. The files are in .pdf format as well as .ps format. If they are difficult to read on your machine, try altering the scale. They should print without any problems.

1A11 Analysis

General Course Information (.ps)

Contents: Introduction: what are the real numbers? Convergence of sequences and definition of limits. Quantified statements and how to manipulate them. Algebra of convergent sequences. Convergent sequences are bounded. Nested interval theorem. Series: convergence tests and standard examples (including use of integral test).

  • Students on the course: 1A11:

    Lecture Summary Sheets

    Section 1: real numbers and sequences (.ps)

    Section 2: convergent sequences (.ps)

    Section 3: series (.ps)

    Revision Materials

    Review sheet on convergence tests for series (.ps)

    Selected recent exam questions

    Selected questions 1970-1997 (.ps)

    1A22 Analysis

    General Course Information (.ps)

    Contents: Introduction: what do we want to be able to do with functions? Continuous functions and their properties. Sequence test for continuity. Bolzano-Weierstrass property and Intermediate Value Theorem. Differentiability and the Mean Value Theorem. Applications to maxima and minima. Cauchy Mean Value Theorem. Riemann integration. The fundamental theorem of calculus. ``Elementary'' calculus.

    Aims and Objectives: Students on this course should learn how the naive ideas about the behaviour of functions and numbers may be made rigorous. For example, starting with a ``definition'' that the real numbers comprise the things that can be written as infinite decimals, we prove that a continuous function that crosses the x-axis must have a zero. Throughout the emphasis is on giving precise definitions of convergence, continuity and so on. The last part of the course (end of 1A22) deals with Riemann integration and the fundamental theorem of calculus.

  • Students on the course: 1A22

    Lecture Summary Sheets

    Section 1: What is analysis? (.ps)

    Section 2: The limit of a function at a point (.ps)

    Section 3: Other limits of functions (.ps)

    Section 4: Continuity - basic results (.ps)

    Section 5: Continuity - Intermediate Value Theorem (.ps)

    Section 6: Differentiability (.ps)

    Section 7: Applications (.ps)

    Section 8: The Riemann integral (.ps)

    Section 9: General theorems on integration (.ps)

    Section 10: The fundamental theorem(s) of calculus (.ps)

    Section 11: ``Elementary'' calculus (.ps)

    Review problems (for both 1A11 and 1A22) (.ps)

    Selected exam questions

    Tom Ward; Email: T.Ward@uea.ac.uk