 |
MAST10005 |
Semester I 2025 |
Arun Ram, 174 Peter Hall, email: aram@unimelb.edu.au
I am not teaching Calculus in 2023. I am using this page to try to organize various notes on Calculus that I have written over the years.
Notes written by Arun Ram
- Numbers and Intervals Summary pdf file
- Complex numbers summary pdf file
- Circular and Hyperbolic functions
- Exponential and trig functions
-
Binomial theorem
- Sets summary pdf file
- Functions pdf file
- Composition of functions
-
Cardinality
- Derivatives and Integrals pdf file
- Fundamental Theorem of Change
- Fundamental Theorem of Calculus
-
Why the Fundamental Theorem of Calculus works
- Continuity
- Limits
- Limits and operations
Lectures Semester 1, 2025
- Week 1: Notation, numbers, proofs; Tutorial Topic: Trigonometric functions
- 3 March 2025 Lecture 1: Numbers and intervals, Calculus and complex numbers
- 5 March 2025 Lecture 2: Complex numbers, exponentials, sine and cosine;
- 7 March 2025 Lecture 3:
Proof Machine
- Week 2: Numbers, sets and graphing; Tutorial Topic: Complex numbers
- 10 March 2025 Lecture 4: Complex numbers and vectors
- 12 March 2025 Lecture 5: Sets and functions
- 14 March 2025 Lecture 6:
Graphing: Shifting, scaling and flipping
- Week 3: Graphing and vectors; Tutorial topic: Complex numbers
- 17 March 2025 Lecture 7: Graphing: exponentials, logs and inverse trig functions;
- 19 March 2025 Lecture 8: Complex numbers, ℝ2 and ℝn;
- 21 March 2025 Lecture 9:
Graphing and parametric equations
Vectors
- Week 4: Algebra and factoring; Tutorial topic: Complex numbers and triangle inequality
- 24 March 2025 Lecture 10: Quadratic formula and partial fractions
- 26 March 2025 Lecture 11: Injective, surjective, bijective
- 28 March 2025 Lecture 12:
Roots of unity, and factoring polynomials
- Week 5: Derivatives; Tutorial topic: Complex numbers and factoring polynomials
- 31 March 2025 Lecture 13: ℂ((x)) and derivatives
- 2 April 2025 Lecture 14: Derivatives, composition and the chain rule
- 4 April 2025 Lecture 15:
Derivatives and inverse functions
- Week 6: Dreivatives and graphing; Tutorial topic: Composition of functions
- 7 April 2025 Lecture 16: Derivatives and slope
- 9 April 2025 Lecture 17: Derivatives and optimisation
- 11 April 2025 Lecture 18:
Derivatives, tangents and normals
- Week 7: Tutorial topic: Vectors, dot product and length
- 14 April 2025 Lecture 19: Vector examples;
- 16 April 2025 Lecture 20: Orders and numbers systems
- 18 April 2025 Lecture 21:
Integrals
- Non Teaching week
- Week 8: Integrals; Tutorial topic: Parametric equations and derivatives
- 28 April 2025 Lecture 22: Fundamental theorem of Calculus
- 30 April 2025 Lecture 23: Computing areas
- 2 May 2025 Lecture 24:
Differential equations
- Week 9: Exam preparation bootcamp; Tutorial Topic: Optimisation
- 5 May 2025 Lecture 25: Differential equations examples
- 7 May 2025 Lecture 26: Integrals examples;
- 9 May 2025 Lecture 27:
Graphing examples;
- Week 10: Exam preparation bootcamp; Tutorial topic: Tangent vectors and tangent lines
- 12 May 2025 Lecture 28: Derivatives examples;
- 14 May 2025 Lecture 29: Vectors examples;
- 16 May 2025 Lecture 30:
Complex numbers examples;
- Week 11: Exam preparation bootcamp; Tutorial topic: Fundamental theorem of Calculus
- 19 May 2025 Lecture 31: Assorted examples;
- 21 May 2025 Lecture 32: Graphing examples;
- 23 May 2025 Lecture 33:
Vector examples;
- Week 12: Final week; Tutorial topic: Differential equations, initial value problems
- 26 May 2025 Lecture 34: Differential equations examples;
- 28 May 2025 Lecture 35: Blow your mind topics like 3d-space-time;
- 30 May 2025 Lecture 36: Last lecture;
Handwritten Lecture notes from Semester 2, 2019
- Week 1: Subsets and proofs
- 30 July 2019 Lecture 1:
Admin, Real Numbers, Set Descriptions;
handwritten lecture notes (pdf file)
- 31 July 2019 Lecture 2: Subsets, Proofs, Intersections, Products; handwritten lecture notes (pdf file).
- 02 August 2019 Lecture 3:
Inequalities, Monotone functions;
handwritten lecture notes (pdf file).
- Week 2: Irrationals, complex arithmetic
- 6 August July 2019 Lecture 4: Complex numbers, Addition, Multiplication, Conjugate; handwritten lecture notes (pdf file).
- 7 August 2019 Lecture 5: Properties of conjugate, Division, Exponential Polar Form; handwritten lecture notes (pdf file).
- 9 August 2019 Lecture 6:
Complex Exponential form with calculations;
handwritten lecture notes (pdf file).
- Week 3: Roots and factorization
- 13 August 2019 Lecture 7: Properties of modulus and argument, Trig functions in exponential form; handwritten lecture notes (pdf file).
- 14 August 2019 Lecture 8: Sketching sets, Powers and roots; handwritten lecture notes (pdf file).
- 16 August 2019 Lecture 9:
Factorising complex polynomials;
handwritten lecture notes (pdf file).
- Week 4: Functions
- 20 August 2019 Lecture 10: Factorising complex polynomials, functions; handwritten lecture notes (pdf file).
- 21 August 2019 Lecture 11: Images, bijectivity;
- 23 August 2019 Lecture 12:
Composition, inverses;
- Week 5: Vector arithmetic, length, etc
- 27 August 2019 Lecture 13: Inverses, Domain & range calculations, Absolute value;
- 28 August 2019 Lecture 14: Vector arithmetic & geometry, length; handwritten lecture notes (pdf file).
- 30 August 2019 Lecture 15:
Vectors as arrows, distance, unit vectors, basis vectors;
handwritten lecture notes (pdf file).
- Week 6: Angles, projections
- 3 September 2019 Lecture 16: Scalar product, angles; handwritten lecture notes (pdf file).
- 4 September 2019 Lecture 17: Projections;
- 6 September 2019 Lecture 18:
Parametric curves;
- Week 7: Graphing
- 10 September 2019 Lecture 19: Differentiation revision, linearity, product rule, quotient rule; handwritten lecture notes (pdf file).
- 11 September 2019 Lecture 20: Chain rule,Stationary points; handwritten lecture notes (pdf file).
- 13 September 2019 Lecture 21:
Concavity, Inflection Points;
handwritten lecture notes (pdf file).
- Week 8: Implicit differentiation and parametric curves
- 17 September 2019 Lecture 22: Asymptotes, Graph sketching, Implicit Differentiation; handwritten lecture notes (pdf file).
- 18 September 2019 Lecture 23: Inverses, Differentiating parametric curves; handwritten lecture notes (pdf file).
- 20 September 2019 Lecture 24:
Differentiating parametric curves, Projectile motion;
handwritten lecture notes (pdf file).
- Week 9: Integration
- 24 September 2019 Lecture 25: Integration revision, definite integrals;
- 25 September 2019 Lecture 26: Integration by substitution, linear substitutions; handwritten lecture notes (pdf file).
- 27 September 2019 Lecture 27:
NO LECTURE, AFL GRAND FINAL EVE;
- Week 10: Integration and Area
- 8 October 2019 Lecture 28: Integration by substitution, linear substitutions; Powers of trig functions, simple rational functions; handwritten lecture notes (pdf file).
- 9 October 2019 Lecture 29: Partial fractions; handwritten lecture notes (pdf file).
- 11 October 2019 Lecture 30:
Partial fractions, Areas between curves;
handwritten lecture notes (pdf file).
- Week 11: Differential equations
- Week 12: Revision
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.
- Topic 1: Numbers and sets
- Sets
- Inequalities
- Topic 1: Complex Numbers
- Complex numbers (pdf file)
- Complex arithmetic
- Modulus and argument
- Regions in the complex plane
- Complex exponential
- Finding complex solutions
- Powers of sin and cos
- Topic 2: Functions
- Basics of functions
- Inverse trigonometric functions
- Domain and range
- Topic 2: Vectors
- Vector algebra
- Position and length
- Dots and angles
- Vector geometry
- Scalar and vector projections
- Parametric curves
- Topic 3: Differential calculus
- second and higher derivatives
- Graph sketching
- Imiplicit differentiation
- Inverse trigonometric functions
- Applications of differentiation
- Differentiating parametric functions
- Topic 4: Integration (pdf file)
- Integration from standard integrals
- Integration by substitution
- Integration using trigonometric functions
- Integration by partial fractions
- Definite integrals
- Areas
- Topic 4: Differential Equations (pdf file)
- Verification of solutions
- Solving by direct antidifferentiation
- Separable differential equations
- Applications of differential equations
- Population models
- Newton's law of cooling