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MAST10006
Calculus 2

Semester II 2024

Course Coordinator: Anthony Morphett. a.morphett@unimelb.edu.au. Room G43, Peter Hall Building.

Lecturer for Stream 2: Arun Ram, 174 Peter Hall, email: aram@unimelb.edu.au

Consultation hourse for Arun Ram: Monday 12:00-13:00 and 15:15-17:15 in Peter Hall 174

Time and Location for Stream 2:
       Lecture: Monday-Wednesday-Friday 13:00-14:00 Glyn Davis 117,
       Practice class: One per week, see the timetable.

You should also take advantage of the consultation hours of the other members of our teaching team for this course as listed on Canvas.

The student representatives are
Yitian (Nicole) Zhou   email: yitian.zou@student.unimelb.edu.au   and
Hannah Wilson   email: hannahhpw@student.unimelb.edu.au

Some of my thoughts about teaching have been written down in the Lecture script: Teaching Math in the Next Life
I am very interested in listening to and discussing any thoughts/reaction/improvements that you have in relation to this content.

Announcements

  • This page is a only a supplement consisting of resources produced by Arun Ram for delivery by public webpage. If you are actually enrolled in the course, then you need to be fully engaging with the resources on Canvas (for Melbourne University enrolled students).
  • Prof. Ram reads email but generally does not respond by email. Usually these are collated and reponses to email queries are provided in the first few minutes of lectures. That way all students can benefit from the answer to the query.
  • The lectures will be live streamed and recorded via the usual LectureCapture.

Subject Outline

The handbook entry for this course is at https://handbook.unimelb.edu.au/view/2024/MAST10006.

Notes

The following notes are written by Arun Ram.

Number systems

  • Numbers and intervals
  • The complex numbers
  • The binomial theorem
  • The number system ℚ[[x]], the exponential and the logarithm
  • Circular and hyperbolic functions
  • The interest sequence
  • Harmonic series, p-series and the Riemann zeta function
  • Some Series problems
  • Constant coefficient differential equations

Graphing

  • Basic graphs
  • Asymptotes
  • Shifting, scaling flipping
  • Continuity and differentiability -- jumps and slopes
  • The fundamental theorems of change and space
  • Why the fundamental theorem of space works

Algebra skills

  • Completing the square, the quadratic formula, and multiplying by the conjugate
  • Derivatives and integrals
  • Polynomial long division
  • Partial fractions

Limits, sequences and series

Limits       Limits and operations       Continuity      
Limits and addition: proof       Limits and scalar multiplication: proof       Limits and multiplication: proof      
Limits and composition of functions: proof       Limits and order: proof       Limits and order for sequences: proof      
     
Sequences Series Absolute convergence
Harmonic series Derivatives and Integrals Fundamental Theorem of Change
Areas and limits Fundamental Theorem of Calculus
     
The binomial theorem       Exponential and logarithm       Circular and hyperbolic functions      
     
Sets       Functions       Composition of functions
     

Constant Coeff. ODEs       Proof machine (How to do proofs)      

Handwritten lecture notes from 2019

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 and Continuity
    • 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: Sequences and Series
    • 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: Differential equations and vibrations
    • 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 get the bulk of the marks for my solution. I generally do not try to get full marks, as this is a very unpredicatble effort because it depends on too many uncontrollable parameters.

THE FOLLOWING EXMPLE SOLUTIONS ARE NOT DESIGNED TO BE AWARDED FULL MARKS. In most cases they would get the bulk of the marks for that question.

  • 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