MAST10006 Calculus 2
The Complete Exam Bible · S1 2026MAST10006 Overview
- University of Melbourne
- S1 2026
- 8 chapters
- Science
Calculus 2 is the University of Melbourne's broad, procedure-heavy first-year analysis subject — limits & continuity, sequences & series, Taylor series, hyperbolic functions and the complex exponential, six techniques of integration, first- and second-order differential equations, and multivariable calculus.
- Assessed by Final written exam 60% · Mid-semester test 20% · Assignments (best 6 of 7) 20%
- Key terms Limit, L'Hopital's rule, Convergence test, Integrating factor
- How to prepare Treat MAST10006 as a set of technique chains you drill until they run on autopilot…
- Most asked Can I bring notes or a formula sheet into the MAST10006 exam?
How MAST10006 is assessed
| Component | Weight | Format |
|---|---|---|
| Final written exam | 60% | End of semester · 3 hours, closed-book, in person, no calculator · covers all content |
| Mid-semester test | 20% | In semester · 45 minutes |
| Assignments (best 6 of 7) | 20% | Across the semester — one free miss; confirm the exact dates in your subject guide |
What MAST10006 covers
Eight teaching blocks → one exam-ready map. Each links to its free chapter guide.
Limits and Continuity
Limit laws · squeeze theorem · continuity · L'HopitalA limit describes where a function is heading as the input closes in on a point — whether or not the function is even defined there.
- 1.1 The working definition — one-sided limits and existence
- 1.2 The limit laws (and why both pieces must exist first)
- Standard limits you must know cold
- The squeeze (sandwich) theorem
- Continuity — the three conditions and the failure modes
- L'Hopital's rule — verify the form before differentiating
Sequences and Series
Convergence · geometric · the convergence-test toolkitA sequence (an) is an ordered list; a series ∑an is what you get by adding them all up.
- 2.1 Limit of a sequence — laws and standard limits
- Monotone + bounded ⇒ converges (without finding the limit)
- 3.1 The geometric series and the p-series cutoff
- 3.2 The divergence (nth-term) test — do this first
- The convergence-test toolkit and the decision tree
- Absolute vs conditional convergence
Taylor Series
Maclaurin catalogue · building new series · remainder boundA Taylor polynomial is the best polynomial that agrees with f at a chosen centre x = a: it shares the same value, slope, curvature and all higher derivatives up to order n there.
- 4.1 The Taylor / Maclaurin polynomial and the AHA overlay
- The standard Maclaurin catalogue — six series to memorise
- Building new series by substitute / multiply / differentiate / integrate
- 4.3 The Lagrange remainder and the error bound
- The bounding recipe (maximise over the whole interval)
Hyperbolic Functions
sinh / cosh / tanh · log forms · Euler · complex-exp integrationSplit ex into its even and odd parts and you get the two hyperbolic functions: cosh x = (ex + e−x)/2 and sinh x = (ex − e−x)/2. They look like cosine and sine, obey almost the same identities — the master identity is cosh² x − sinh² x = 1, with a MINUS sign — and they appear whenever a √(x² + a²) shows up in an integral.
- 5.1 Definitions and the master identity cosh² − sinh² = 1
- Derivatives and integrals — like trig, minus the sign flips
- 5.4 Inverse hyperbolic functions and their log forms
- 5.5 Hyperbolic substitution for √(x² ± a²)
- 6.1 Euler's formula and the Argand diagram
- 6.2 The complex-exponential integration method
Integration Techniques
By parts (LIATE) · partial fractions · trig / hyperbolic sub · improperDifferentiation is mechanical; integration is pattern recognition. There is no single algorithm — you read the shape of the integrand and reach for the matching tool, and this chapter is the decision tree.
- 7.2 Integration by parts and the LIATE priority order
- Worked by-parts: power×exp, the lone log, return-of-integral
- 7.6 Partial fractions → logs and arctans
- 7.3 Trigonometric substitution
- 7.4-7.5 Hyperbolic substitution and powers of sin/cos/sinh/cosh
- Improper integrals — replace the bad point with a limit
First-Order Differential Equations
Separable · integrating factor · phase line · logisticA first-order ODE relates y and its rate of change y′. The chapter works through the standard solving methods and the qualitative tools, in roughly the order of difficulty.
- 8.1 Separable equations
- 8.2 Linear ODEs and the integrating-factor method
- 8.3 Substitution methods (homogeneous, Bernoulli)
- 8.4 The direction (slope) field
- Qualitative analysis and the phase line (stability)
- 8.5 Applications: logistic populations and mixing tanks
Second-Order Differential Equations
Characteristic equation · undetermined coefficients · springsA constant-coefficient second-order linear ODE a y′′ + b y′ + c y = g(x) is the workhorse of oscillation, circuits and the mass-spring system.
- 1 From the ODE to the auxiliary equation
- 2 The three cases of y_h (real distinct / repeated / complex)
- 3 The homogeneous procedure, one worked example per case
- 4 Particular solutions by undetermined coefficients
- 5 The resonance / modification rule
- 6-7 The mass-spring-damper and the damping regimes
Multivariable Calculus
Level curves · partials · gradient · Hessian test · double integralsA function z = f(x, y) takes a point in the plane and returns a height; its graph is a surface — a landscape of hills, valleys and passes. The whole chapter is the calculus of that landscape.
- 1-2 Surfaces, cross-sections and level-curve / contour maps
- 3 Limits and the path test
- 4-5 Continuity and partial derivatives (Clairaut)
- 6 The tangent plane (linear approximation)
- 7-8 The chain rule, gradient and directional derivative
- 9 Stationary points and the Hessian test
The final exam is 60% of your grade, closed-book, in person, with no calculator, covering the whole subject, so the marks come not just from the right answer but from choosing the right technique, justifying it (name the theorem, check its conditions) and using correct notation.
This guide teaches each method to exam standard: the trigger that tells you which tool to reach for, the worked procedure, and the trap examiners watch for.
Worked example · free
L'Hopital's rule — the signature limit, mark by mark
- +1Substitute first and name the form. At x = 0 the top is e0 − 1 − 0 = 0 and the bottom is 0, so the limit is the indeterminate form 0/0 — L'Hopital's rule is legal.
- +1Differentiate top and bottom separately: the limit becomes limx→0 (ex − 1) / (2x).
- +1Check the form again: at x = 0 this is still 0/0, so apply the rule a second time: limx→0 ex / 2.
- +1Evaluate: e0 / 2 = 1/2. State the answer.
Key terms
- Limit
- Where a function f(x) is heading as x closes in on a point a — whether or not f is even defined there. The two-sided limit exists if and only if the left and right one-sided limits agree.
- L'Hopital's rule
- For an indeterminate quotient 0/0 or ∞/∞, the limit of f/g equals the limit of f′/g′ — provided that right-hand limit exists. You must verify the indeterminate form before differentiating, and may iterate while it stays indeterminate.
- Convergence test
- A named rule (divergence, comparison, limit comparison, ratio, root, integral, alternating) for deciding whether an infinite series adds up to a finite total. The exam skill is reading the shape of the terms and choosing the test that resolves the series.
- Integrating factor
- The multiplier μ = e∫P dx that turns a linear first-order ODE y′ + P(x)y = Q(x) into an exact derivative d/dx(μy) = μQ, which then integrates directly.
- Hessian test
- For a function of two variables, the determinant D = fxxfyy − (fxy)2 at a stationary point classifies it: D > 0 with fxx > 0 is a local minimum, D > 0 with fxx < 0 a local maximum, D < 0 a saddle, and D = 0 inconclusive.
MAST10006 FAQ
Is MAST10006 hard?
It is broad and procedure-heavy rather than conceptually deep: limits, sequences/series, Taylor, hyperbolic functions, the complex exponential, six integration techniques, first- and second-order ODEs and two-variable calculus are all examined in one 3-hour closed-book paper.
The difficulty is doing the right method by hand, accurately, under time — so the safest preparation is drilling the worked procedures until they are automatic.
How is MAST10006 assessed?
The final exam is 60% of your grade: 3 hours, closed-book, in person, with no calculator, covering the whole subject. The remaining 40% is a mid-semester test (about 20%) and weekly assignments (about 20%, best 6 of 7 counting). Confirm this year's exact dates and weights in your subject guide.
Can I bring notes or a formula sheet into the MAST10006 exam?
The exam is closed-book and allows no calculator, but you may bring one double-sided A4 sheet of your own notes, and you are given the official Calculus 2 formula sheet. The smart play is to put on your A4 only what the provided sheet does not already give you — the decision rules for which test or substitution to use, your re-usable templates, and the trap reminders.
What is on the MAST10006 final exam?
Short- and long-answer questions across the whole subject: evaluating limits (often with L'Hopital), deciding series convergence with the right test, Taylor series and remainder bounds, hyperbolic and complex-exponential integrals, the six integration techniques, solving first- and second-order ODEs, and multivariable stationary-point classification with the Hessian. If a question specifies a method, you must use it.
Is using AskSia for MAST10006 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 MAST10006 chapter
- Does a limit existing mean the function is defined there?
- No — this is the central distinction of the chapter. A limit watches what happens near a, never at a, so it can exist where f(a) is undefined (a hole), and conversely f(a) can exist but differ from the limit. Sorting out exactly when limit equals value is the whole content of continuity. Chapter 01 · Limits and Continuity
- What is the hardest part of series questions?
- Choosing the right test. There is no single algorithm — you read the shape of an and match it to a test: factorials or exponentials point to the ratio test, an nth power to the root test, a polynomial ratio to limit comparison with a p-series, an (−1)n to the alternating test. The marks are in the justified choice plus the condition check, not the one-word verdict. Chapter 02 · Sequences and Series
- Do I have to derive each Maclaurin series from scratch in the exam?
- Almost never. The six catalogue series (ex, sin, cos, 1/(1−x), log(1+x), (1+x)α) are expected to be quoted, not re-derived. The exam then asks you to produce a series for a function not in the catalogue, which you do by substituting, multiplying, differentiating or integrating one of the six — far faster and far less error-prone than tabulating derivatives. Chapter 03 · Taylor Series
- Why does differentiating cosh not give a minus sign?
- Because cosh is built purely from ex and e−x, and differentiating real exponentials can never introduce a minus. So d/dx cosh x = +sinh x, unlike d/dx cos x = −sin x. Carrying the trig minus across to the hyperbolic case is the most-marked error in this chapter — it silently corrupts every hyperbolic integral. Chapter 04 · Hyperbolic Functions
- How do I decide which integration technique to use?
- Read the integrand first. A bare log or arctan with no other factor means parts with dv = dx. A polynomial times an exponential, sine or cosine means parts (or the tabular shortcut). A product of two cycling functions like ex sin x means parts twice and solve for I. A rational function means partial fractions. A square root of a quadratic means a trig or hyperbolic substitution. A single composite f(g)·g′ is a plain substitution, not parts. Chapter 05 · Integration Techniques
- How do I know which first-order method to use?
- Check the form. If the right side factorises into (function of x)(function of y), it is separable. If it can be written y′ + P(x)y = Q(x), it is linear — use the integrating factor. If it is neither but has a recognisable structure (depends only on y/x, or has the form F(ax+by+c), or is Bernoulli y′ + Py = Qyn), a substitution converts it to a separable or linear equation. Chapter 06 · First-Order Differential Equations
- How do I choose the form of the homogeneous solution?
- Compute the discriminant Δ = b² − 4ac of the auxiliary equation. Δ > 0 gives two real distinct roots and yh = A eλ₁x + B eλ₂x; Δ = 0 gives a repeated root and yh = (A + Bx)eλx; Δ < 0 gives complex roots α ± iβ and the real form eαx(A cos βx + B sin βx). The discriminant is the only decision; the procedure is otherwise identical. Chapter 07 · Second-Order Differential Equations
- How do I show a two-variable limit does not exist?
- Use the path test: find two paths to the point that give different limiting values — common choices are y = 0, y = x, y = mx and y = x². For xy/(x²+y²) at the origin, the path y = 0 gives 0 while y = x gives ½, so the limit does not exist. To CONFIRM a limit exists, agreeing along straight lines is not enough; switch to polar coordinates and show the limit as r → 0 is independent of the angle. Chapter 08 · Multivariable Calculus
How to study for the exam
Treat MAST10006 as a set of technique chains you drill until they run on autopilot, because under exam pressure you only do reliably what you have done many times. The recurring chains are: indeterminate form → name it, then L'Hopital; series?
→ read the shape, choose the test, check its conditions; √(x² ± a²) → trig or hyperbolic substitution; linear ODE → integrating factor; second-order ODE → discriminant decides the case; two-variable stationary point → Hessian determinant.
Because the exam is closed-book with no calculator, the examiner can only test whether you can execute the method by hand and justify it — so show every line, name every theorem, and use the past and sample exams (timed, by hand) as your highest-leverage practice while you build your one double-sided A4.
