MAT9004 Mathematical Foundations for Data Science and AI
The Complete Exam Bible · S1 2026MAT9004 Overview
- Monash University
- S1 2026
- 6 chapters
- Information Technology
Mathematical Foundations for Data Science and AI is the mathematical toolkit a data-science or AI degree assumes — six disjoint maths worlds in one paper: single-variable calculus, linear algebra, multivariable optimisation, combinatorics, probability & Bayes, and graph theory.
- Assessed by Final exam 60% · Applied-class quizzes 20% · Assignments 20% — Final exam is a hurdle
- Key terms Stationary point, Gaussian elimination, Gradient, Bayes' theorem
- How to prepare Because the exam is closed-book, no calculator, and a 60% hurdle that samples all six worlds…
- Most asked What is on the MAT9004 final exam?
How MAT9004 is assessed
| Component | Weight | Format |
|---|---|---|
| Final exam · hurdle | 60% | Closed book · no calculator · formula sheet provided · must score ≥45% on it to pass the unit (hurdle) |
| Applied-class quizzes | 20% | Five short quizzes across the semester — confirm the exact schedule in your unit guide |
| Assignments | 20% | Two assignments submitted across the semester — confirm the exact split in your unit guide |
What MAT9004 covers
Six examined topics → one exam-ready map. Each links to its free chapter guide.
Calculus
Functions · limits · derivative rules · stationary points · integrationSingle-variable calculus is the largest single block of the MAT9004 exam, and it is pure technique: take a function, apply the right rule, give the exact value.
- 1.1 Functions and the function types you must recognise
- 1.2 Fitting data: log–log and semi-log plots
- 1.3 Limits — stated, not laboured
- 1.4 The derivative as slope and rate of change
- 1.5 All the differentiation rules (power, product, quotient, chain)
- 1.6 Tangent lines
Linear Algebra
Vectors · matrices · Gaussian elimination · determinants · eigenvaluesLinear algebra is the algebra of data — the maths that lets you handle many numbers at once.
- 2.1 Vectors: notation and arithmetic
- 2.2 Length and orthogonality
- 2.3 Projection: the shadow of one vector on another
- 2.4 Linear independence
- 2.5 Matrices and matrix multiplication
- 2.6 Systems Ax = b solved by Gaussian elimination
Multivariable Calculus
Partial derivatives · the gradient · the Hessian · two-variable optimisationMultivariable calculus is single-variable calculus done in two variables at once — the maths behind optimising a model with more than one parameter.
- 3.1 Functions of two variables and their surfaces
- 3.2 Level sets and contour maps
- 3.3 Partial derivatives
- 3.4 Second-order and cross-partials (Clairaut)
- 3.5 The gradient ∇f — direction of steepest ascent
- 3.6 First-order approximation and the tangent plane
Combinatorics
Product & sum rules · the four selection types · binomial · inclusion-exclusionCombinatorics is counting without listing — working out how many ways something can happen without writing them all out.
- 4.1 The three fundamental counting rules (product, sum, complement)
- 4.2 The four selection types — the core of the topic
- 4.3 One worked example for each cell of the table
- 4.4 The decision table — the biggest exam trap
- 4.5 A multi-rule worked count
- 4.6 Binomial coefficients and Pascal’s triangle
Probability and Bayes
Events & axioms · conditional probability · Bayes' theorem · random variablesProbability is the language of uncertainty — and of machine learning — and MAT9004 emphasises it.
- 5.1 Probability spaces, events and the axioms
- 5.2 Equally-likely outcomes: counting becomes probability
- 5.3 Complement, union, intersection and mutually exclusive
- 5.4 Conditional probability and independence
- 5.5 Law of total probability and Bayes’ theorem
- 5.6–5.7 The signature two-stage Bayes example and the base-rate trap
Graph Theory
Degrees & handshaking · trees · adjacency matrices · Euler & HamiltonGraph theory is the maths of connection — networks of vertices and edges that model everything from social networks to dependencies in a computation.
- 6.1 What is a graph? Vertices and edges
- 6.2 Degree and the handshaking lemma Σ deg(v) = 2|E|
- 6.3 Walks, paths, cycles and connectivity
- 6.4 Trees and spanning trees
- 6.5 Adjacency matrices and counting walks with Aᵏ
- 6.6 Reading A and counting walks (worked)
They don't build on one another the way a normal maths unit does, so you can't coast on a single strength. The final exam is 60% of your grade and a hurdle — you must score at least 45% on the exam itself to pass the unit, whatever your assignment marks.
It is closed-book with no calculator, but a formula sheet is provided, so the exam tests one thing: can you execute each method by hand on fresh numbers. This guide teaches every examined technique to that standard — the definition stated plainly, the method on a worked example, and the trap that loses marks.
MAT9004 cheatsheet for exam revision
AskSia publishes a free MAT9004 cheatsheet for exam revision, an exam cheat sheet that follows the official permitted-materials rule for this assessment and condenses the topics in this guide. Open the MAT9004 cheatsheet.
Worked example · free
Stationary points & the second-derivative test — the calculus staple, step by step
- +1Differentiate: f′(x) = 3x2 − 12x + 9.
- +1Set f′(x) = 0: 3x2 − 12x + 9 = 0 ⇒ x2 − 4x + 3 = 0 ⇒ (x − 1)(x − 3) = 0, so x = 1 and x = 3.
- +1Second derivative: f″(x) = 6x − 12.
- +1Classify x = 1: f″(1) = −6 < 0 ⇒ local maximum. The value is f(1) = 6.
- +1Classify x = 3: f″(3) = +6 > 0 ⇒ local minimum. The value is f(3) = 2.
Key terms
- Stationary point
- A point where the first derivative is zero, f′(x) = 0 — a candidate for a local maximum, local minimum, or a point of inflection. The second-derivative test then settles which it is.
- Gaussian elimination
- The row-reduction method for solving a linear system Ax = b: use elementary row operations to reach an upper-triangular form, then back-substitute. The flagship by-hand technique of the linear-algebra block.
- Gradient
- The vector of first partial derivatives, ∇f = (∂f/∂x, ∂f/∂y). It points in the direction of steepest ascent, and setting ∇f = 0 locates the stationary points of a two-variable function.
- Bayes' theorem
- The rule for reversing a conditional probability: P(A|B) = P(B|A)P(A) / P(B). It is the signature long-answer of the probability block, usually set up with the law of total probability in the denominator.
- Handshaking lemma
- In any graph the sum of the vertex degrees equals twice the number of edges, Σ deg(v) = 2|E|. A direct corollary: every graph has an even number of odd-degree vertices.
MAT9004 FAQ
Is MAT9004 hard?
It is broad rather than deep: six disjoint maths worlds — calculus, linear algebra, multivariable optimisation, combinatorics, probability and graphs — in one exam. The difficulty is coverage and speed: each topic is procedural, but you must execute the method by hand, with no calculator, across all six. You can't coast on one strength, so the trap is leaving a whole world under-revised.
How is MAT9004 assessed?
The final exam is 60% of the unit mark and a hurdle — you must score at least 45% on the exam itself to pass, whatever your assignment marks. The remaining 40% is continuous assessment: applied-class quizzes (about 20%) and assignments (about 20%) across the semester. Confirm this year's exact split and schedule in your unit guide.
What is on the MAT9004 final exam?
All six examined topics: single-variable calculus (derivatives, stationary points, integration), linear algebra (vectors, Gaussian elimination, determinants, eigenvalues), multivariable optimisation (gradient and Hessian), combinatorics (counting rules, binomial, inclusion-exclusion), probability and Bayes (conditional probability, the law of total probability, random variables), and graph theory (degrees, trees, adjacency matrices).
The exam is closed-book with no calculator, but a formula sheet is supplied inside the paper.
Do I need a strong maths background for MAT9004?
It is a foundation unit, so it starts from first principles, but it moves fast and assumes comfort with school-level algebra and functions. The work is procedural rather than proof-heavy: the marks reward applying the right technique cleanly by hand, line by line, not abstract derivation.
Is using AskSia for MAT9004 cheating?
No. AskSia is a study reference written in our own words — we host none of your lecturer's files, and Sia teaches you the method to earn the marks; it does not complete or sit your assessments.
Questions students ask in each MAT9004 chapter
- Which differentiation rule do I reach for first?
- Read the structure of the function before differentiating. A product of two x-terms (like x·e2x) needs the product rule; a quotient needs the quotient rule; anything 'inside' something else (like e2x or (3x+1)5) needs the chain rule for the inner derivative. Most marks are lost by jumping in — name the structure first, then apply the matching rule, and remember the chain rule's inner derivative. Chapter 01 · Calculus
- What's the reliable way to solve Ax = b by hand?
- Write the augmented matrix [A | b] and row-reduce to upper-triangular (echelon) form using elementary row operations, then back-substitute from the bottom row up. Work one column at a time, creating zeros below each pivot. Show every row operation — method marks are awarded for the reduction even if a number slips. Chapter 02 · Linear Algebra
- How do I take a partial derivative correctly?
- Differentiate with respect to the named variable and treat every other variable as a constant. For fx, treat y as a fixed number, so a pure-y term like 3y2 differentiates to zero. The most common slip is forgetting to hold the other variable constant, or dropping a term that survives. Chapter 03 · Multivariable Calculus
- How do I pick the right counting formula?
- Answer two questions in order. First: does order matter? If yes you are permuting, if no you are combining. Second: can items repeat? If yes use the with-repetition formula, if no the without-repetition one. Those two answers point to exactly one cell of the four-selection-types table — choosing the wrong cell is the single biggest source of lost marks. Chapter 04 · Combinatorics
- What's the difference between mutually exclusive and independent?
- Mutually exclusive means the two events cannot both happen, so P(A ∩ B) = 0. Independent means one happening tells you nothing about the other, so P(A ∩ B) = P(A)P(B). They are almost opposites: two events with positive probability that are mutually exclusive are necessarily dependent, because if one occurs the other definitely does not. Mixing these up is a classic exam slip. Chapter 05 · Probability and Bayes
- What is the handshaking lemma good for?
- It links the degrees to the edge count: Σ deg(v) = 2|E|. Use it to count edges from a degree sequence (sum the degrees, halve it) and to rule out impossible graphs — if the degree sum is odd, no such graph exists, because 2|E| is always even. That impossibility argument is a recurring exam question. Chapter 06 · Graph Theory
How to study for the exam
Because the exam is closed-book, no calculator, and a 60% hurdle that samples all six worlds, the winning move is to drill the recurring procedural chains until they are automatic: differentiate → set f′ = 0 → classify with f″; row-reduce → back-substitute; ∇f = 0 → Hessian test; condition → Bayes; degree sequence → count edges.
Don't cram what the formula sheet already gives you — standard derivative tables, counting formulas and distribution facts are supplied — spend the time instead on executing the methods cold on fresh numbers. Show every line: method marks are real, and they are the safest marks when you can't reach for a calculator.
Above all, spread your revision so no single world is left blank, because the hurdle is on the whole paper.
