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MAST10006
Calculus 2

Semester II 2019

Course Coordinator: Dr Anthony Morphett. a.morphett@unimelb.edu.au. Room G43, Peter Hall Phone 8344 3879.

Lecturer for Stream 3: Arun Ram, 174 Richard Berry, email: aram@unimelb.edu.au

Time and Location for Stream 3:
       Lecture: Monday-Wednesday-Friday 13:00-14:00 PAR-Peter Hall-G01 (JH Michell Theatre),
       Practice class: One per week, see the timetable.


See https://ms.unimelb.edu.au/study/mslc/consultation

The student representatives are
Raymond Li   email: raymond1@student.unimelb.edu.au   and
Dominique Cera   email: dcera@student.unimelb.edu.au

Announcements

  • Prof. Ram reads email but generally does not respond by email.
  • The lectures will be recorded and made available in the LMS. The lecture recordings are NOT a complete resource and are usually a very incomplete record of the lecture. It is unreasonable to expect that you can do well on the assessment without more comprehensive and thorough utilisation of all resources.
  • Plagiarism declaration: It is required to complete the plagiarism declaration in LMS, by clicking on the Plagiarism declaration link for this subject and completing the submission of the Plagiarism declaration through LMS for this subject.
  • Academic Misconduct The start of semester pack includes a statement about Academic Misconduct (pdf file), Further information regarding academic misconduct and plagiarism is available at http://academichonesty.unimelb.edu.au/
  • Generic Skills statement: In addition to learning specific technical skills that will assist you in your future careers in science, engineering, commerce, education or elsewhere, you will have the opportunity to develop, in this subject, generic skills that will assist you in whatever your future career path:
    • You will develop problem-solving skills (especially through tutorial exercises) including engaging with unfamiliar problems, and identifying relevant strategies.
    • You will develop analytical skills - the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of the analysis.
    • Through tutorials and other interactions with fellow students, you will develop the ability to work in a team. The school distinguishes between ethical collaboration, which is strongly encouraged, and plagiarism, which is prohibited.
  • Printing arrangements for Peter Hall building Students must use UNICARD to print documents. The UNICARD printer is located near the G70 computer lab. Please note: there is not a money uploader in the Peter Hall Building. For more information about printing at the University and for locations of UNICARD uploaders direct students to Student IT Support: http://studentit.unimelb.edu.au/
  • Housekeeping: The start of semester pack includes: Housekeeping (pdf file), SSLC responsibilities (pdf file).
  • Stop1 Quick Reference Guide pdf file contains information on Requisite waivers, Enrolment after week 2, Electives, subject and major selection, study abroad, credit exemptions, academic adjustments, special consideration, exam reviews, Stop1 services, Healthand well being, further information.
  • it is required to complete the plagiarism declaration in LMS, by clicking on the Plagiarism declaration link for this subject and completing the submission of the Plagiarism declaration through LMS for this subject.
  • It is University Policy that:           

    “a further component of assessment, oral, written or practical, may be administered by the examiners in any subject at short notice and before the publication of results.  Students must therefore ensure that they are able to be in Melbourne at short notice, at any time before the publication of results” (Source: Student Diary).

    Students who make arrangements that make them unavailable for examination or further assessment, as outlined above, are therefore not entitled to an alternative opportunity to present for the assessment concerned (i.e. a ‘make-up’ examination).

Subject Outline

The handbook entry for this course is at https://handbook.unimelb.edu.au/view/2019/MAST10006.

This subject will extend knowledge of calculus from school. Students are introduced to hyperbolic functions and their inverses, the complex exponential and functions of two variables. Techniques of differentiation and integration will be extended to these cases. Students will be exposed to a wider class of differential equation models, both first and second order, to describe systems such as population models, electrical circuits and mechanical oscillators. The subject also introduces sequences and series including the concepts of convergence and divergence.

Calculus topics include: intuitive idea of limits and continuity of functions of one variable, sequences, series, hyperbolic functions and their inverses, level curves, partial derivatives, chain rules for partial derivatives, directional derivative, tangent planes and extrema for functions of several variables. Complex exponential topics include: definition, derivative, integral and applications. Integration topics include: techniques of integration and double integrals. Ordinary differential equations topics include: first order (separable, linear via integrating factor) and applications, second order constant coefficient (particular solutions, complementary functions) and applications.

Assessment

Eight to ten assignments (written or online) due at regular intervals during semester amounting to a total of up to 50 pages (20%), and a 3-hour written examination in the examination period (80%). Up to one third of the assignment based assessment will be completed online.

Assignments

Eight to ten assignments (written or online) due at regular intervals during semester amounting to a total of up to 50 pages (20%), and a 3-hour written examination in the examination period (80%). Up to one third of the assignment based assessment will be completed online.

Prerequisites

The prerequisites are listed in the handbook entry for this course at https://handbook.unimelb.edu.au/view/2019/MAST10006.

Lecture notes

Lecture notes and the Problem booklet (together in a single shrink wrapped set) will be available for sale at the University Coop. These contain information on lecture schedule, supplementary references, and course procedures.

Lectures this semester

I like to prepare by writing my own handwritten lecture notes (even though, in class, I usually follow the Melbourne University tradition and annotate the course note book on the document projector. I will post my handwritten lecture notes below.

  • Week 1: Limits, Continuity, Sequences and Series
    • 29 July 2019 Lecture 1: Intuitive idea of limits. Limit laws. Limits involving infinity;
      handwritten lecture notes (pdf file) Admin slides(pdf file)
    • 31 July 2019 Lecture 2: Evaluating limits. Sandwich theorem;
      handwritten lecture notes (pdf file). Admin slides (pdf file).
    • 02 August 2019 Lecture 3: Continuity of functions. Limits of compositions of functions; handwritten lecture notes (pdf file).

  • Week 2: Irrationals, complex arithmetic
    • 5 August 2019 Lecture 4: L'Hopital's rule. Sequences: definition, standard limits, connecting limits of functions and limits of sequences; handwritten lecture notes (pdf file).
    • 7 August 2019 Lecture 5: Limits of sequences examples. Series: definition. Geometric series. Divergence Test; handwritten lecture notes (pdf file).
    • 9 August 2019 Lecture 6: Ratio Test. Comparison test; handwritten lecture notes (pdf file).

  • Week 3: Hyperbolic Functions
    • 13 August 2019 Lecture 7: Hyperbolic functions: definition, basic properties, sketching graphs; handwritten lecture notes (pdf file).
    • 14 August 2019 Lecture 8: Hyperbolic functions: identities and applications, differentiation;
    • 16 August 2019 Lecture 9: Inverse hyperbolic functions: definition, sketching graphs, manipulation;

  • Week 4: Complex numbers
    • 20 August 2019 Lecture 10: Inverse hyperbolic functions: differentiation. Cartesian and polar form of a complex number; handwritten lecture notes (pdf file).
    • 21 August 2019 Lecture 11: Complex exponential: definition, properties. De Moivre's theorem;
    • 23 August 2019 Lecture 12: Differentiation and integration using the complex exponential;

  • Week 5: Integrals
    • 27 August 2019 Lecture 13: Substitution with derivative present. Trigonometric and hyperbolic substitutions; handwritten lecture notes (pdf file).
    • 28 August 2019 Lecture 14: Hyperbolic substitutions and products of hyperbolic functions; handwritten lecture notes (pdf file).
    • 30 August 2019 Lecture 15: Partial fractions and polynomial long division; handwritten lecture notes (pdf file), additional handwritten lecture notes (pdf file).

  • Week 6: Differential equations
    • 3 September 2019 Lecture 16: Integration by parts; handwritten lecture notes (pdf file).
    • 4 September 2019 Lecture 17: Definitions: ODE, order, general solution, IVP. Separable ODEs; handwritten lecture notes (pdf file).
    • 6 September 2019 Lecture 18: Linear ODE using integrating factors.

  • Week 7: Applications of Differential equations
    • 10 September 2019 Lecture 19: Making a substitution to reduce ODE to linear or separable. Equilibrium points. Applications: Doomsday population models without harvest; handwritten lecture notes (pdf file).
    • 11 September 2019 Lecture 20: Applications: Doomsday population models with harvesting. Logistic population models with and without harvesting; handwritten lecture notes (pdf file).
    • 13 September 2019 Lecture 21: Applications: Mixing problems with constant and variable volume;

  • Week 8: Second order differential equations
    • 17 September 2019 Lecture 22: Definitions: homogeneous, inhomogeneous, linear/non-linear. General solutions. Solution of homogeneous constant coefficient linear ODE; handwritten lecture notes (pdf file).
    • 18 September 2019 Lecture 23: Solution of homogeneous/inhomogenous constant coefficient linear ODEs. Particular solutions using method of undetermined coefficients; handwritten lecture notes (pdf file).
    • 20 September 2019 Lecture 24: Particular solutions continued. Superposition of particular solutions; handwritten lecture notes (pdf file).

  • Week 9: Integration
    • 24 September 2019 Lecture 25: Applications: Free vibrations of hanging spring-mass systems including air resistance; handwritten lecture notes (pdf file).
    • 25 September 2019 Lecture 26: Applications: Forced vibrations of hanging spring-mass systems; handwritten lecture notes (pdf file).
    • 27 September 2019 Lecture 27: NO LECTURE, AFL GRAND FINAL EVE;

  • Week 10: Functions of two variables
    • 8 October 2019 Lecture 28: Introduction to functions of two variables. Level curves and cross sections. Sketching surfaces; handwritten lecture notes (pdf file).
    • 9 October 2019 Lecture 29: Sketching surfaces (continued). Limits and continuity; handwritten lecture notes (pdf file).
    • 11 October 2019 Lecture 30: First and second order partial derivatives; handwritten lecture notes (pdf file).

  • Week 11: Derivatives
    • 15 October 2019 Lecture 31: Tangent planes. Linear approximations; handwritten lecture notes (pdf file).
    • 16 October 2019 Lecture 32: Chain rule for partial derivatives. Directional derivatives; handwritten lecture notes (pdf file).
    • 18 October 2019 Lecture 33: Directional derivatives (continued). Gradient. Steepest descent;

  • Week 12: Critical points and double integrals
    • 22 October 2019 Lecture 34: Stationary points. Classification of stationary points using second derivative test; handwritten lecture notes (pdf file).
    • 23 October 2019 Lecture 35: Partial integration. Double integrals over rectangular domains;
    • 25 October 2019 Lecture 36: Revision; handwritten lecture notes (pdf file).

Exercises

I like to understand the curriculum by understanding what skills are required, i.e. what problems could be asked on the exam. I do this by doing the problems that are available, from lectures, from the problem books, and the past exams, and thinking about how I can write the solutions so that, no matter which marker is marking my solution, I will maximise the marks that I get for my solution.

  • Sheet 1: Limits, Continuity, Sequences and series
    • Limits and continuity
    • Sequences and series Problem sheet 1 some examples
  • Sheet 2: Hyperbolic functions
  • Sheet 3: Complex numbers
  • Sheet 4: Integral calculus
    • Techniques of integration
    • Applications of integration
  • Sheet 5: First order differential equations
    • Solution of first order differential equations
    • Applications of first order differential equations
  • Sheet 6: Second order differential equations
    • Solution of second order differential equations
    • Applications of second order differential equations
  • Sheet 7: Functions of two variables
    • Sketching functions of two variables
    • Limits and continuity
    • Partial derivatives and applications
    • Double integrals