MAST10006: pass the exams, not just read the notes
Your complete guide to University of Melbourne's calculus 2 unit. See where the marks are, work real practice questions, and study with an AI tutor that knows MAST10006.
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Worked example
Evaluate the limit lim as × -> 0 of (sin × − x) / x^3.
Check the form: as × -> 0, sin × − × -> 0 and x^3 -> 0, so this is the indeterminate form 0/0 and L'Hopital's rule applies.
Second application: (-sin x) / (6x). At × = 0 this is again 0/0, so apply L'Hopital a third time.
Third application: (-cos x) / 6. Now substitute × = 0 directly: (-cos 0)/6 = -1/6, a finite (non-indeterminate) value, so stop.
Therefore the limit equals -1/6 (option index 1). A numerical check confirms it: at × = 0.01 the quotient is -0.16666583..., approaching -1/6 = -0.16666....
The trap: Stopping too early or never checking the form. After the first L'Hopital step you still have (cos × − 1)/(3x²), which is 0/0, not a finite answer; differentiating once and reading off 0 gives the wrong value. You must re-verify the 0/0 form before each application and only substitute once the expression is no longer indeterminate. (Alternatively, the Maclaurin series sin × = × − x^3/6 + ... gives (sin × − x)/x^3 = -1/6 + ... -> -1/6 directly.) classic slip!
One exam decides 60% of your grade. Covers all MAST10006 content. If a question specifies a method you must use that method; otherwise any MAST10006 method is acceptable. This whole page is built around that.
Overview
What MAST10006 is, and where it sits
MAST10006 Calculus 2 is the University of Melbourne's core first-year calculus subject in the School of Mathematics and Statistics, and the foundation for further study in mathematics, statistics and the quantitative sciences. It extends VCE Specialist Mathematics and MAST10005 Calculus 1 into a single-variable-and-beyond toolkit: limits and continuity, sequences and series and their convergence tests, Taylor series, hyperbolic functions and the complex exponential, the full set of integration techniques, first and second-order differential equations with real applications, and a closing module on functions of two variables. The stated aim is not just computation but logical thinking and the ability to communicate a mathematical argument.
The subject runs at three 50-minute lectures plus one tutorial each week across the full semester (34 lectures), and it moves fast. The first half (Weeks 1 to 5) front-loads the theory-heavy material: the definition of a limit, L'Hopital's rule, the divergence, comparison and ratio tests for series, Taylor polynomials and the Lagrange remainder, hyperbolic identities and inverses, and Euler's formula. The second half (Weeks 6 to 12) is technique-dense and where most of the marks live: integration by parts and trigonometric/hyperbolic substitution and partial fractions, separable and linear (integrating-factor) first-order ODEs with population and mixing applications, constant-coefficient second-order ODEs with spring/oscillation applications, and finally partial derivatives, the gradient, stationary points classified by the Hessian, and double integrals.
It is assessed almost entirely under closed-book, no-calculator exam conditions, so fluency matters as much as understanding: 80% of the grade is the mid-semester test (20%) plus the final (60%), with a best-6-of-7 assignment stream making up the other 20%. There is no required textbook; comprehensive lecture slides, a provided formula sheet and exercise sheets are the source material. Note the local notation: natural log is written log (not ln), and inverse hyperbolics are written arcsinh and so on.
Difficulty & time commitment
Is MAST10006 hard, and how much time does it take?
MAST10006 is manageable if you keep a weekly rhythm and treat the back half as the main event. Across student reviews the pattern is consistent: it starts gently and steepens, and the heaviest assessment is the part that separates grades.
A read across student reviews and course feedback. See what students say ↓
The difficulty curve and the assessment weighting point the same way: the back half is harder and worth more. Front-loading effort there is the highest-return decision in the unit.
Is this unit for you
Who tends to do well, and who tends to struggle
You will likely do well if
- You did VCE Specialist Mathematics (or equivalent) well: your algebra, trigonometry, differentiation and basic integration are fluent and automatic, so the new material is the only thing you are learning.
- You work the exercise sheets and past papers by hand, closed-book and without a calculator, every week, rather than only reading solutions, because that is exactly how you are tested.
- You keep up with the fast first half (limits, series tests, Taylor, hyperbolics, the complex exponential) so the heavier technique-and-ODE second half has a foundation to stand on.
- You write full, justified solutions (state the rule, check its conditions, label your diagrams, use correct notation), since marks are awarded for method and communication, not just the final number.
You may struggle if
- Your algebra, trigonometry or integration is shaky; the unit moves too fast to repair foundations mid-semester and the no-calculator exams expose any gaps.
- You rely on a calculator or a formula lookup; both exams are closed-book and calculator-free with only one A4 page of notes, so everything has to be in your hands.
- You leave the integration techniques and the first and second-order ODEs to cram, even though that technique-dense block is where most of the final-exam marks sit.
- You skip stating rules and their conditions and skip labelling diagrams; full-working marks vanish and partial credit dries up even when your final answer is right.
- Drill until each standard move is reflexive: when to use the ratio versus comparison test, the integrating factor for a linear ODE, the resonance rule for second-order particular solutions, and the Hessian classification for stationary points.
- Build your one A4 double-sided notes page early and refine it as you go, so it is a genuine fast-lookup of formulas, standard limits/series and method checklists, not a wall of text you cannot navigate under time pressure.
- Sit every past and sample exam timed, closed-book and without a calculator, then mark yourself on method and justification, not just the answer key.
- Master the recurring traps: verify the indeterminate form before each L'Hopital step, remember the derivative of cosh has no minus sign, keep equilibrium solutions when separating an ODE, and normalise the direction vector before a directional derivative.
Syllabus
The 10 topics, topic by topic
The exam-weight marker on each topic shows where the marks concentrate. The amber topics carry the highest exam weight.
T1 · Limits and continuity of one variable
The definition of a limit, the limit toolkit and standard limits, the sandwich (squeeze) theorem, continuity and differentiability, and L'Hopital's rule for indeterminate forms.
T2 · Sequences
Convergence of a sequence, limit techniques and standard sequence limits (r^n, n^(1/n), (1+x/n)^n -> e^x), and the bridge theorem that lets you use L'Hopital on the continuous version.
T3 · Series and convergence tests
Geometric, harmonic and p-series, then the divergence, comparison, ratio, integral and alternating-series tests, with the recurring trap that limit of the terms equal to zero does not prove convergence.
T4 · Taylor series and polynomials
Taylor and Maclaurin polynomials, the standard Maclaurin series (e^x, sin x, cos x, 1/(1-x), log(1+x)), and the Lagrange remainder used to bound the approximation error.
T5 · Hyperbolic functions and inverses
Definitions and identities of sinh, cosh, tanh, their derivatives (note no minus sign on the derivative of cosh), and the inverse hyperbolic functions in log form with their key integrals.
T6 · Complex numbers and the complex exponential
Euler's formula, the complex exponential, and the canonical MAST10006 method of integrating products like e^(ax)cos(bx) by taking the real or imaginary part of an integral of e^((a+ib)x).
T7 · Techniques of integration
Derivative substitution, integration by parts (LIATE and recurrence), trigonometric and hyperbolic substitution, powers of hyperbolic functions, and partial fractions including long division for top-heavy integrands.
T8 · First-order differential equations
Separable equations, linear ODEs by the integrating-factor method, solving by substitution, qualitative (phase-line) analysis of autonomous equations, and applications to logistic population models and mixing problems.
T9 · Second-order linear ODEs and springs
Constant-coefficient homogeneous solutions by the characteristic equation, particular solutions by undetermined coefficients (with the resonance rule), and mass-spring-damper applications across the damping regimes.
T10 · Functions of two variables
Surfaces and level curves, two-variable limits and continuity (same value along every path), partial and directional derivatives, the gradient, tangent planes, the chain rule, stationary points classified by the Hessian, and double integrals over rectangles.
How it's assessed
Assessment structure
| Component | Weight | Format & timing |
|---|---|---|
| Assignments (best 6 of 7) | 20% | 7 assignments (5 written, submitted as a PDF in Canvas; 2 online in WebWork). Your best 6 count, each worth 3.3%. Written ones are marked on technique, accuracy, justification, diagrams and notation; online ones on correct answers only. Due 12pm Monday of Weeks 4, 5, 6, 7, 10, 11 and 12. Best 6 of 7 count, so one missed assignment does not affect the grade. Late submissions are not accepted (0 for a late or missing assignment). |
| Mid-semester test | 20% | In-person written test, 45 minutes writing time, no reading time. Closed book except 1 double-sided A4 page of notes; the Calculus 2 formula sheet is provided. No calculator. Pen or pencil. Held on campus during the lecture slot around Week 7 (S1 2026 sitting was Wednesday 22 April; the S2 date is set by the subject). Covers material up to and including integration of powers of hyperbolic functions. With successful special consideration the 20% can be re-weighted onto the final exam. |
| Final exam | 60% | In-person invigilated written exam, 15 minutes reading time plus 3 hours writing time. Closed book except 1 double-sided A4 page of notes; the Calculus 2 formula sheet is provided. No calculator. A mix of short-answer and long-answer questions. End-of-semester examination period (confirm the date, time and venue in my.unimelb). Covers all MAST10006 content. If a question specifies a method you must use that method; otherwise any MAST10006 method is acceptable. |
- Pass on a weighted average of at least 50% across the three components. No separate hurdle or attendance requirement is stated in the subject materials reviewed.
- Both the mid-semester test and the final are handwritten, closed-book (one A4 double-sided notes page allowed) and calculator-free. Marks are awarded for correct technique, accurate calculation, clear justification of rules and their conditions, well-labelled diagrams, and correct notation, not just the final answer, so full working is essential.
- Calculator policy: Calculators are NOT permitted in either the mid-semester test or the final exam. There is no formal requirement to own a calculator for the subject; the assessment tests concepts and procedures in simple cases. Scientific calculators are fine for some exercise-sheet questions only.
This is an exam-cram unit. With the exams at 80% of the grade and the final exam alone at 60%, your result is overwhelmingly decided by how well you perform under time pressure. Covers all MAST10006 content. If a question specifies a method you must use that method; otherwise any MAST10006 method is acceptable.
Final exam timing: approx mid-to-late November 2026 (S2 offering, confirm the date, time and venue against the official my.unimelb exam timetable). Confirm the exact date and venue on the official exam timetable.
How to actually pass it
A weekly rhythm, two checklists, and the traps to avoid
The unit rewards consistency over cramming, and practice over re-reading. Here is the loop that works, then what to have nailed before each exam.
The weekly loop
Before the mid-semester checklist
- The mid-semester test covers up to and including integration of powers of hyperbolic functions: drill limits and L'Hopital, the sequence and series convergence tests, Taylor polynomials and the remainder, hyperbolic identities and derivatives, the complex-exponential integration method, and the integration techniques.
- Practise the 45-minute test closed-book with only your A4 page and no calculator, since the real constraint is speed under those exact conditions.
- Make sure the standard limits, standard Maclaurin series and hyperbolic identities are memorised or on your notes page, because there is no time to re-derive them.
- Sit the provided practice mid-semester test and check your working against the official solutions.
Before the final heaviest topics
- Prioritise the second-half technique block, where most marks sit: integration by parts and trig/hyperbolic substitution and partial fractions, first-order ODEs (separable, integrating factor, qualitative analysis, population and mixing models), second-order ODEs and springs, and the multivariable module.
- Work all available past and sample exams timed (3 hours), closed-book and calculator-free, then review method, not just the answer.
- Rehearse the full ODE workflows end to end: classify, choose the method, solve, apply initial conditions, and interpret the application (carrying capacity, damping regime, steady state).
- Drill the multivariable finale: stationary points solved as a system then classified by the Hessian determinant D, directional derivatives with a normalised direction, tangent planes, and double integrals over rectangles.
- Practise writing solutions that state each rule, check its conditions and label diagrams, because marks are awarded for justification and notation.
The mistakes that cost marks
Not re-checking the indeterminate form before each L'Hopital step. L'Hopital only applies to 0/0 or infinity/infinity. Students apply it once and read off a finite-looking expression that is still 0/0, or apply it to a form that was never indeterminate (a marked error). Verify the form before every application and stop only when the expression is no longer indeterminate.
Treating the assignments as where the marks are. The assignments are only 20% combined (best 6 of 7), while the two closed-book exams are 80%. Polishing assignment marks while neglecting timed, no-calculator exam practice misreads where the grade actually lives.
Losing the cushion by skipping setup steps. Forgetting to keep equilibrium solutions when separating an ODE, dropping the +C inside the integrating factor, omitting the resonance x-factor when the trial particular solution duplicates a homogeneous one, or forgetting the derivative of cosh has no minus sign. Each is a small slip that cascades into a wrong final answer and lost method marks.
Relying on a calculator you will not have. Both the test and the final are calculator-free with only one A4 notes page. If your practice uses a calculator or constant formula lookup, exam conditions will feel much harder than they should. Train under the real constraints from week one.
Teaching team
Who teaches MAST10006
The bios below are factual. The star ratings are not ours: they are impressions from students who have taken the unit, so you can hear from people who sat in the lectures.
Dr Alba Santin Garcia
Coordinates and lectures MAST10006 Calculus 2 in the School of Mathematics and Statistics, University of Melbourne.
Prof Mark Holmes
Professor in the School of Mathematics and Statistics, University of Melbourne, and a lecturer on Calculus 2.
Dr Paul Keeler
Lecturer in the School of Mathematics and Statistics, University of Melbourne, teaching Calculus 2.
Dr Gabor Somlai
Lecturer in the School of Mathematics and Statistics, University of Melbourne, teaching Calculus 2.
Teaching team as listed in the unit materials reviewed. AskSia does not rate lecturers; star ratings are submitted by students who have taken MAST10006.
Formula & concept sheet
The vocabulary and formulas you must own
- L'Hopital's rule
- If lim f/g is the indeterminate form 0/0 or infinity/infinity (and g' is nonzero near a), then lim f/g = lim f'/g', provided the right-hand side exists. Re-check the form before each application.
- Sandwich (squeeze) theorem
- If g(x) <= f(x) <= h(x) near a and lim g = lim h = L, then lim f = L. Used to handle bounded-times-vanishing terms such as × sin(1/x) -> 0.
- Ratio test
- With L = lim |a_(n+1)/a_n|: the series converges absolutely if L < 1, diverges if L > 1, and is inconclusive if L = 1. Best for factorials and exponentials.
- Standard Maclaurin series
- e^x = sum x^n/n!, sin × = sum (-1)^n x^(2n+1)/(2n+1)!, cos × = sum (-1)^n x^(2n)/(2n)!, 1/(1-x) = sum x^n (|x|<1), log(1+x) = sum (-1)^(n+1) x^n/n (|x|<1). New series are built by substituting into these, not differentiating from scratch.
- Hyperbolic functions
- sinh × = (e^x − e^(-x))/2, cosh × = (e^x + e^(-x))/2; identity cosh^2 × − sinh^2 × = 1; derivatives d/dx sinh × = cosh × and d/dx cosh × = +sinh × (no minus sign).
- Euler's formula and complex exponential
- e^(i.theta) = cos theta + i sin theta. To integrate e^(ax)cos(bx) or e^(ax)sin(bx), integrate e^((a+ib)x) and take the real or imaginary part, rationalising 1/(a+ib) = (a-ib)/(a^2+b^2).
- Integrating factor (linear first-order ODE)
- For dy/dx + P(x) y = Q(x), use mu(x) = e^(integral of P dx) (no +C in the exponent). Then d/dx (mu y) = mu Q, so y = (1/mu)(integral of mu Q dx + C).
- Logistic population model
- dP/dt = rP(1 − P/K) with carrying capacity K; with constant harvesting, dP/dt = rP(1 − P/K) − h. Equilibria solve the right-hand side equal to zero; stability comes from the sign of the derivative there.
- Second-order constant-coefficient ODE
- For a y'' + b y' + c y = 0 the auxiliary equation a.lambda^2 + b.lambda + c = 0 gives: real distinct roots -> A e^(l1 x) + B e^(l2 x); repeated root -> (A + Bx) e^(l x); complex alpha +/- i.beta -> e^(alpha x)(A cos beta × + B sin beta x).
- Stationary points and the Hessian test
- Solve f_x = 0 and f_y = 0 together; with D = f_xx f_yy − (f_xy)^2: D>0 and f_xx>0 is a local minimum, D>0 and f_xx<0 a local maximum, D<0 a saddle, D=0 inconclusive.
- Directional derivative and gradient
- grad f = (f_x, f_y); the directional derivative is grad f dotted with a UNIT vector u-hat. grad f points in the direction of steepest ascent, the maximum rate is |grad f|, and grad f is perpendicular to level curves.
- Double integral over a rectangle
- Over R = [a,b] × [c,d], the double integral equals an iterated integral (Fubini), doing the inner integral first with the outer variable held constant. For separable f(x,y) = g(x)h(y) it factorises into the product of two single integrals.
Common acronyms: ODE · IC · MST · SHM · LIATE · VCE.
What students say
What students actually say about MAST10006
Recurring themes from student reviews, paraphrased in our own words.
- Widely described as content-heavy and fast: three lectures a week and a long list of techniques to master, with the second half (integration and differential equations) seen as the hardest stretch.
- Manageable for students with a strong Specialist Mathematics foundation; much harder for anyone whose algebra, trigonometry or integration is rusty.
- The closed-book, no-calculator exams are a recurring talking point: fluency and the one allowed A4 notes page matter a lot.
- Heavy reliance on working the exercise sheets and past/sample exams by hand, since there is no set textbook and the exams mirror that practice.
- Students value worked-solution walkthroughs of the harder ODE and integration questions, and build their A4 notes page as a study artefact in its own right.
- Demand for timed, closed-book practice and for step-by-step explanations of the recurring exam techniques (L'Hopital, convergence tests, integrating factors, second-order ODEs, the Hessian).
Recurring student opinions, paraphrased and aggregated, not official course information.
Set texts
The prescribed reading
The syllabus references map straight onto these.
MAST10006 lecture slides, Calculus 2 formula sheet and exercise sheets (provided on Canvas)
School of Mathematics and Statistics, University of Melbourne.
Where it fits
Prerequisites, related units & why it matters
Prerequisite: a study score of at least 29 in VCE Specialist Mathematics (or equivalent), or completion of MAST10005 Calculus 1. You may not enrol in Calculus 1 and Calculus 2 concurrently. Credit exclusion: you can gain credit for only one of MAST10006 Calculus 2, MAST10009 Accelerated Mathematics 2, MAST10019 Calculus Extension Studies and MAST10021 Calculus 2: Advanced.
Your MAST10006 study toolkit
Study the unit with Sia, not just read about it
Each tool already knows MAST10006: your syllabus, your texts, and where the marks are. Grouped by how you study, from first contact to exam week.
FAQ
Frequently asked questions
Is MAST10006 hard?
It is one of the harder first-year subjects, mostly because 80% of the grade is two closed-book, no-calculator written assessments (a 20% mid-semester test and a 60% final) and the content moves quickly across three lectures a week. With a solid VCE Specialist Mathematics background (the study-score-29 prerequisite is a real filter) and consistent weekly practice on the exercise sheets it is very manageable, but it is unforgiving of weak algebra, trigonometry and integration fluency.
How is MAST10006 assessed?
Three components: assignments worth 20% (7 assignments, 5 written and 2 online in WebWork, best 6 of 7 counting at 3.3% each), a 20% in-person mid-semester test (45 minutes), and a 60% in-person final exam (3 hours writing plus 15 minutes reading). You pass on a weighted average of at least 50%, with no separate hurdle stated in the materials reviewed.
Can I use a calculator in the exam?
No. Calculators are not permitted in either the mid-semester test or the final exam. Both are closed-book apart from one double-sided A4 page of your own notes (handwritten or printed), and the official Calculus 2 formula sheet is provided with the paper. There is no requirement to own a calculator for the subject.
Is there a textbook?
No required textbook. The subject runs entirely off comprehensive lecture slides, a provided formula sheet and exercise sheets (with solutions). A few suggested reference texts are listed on Canvas but none is needed to do well; the lecture material is self-contained.
What topics does MAST10006 cover?
Limits and continuity (including L'Hopital), sequences and series with their convergence tests, Taylor series, hyperbolic functions and their inverses, the complex exponential, the full set of integration techniques, first-order differential equations (separable, integrating factor, qualitative analysis, population and mixing models), second-order constant-coefficient ODEs with spring applications, and functions of two variables (partial derivatives, the gradient, stationary points via the Hessian, and double integrals).
What is the difference between MAST10006, MAST10009 and MAST10021?
MAST10006 Calculus 2 is the standard stream. MAST10009 Accelerated Mathematics 2 and MAST10021 Calculus 2: Advanced are the accelerated and advanced streams covering related material at a higher level or pace, and MAST10019 is the Calculus Extension Studies stream. They share a credit exclusion, so you can gain credit for only one of the four.
What's the best way to study for the exam?
Work the exercise sheets and past/sample exams by hand under closed-book, no-calculator conditions, because that is exactly how you are tested. Aim to have attempted every problem-booklet question before the exam, and build your one allowed A4 notes page early so you know exactly what is on it. Because marks reward justification and notation, practise writing full solutions (stating the rule, checking its conditions, labelling diagrams), not just getting to the answer.
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