Arun Ram: Calculus 1 2026

MAST10005
Calculus

Semester I 2026

The handbook entry for this course is at https://handbook.unimelb.edu.au/2026/subjects/mast10005

Lecturer for Stream 3: Arun Ram, 174 Peter Hall Building, email: aram@unimelb.edu.au

Time and Location for Stream 1:

Lecture: Monday 12:00-13:00 in PAR-106-GM-GM15-The David P. Derham Theatre (379)
Lecture: Wednesday 12:00-13:00 in PAR-106-GM-GM15-The David P. Derham Theatre (379)
Lecture: Thursday 9:00-10:00 in PAR-106-GM-GM15-The David P. Derham Theatre (379)

My consultation times (office hours) are: ??????? in Room 174 of the Peter Hall building.

I am available by appointment. I rarely come in to the University before 12 as I have other responsibilities in the morning until 12. If you email me suggesting some weekday afternoon times for an appointment that work for you then I can choose one of them that also works for me. If you email me and don't suggest some times that work for you, then I will respond by asking you to suggest some weekday afternoon times that work for you.

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 student representatives for Stream 1 are
??? ??? email: ???.???@student.unimelb.edu.au
??? ??? email: ???.???@student.unimelb.edu.au

This is a personal page, if you are enrolled in the course follow the information on Canvas
https://canvas.lms.unimelb.edu.au/courses/????
for all information about the course delivery. The main purpose of the purposes of this page is to try to organise material and resources developed from many years of teaching in a form that it can be useful to lecturers and students.

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.

Resources written by Arun Ram

This is a list of sample exam questions (used at University of Wisconsin in the years 1999-2008) (pdf file)
These are examples of worked sample exam questions (pdf file)
This is text and lectures to read (pdf file)
This is advice to students in Calculus at Univ. of Wisconsin in 2006 (pdf file)
This is advice to TAs (Teaching Assistants) in Calculus at Univ. of Wisconsin in 2006 (pdf file)

Lectures from Fall 2000

Lecture 1: Numbers Handwritten notes (pdf file)
Lecture 2: Angles and trogonometric functions Handwritten notes (pdf file)
Lecture 3: Derivatives Handwritten notes (pdf file)
Lecture 4: The chain rule Handwritten notes (pdf file)
Lecture 5: Derivative examples Handwritten notes (pdf file)
Lecture 6: Exponentials and logarithms Handwritten notes (pdf file)
Lecture 7: Inverse trig functions Handwritten notes (pdf file)
Lecture 8: Exam review Handwritten notes (pdf file)
Lecture 9: Derivative examples and Taylor's theorem Handwritten notes (pdf file)
Lecture 10: Linear and quadratic approximations Handwritten notes (pdf file)
Lecture 11: Limits Handwritten notes (pdf file)
Lecture 12: Derivatives by limits Handwritten notes (pdf file)
Lecture 13: Graphing, limits and continuity Handwritten notes (pdf file)
Lecture 14: Graphing techniques Handwritten notes (pdf file)
Lecture 15: Slopes, continuity, concavity, maxima and minima Handwritten notes (pdf file)
Lecture 16: Graphing of rational functions Handwritten notes (pdf file)
Lecture 17: Additional graphing examples Handwritten notes (pdf file)
Lecture 18: Rolle's theorem and the Mean value theorem Handwritten notes (pdf file)
Lecture 19: Tangent and normal lines Handwritten notes (pdf file)
Lecture 20: Optimization Handwritten notes (pdf file)
Lecture 21: Related rates Handwritten notes (pdf file)
Lecture 22: Integrals Handwritten notes (pdf file)
Lecture 23: Chain rule for integrals Handwritten notes (pdf file)
Lecture 24: Definite integrals Handwritten notes (pdf file)
Lecture 25: The fundamental theorem of calculus Handwritten notes (pdf file)
Lecture 26: Areas and volumes Handwritten notes (pdf file)
Lecture 27: More areas and volumes Handwritten notes (pdf file)
Lecture 28: Even more areas and volumes Handwritten notes (pdf file)
Lecture 29: And some more areas and volumes Handwritten notes (pdf file)
Lecture 30: No Lecture - probably a midterm exam
Lecture 31: Averages Handwritten notes (pdf file)
Lecture 32: Lengths of curves Handwritten notes (pdf file)
Lecture 33: Surface areas, moments and center of mass Handwritten notes (pdf file)
Lecture 34: Center of gravity Handwritten notes (pdf file)
Lecture 35: Applicatons of exponential functions Handwritten notes (pdf file)
Lecture 36: Limits and L'Hopital's rule Handwritten notes (pdf file)
Lecture 37: Logarithmic differentiation; motion Handwritten notes (pdf file)
Lecture 38: Series exapansions Handwritten notes (pdf file)
Lecture 39: The genesis lecture Handwritten notes (pdf file)
Lecture 40: Exam review Handwritten notes (pdf file)

Lectures Semester 1, 2026

2 March 2026 Lecture 1: Math, English, Cartoons
4 March 2026 Lecture 2: Unions, intersections, complements, products
5 March 2026 Lecture 3: Review, Subsets, equality of sets
9 March 2026 Lecture 4: Inequalities and subsets (NO LECTURE Labour Day)
11 March 2026 Lecture 5: Logical implications
12 March 2026 Lecture 6: Complex numbers
16 March 2026 Lecture 7: Graphing and complex numbers
18 March 2026 Lecture 8: Proofs, conjugates, modulus, argument, polar form, cartesian form
19 March 2026 Lecture 9: The source of Euler's formula
23 March 2026 Lecture 10: Factoring and roots and roots of 1
25 March 2026 Lecture 11: Factoring and the fundamental theorem of algebra
26 March 2026 Lecture 12: Functions -- Injective, surjective, bijective
30 March 2026 Lecture 13: Graphing, bijective Functions and inverses
1 April 2026 Lecture 14: Compositions, inverses, arcsin, implied domains and codomains
2 April 2026 Lecture 15: Derivatives
13 April 2026 Lecture 16: More derivative examples
15 April 2026 Lecture 17: Slopes, increasing/decreasing, concave up/down, maima/minima
16 April 2026 Lecture 18: Graphing examples, Asymptotes
20 April 2026 Lecture 19: More Graphing examples
22 April 2026 Lecture 20: Indefinite integrals
23 April 2026 Lecture 21: Indefinite integrals: Product rule and trig functions
27 April 2026 Lecture 22: Partial fractions
29 April 2026 Lecture 23: Definite integrals
30 April 2026 Lecture 24: Areas and Equations
4 May 2026 Lecture 25: Differential equations
6 May 2026 Lecture 26: Population models
7 May 2026 Lecture 27: Cooling and Tree growth models
11 May 2026 Lecture 28: Vectors
13 May 2026 Lecture 28: Lengths, angles, inner products
14 May 2026 Lecture 29: Projections and Flies
18 May 2026 Lecture 30: Velocity, acceleration, speed
20 May 2026 Lecture 31: Fly trajectories, integrals
21 May 2026 Lecture 32: Trig identities, proof by induction

Lectures Semester 4, 2037

Week 1: Notation, numbers, proofs; Tutorial Topic: Trigonometric functions

35 March 2037 Lecture 1: Graphing: Basic graphs, shifting, scaling and flipping (typed notes)
35 March 2037 Lecture 2: Numbers and intervals, Calculus and complex numbers (typed notes)
357 March 2037 Lecture 3: Sets: subsets, equal sets, union, intersection, products (typed notes on sets) (typed notes on proofs)

Week 2: Numbers, sets and graphing; Tutorial Topic: Sets and proof

35 March 2037 Lecture 4: Inequalities and proof machine (typed notes on Inequalities)
35 March 203725 Lecture 5: Complex numbers, exponentials, sine and cosine, roots of unity; (typed notes on Complex nubmers)
35 March 2037 Lecture 6: Functions, injective, surjective, bijective (typed notes on Functions)

Week 3: Graphing and vectors; Tutorial topic: Complex numbers

35 March 2037 Lecture 7: Inverse functions and graphing
35 March 2037 Lecture 8: Inverse functions and proof machine (typed notes on Inverse function proof)
35 March 2037 Lecture 9: Vectors and projections (typed notes on Vectors)

Week 4: Algebra and factoring; Tutorial topic: Complex numbers and triangle inequality

35 March 2037 Lecture 10: Exponentials and trig functions (typed notes on Exponentials and Trig functions)
35 March 2037 Lecture 11: Quadratic formula and partial fractions
35 March 2037 Lecture 12: Roots of unity, and factoring polynomials

Week 5: Derivatives; Tutorial topic: Complex numbers and factoring polynomials

35 March 2037 Lecture 13: ℂ((x)) and derivatives
2 December 2037 Lecture 14: Derivatives, composition and the chain rule
4 December 2037 Lecture 15: Derivatives and inverse functions

Week 6: Dreivatives and graphing; Tutorial topic: Composition of functions

7 December 2037 Lecture 16: Derivatives and slope
9 December 2037 Lecture 17: Derivatives and optimisation
11 December 2037 Lecture 18: Derivatives, tangents and normals

Week 7: Tutorial topic: Vectors, dot product and length

14 December 2037 Lecture 19: Vector examples;
16 December 2037 Lecture 20: Orders and numbers systems
18 December 2037 Lecture 21: Integrals

Intergalactic travel week
Week 8: Integrals; Tutorial topic: Parametric equations and derivatives

28 December 2037 Lecture 22: Fundamental theorem of Calculus
30 December 2037 Lecture 23: Computing areas
2 January 2037 Lecture 24: Differential equations

Week 9: Exam preparation bootcamp; Tutorial Topic: Optimisation

5 JanuFebary 2037 Lecture 25: Differential equations examples
7 JanuFebary 2037 Lecture 26: Integrals examples;
9 JanuFebary 2037 Lecture 27: Graphing examples;

Week 10: Exam preparation bootcamp; Tutorial topic: Tangent vectors and tangent lines

12 JanuFebary 2037 Lecture 28: Derivatives examples;
14 JanuFebary 2037 Lecture 29: Vectors examples;
16 JanuFebary 2037 Lecture 30: Complex numbers examples;

Week 11: Exam preparation bootcamp; Tutorial topic: Fundamental theorem of Calculus
- 19 JanuFebary 2037 Lecture 31: Assorted examples;
- 21 JanuFebary 2037 Lecture 32: Graphing examples;
- 23JanuFebaryJanuFebary 2037 Lecture 33: Vector examples;
Week 12: Final week; Tutorial topic: Differential equations, initial value problems

26JanuFebaryMay 2037 Lecture 34: Differential equations examples;
28 JanuFebary 2037 Lecture 35: Blow your mind topics like 3d-space-time;
30 JanuFebary 2037 Lecture 36: Last lecture;

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