Topic outline

  • WEEK-BY-WEEK LECTURE NOTE PDFs

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  • SAMPLE EXAMINATION

  • REVISION SUPPORT

  • LECTURE NOTES (W1-W4),(W5-W8),(W9-W12)

  • Week 1

    WEEK 1 LECTURES

    We will be reviewing "Continuity" and covering a rigorous introduction to "Differentiation".


    WEEK 1 TASKS

    • Familiarise yourself with the QMPlus page
    • By the end of the week, start solving the problems in course worksheet 1

    • Course Worksheet 1 File
      101.5 KB
    • Solution Sheet 1 File
      122.9 KB
    • Assignment 1 Questions File
      80.2 KB
    • Assignment 1

      The due/cutoff   date for this assignment is February 6th 5pm. 

      Only a SINGLE PDF submission  on QMPlus will be accepted. Emailed submissions will not be accepted under any circumstances. 

      If you have a problem uploading your submission, you will have to submit an EC claim.

      Please put your name and your student number on your submission.

      Tidyness and clarity of the presentation of your answers  counts!

    • Assignment 1 Solutions File
      113.5 KB

  • Week 2

    WEEK 2  LECTURES

    We will be covering the properties of differentiation and the mean value theorem.

    WEEK 2 TASKS

    • By the end of the week, start solving the problems in coursework sheet 2
    • Complete the coursework sheet 1
    • Prepare your solutions to the first assignment (under the week 1 tab)

  • Week 3

    WEEK 3  LECTURES

    We will be covering the Taylor's theorem.

    WEEK 3 TASKS

    • By the end of the week, start solving the problems in coursework sheet 3
    • Complete the coursework sheet 2
    • Submit solutions to the first assignment (under the week 1 tab)

    • Course Worksheet 3 File
      92.7 KB
    • Solution sheet 3 File
      121.2 KB
    • Assessment 2 Questions File
      60.7 KB
    • Assignment 2

      The due/cut-off date for this assignment is February 20th 17.00hrs.

      Only a SINGLE PDF submission  on QMPlus will be accepted. Emailed submissions will not be accepted under any circumstances. 

      If you have a problem uploading your submission, you will have to submit an EC claim.

      Please put your name and your student number on your submission.

      Tidyness and clarity of the presentation of your answers  counts!

    • Assessment 2 Solutions File
      69.4 KB

  • Module Description

    This is a second module in real analysis following on from the Convergence and Continuity module. We will explore in a rigorous way the main concepts, methods and results from the calculus of derivatives and integrals. 

    We formalize the definitions of differentiability and integrability and prove their basic algebraic properties. 

    We then explore some important results in real analysis and calculus such as the Mean Value Theorem and the Fundamental Theorem of Calculus. 

    We discuss Taylor’s Theorem and Improper integrals, widely used in probability and statistics, and close with the introduction of sequences of functions and their convergence.  


    • Add module description

  • Week 4

    WEEK 4  LECTURES

    We will be covering the inverse function theorem.

    • By the end of the week, start solving the problems in coursework sheet 4
    • Complete the coursework sheet 3
    • Start preparing your solutions to the second assignment (under the week 3 tab)

  • Week 5

    WEEK 5  LECTURES

    This week we will be covering the basics of Riemann integration.

    • By the end of the week, start solving the problems in coursework sheet 5
    • Complete the coursework sheet 4 
    • Submit solutions to the second assignment (under the week 3 tab)

    • Course Worksheet 5 File
      113.3 KB
    • Solution sheet 5 File
      157.7 KB
    • Assignment 3 questions File
      58.4 KB
    • Assignment 3

      The due/cut-off date for this assignment is March 12th Tuesday 5pm. 

      Only PDF submissions on QMPlus will be accepted. 

      Emailed submissions will not be accepted under any circumstances. 

      If you have a problem uploading your submission, you will have to submit an EC claim. 

      Remember to include your name and student number on your submission.

      Tidyness and clarity of the presentation of your answers  counts!

       

    • Assignment 3 solutions File
      108.8 KB

  • Week 6

    WEEK 6  LECTURES

    We continue our introduction to Riemann integration and introduce the notion of uniform continuity.

    • By the end of the week, start solving the problems in coursework sheet 6
    • Complete the coursework sheet 5
    • Start preparing your solutions to assignment 3 (under the week 5 tab

  • Week 7 -READING WEEK

  • Week 8

    WEEK 8  LECTURES

    We will be covering basic properties of the Riemann integral.

    • By the end of the week, start solving the problems in coursework sheet 7
    • Complete the coursework sheet 6
    • Submit solutions to the third assignment (under the week 5 tab)

    • Course Worksheet 7 File
      58.8 KB
    • Solution sheet 7 File
      74.3 KB
    • Assignment 4 Questions File
      83.5 KB
    • Assignment 4

      The due date for this assignment is March 26th Tuesday  5pm.

      Only PDF submissions on QMPlus will be accepted. 

      Emailed submissions will not be accepted under any circumstances. 

      If you have a problem uploading your submission, you will have to submit an EC claim. 

      Remember to include your name and student number on your submission.

      Tidyness and clarity of the presentation of your answers  counts!

       

    • Assignment 4 solutions File
      112.0 KB
    • Assignment 4 - solution overview File
      5.1 MB

  • Week 9

    WEEK 9  LECTURES

    We will be covering the Fundamental Theorem of Calculus.

    • By the end of the week, start solving the problems in coursework sheet 8
    • Complete the coursework sheet 7 
    • Start preparing the fourth assignment (under the week 8 tab)

  • Week 10

    We will be covering sequences and series of functions.

    • By the end of the week, start solving the problems in coursework sheet 9
    • Complete the coursework sheet 8
    • Submit solutions to the fourth assignment 4 (under the week 8 tab)

  • Week 11

    We will cover power series.

    • By the end of the week, start solving the problems in coursework sheet 10
    • Complete the coursework sheet 9
    • Start preparing the fifth assignment (under the week 10 tab)

  • Syllabus

    • Module Syllabus


      1. Differentiable functions: Definition of differentiability. Algebra of derivatives, chain rule. Derivative of inverse function. Rolle’s Theorem, Mean Value Theorem and applications. Taylor’s Theorem. 

      2. Integration: Darboux definition of Riemann integral, simple properties. Continuous functions are integrable (via uniform continuity). Fundamental Theorem of the calculus, integral form of Mean Value Theorem and of the remainder in Taylor’s Theorem; applications to some well known series (log, arctan, binomial). Improper integrals. 

      3. Sequences of functions: pointwise and uniform convergence. Weierstrass M-test. Term-by-term integration of power series. 


    • Module Syllabus

      Lecture 1: Topic title;

      Lecture 2: Topic title;

      Lecture 3: Topic title;

      Lecture 4: Topic title;

      Lecture 5: Topic title;

      Lecture 6: Topic title;

      Lecture 7: Topic title;

      Lecture 8: Topic title;

      Lecture 9: Topic title;

      Lecture 10: Topic title;

      Lecture 11: Topic title;

  • Module aims and learning outcomes

    ACADEMIC CONTENT

    This module covers:

    • Differentiable functions, the algebra of derivatives and key theorems.
    • Integration involving the Riemann integral; the Fundamental Theorem of Calculus; applications.
    • Sequences of functions; pointwise and uniform convergence; the Weierstrass M-test; term-by-term integration of power series.

    DISCIPLINARY SKILLS

    At the end of this module, students should be able to:

    • Define the derivative and state the properties of the derivative including the chain rule and inverse function rule.
    • State and use key theorems concerning differentiable functions, such as Rolle's Theorem, the Mean Value Theorem and Taylor's Theorem.
    • Define the Riemann integral, and state its properties.
    • State the Fundamental Theorem of Calculus and apply it to the calculation of limits.
    • Apply Taylor's Theorem to some well-known functions.
    • Distinguish pointwise and uniform convergence.
    • Apply the Weierstrass M-test to determine if an infinite series of functions converges uniformly.

    ATTRIBUTES

    At the end of this module, students should have developed with respect to the following attributes:

    • Grasp the principles and practices of their field of study.
    • Acquire substantial bodies of new knowledge.
    • Explain and argue clearly and concisely.
    • Acquire and apply knowledge in a rigorous way.
    • Connect information and ideas within their field of study.

    • Key Objectives

      • Objective 1
      • Objective 2
      • Objective 3

    • Key Objectives

      • Objective 1
      • Objective 2
      • Objective 3

  • Assessment information

    • Assessment Pattern -  

      There will be five assessed courseworks, released in weeks 1, 3, 5, 8, 10, and due for submission in weeks 3, 5, 8, 10, 12. 

      The asssignments will be made live on QMplus Wednesday 9am for  each of the weeks 1, 3, 5, 8, 10 and the due/cut-off date for submission on QMplus will be Tuesday 5pm for each of the weeks  3, 5, 8, 10, 12.

      Each of the 5 courseworks is worth 4% of the overall credit for the module, making 20% of the final mark .

       A written examination paper in May/June 2024  accounts for the remaining 80%. 

      Look for courseworks under the weeks in which they are released.

       

  • Final Exam

    • Format of the final exam

      The format of the final exam is an in person on campus exam.

      A past paper is available. 

      Solutions to previous years exams will not be provided. 


    • Add information here.

    • MTH5105 Examination 2023 File
      137.9 KB

  • Where to get help

    You are encouraged to seek help in the Learning Cafe. All staff are there to help you with your questions. 

    I will be there for approximately for 2 hours per week, on Mondays and Fridays. 

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  • General course materials

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  • Week 12

    WEEK 12  LECTURES

    We will cover differentiating and integrating power series. 

    • Complete the coursework sheet 10
    • Submit solutions to assignment 5 (under the week 10 tab)

  • Revision

  • Early feedback questionnaire

  • Q-Review

  • Online Reading List