MTH1020 Analysis of Change
The Complete Exam Bible · S2 2026MTH1020 Overview
- Monash University
- S2 2026
- 12 chapters
- Mathematics
MTH1020 Analysis of Change is Monash University's first-year, 6-credit-point unit in the School of Mathematics at Clayton, and it packs a full first-year mathematics diet into twelve teaching weeks.
- Assessed by Applied-class quizzes 15% · Mid-semester tests 30% (all three combined) + 2 more
- Key terms Proof by contradiction, Contrapositive, Dot product (a · b), De Moivre's theorem
- How to prepare Treat MTH1020 as a catalogue of recurring techniques and rehearse them weekly on Moodle material rather than cramming through SWOTVAC.
- Most asked Where can I find past exam papers / practice for MTH1020?
How MTH1020 is assessed
| Component | Weight | Format |
|---|---|---|
| Applied-class quizzes | 15% | 7 short invigilated in-class quizzes across the semester; best 4 of 7 count (mechanic mined; weight from the 2026 Monash Handbook) |
| Mid-semester tests | 30% (all three combined) | 3 invigilated in-seminar written tests: MST1 early in the semester, MST2 covering Weeks 1–4, MST3 in the Week-10 applied class (MST3 = 15% per-test) |
| Problem-solving activities / online assignments | 25% | Continuous problem-solving activities submitted across the semester (weekly online submissions — confirm the exact channel on Moodle) |
| Final examination | 30% | Invigilated end-of-semester exam, 3 hours 10 minutes, held in the Monash Semester-2 examination period (~November 2026); complex numbers were excluded from the S1-2026 final but tested in the mid-semester tests — confirm for your cohort |
What MTH1020 covers
MTH1020 Analysis of Change runs across twelve teaching weeks and seven topic blocks, and this map follows the Monash seminar schedule in order.
Chapters 1–6 build the pure-and-applied foundations — logic and proof, induction and functions, vectors (dot and cross product, lines and planes), and complex numbers (the Argand plane, polar form and De Moivre) — while Chapters 7–12 are the calculus core, from limits and the differentiation rules through integration techniques to areas, volumes of revolution and arc length.
The unit is assessed by invigilated applied-class quizzes, three in-seminar mid-semester tests and a final examination, with marks awarded across every task for rigorous mathematical communication — so use this map to see how each week feeds the tests and the exam.
Logic & Proof: Foundations
Conditionals · converse/contrapositive · direct/contrapositive/contradiction proofs · sets · inequalities · AM–GM (Week 1)Week 1 of Monash MTH1020 Analysis of Change opens the unit with the language of rigorous mathematics: what a proof is, conditional statements and their converse, contrapositive and negation, and the four standard proof techniques — direct, contrapositive, counterexample and contradiction.
- What a proof is: logical certainty vs verifying examples; a single counterexample disproves a 'for all' statement
- Conditional statements P ⇒ Q; hypothesis and conclusion; direct proof (assume P, deduce Q)
- Negation and De Morgan's laws; negating quantifiers: not(∀x P) ≡ ∃x not P
- Converse (Q ⇒ P) vs contrapositive (not Q ⇒ not P); a statement is equivalent to its contrapositive
- Proof by contradiction: assume the negation, derive a contradiction (e.g. √2 and log₂(3) irrational)
- Sets (ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ), builder notation, intervals, and divisibility (m = kn), primes vs composites
Induction & Functions
Σ/Π notation · telescoping · mathematical induction · injective/surjective/bijective · even/odd/periodic · composition & inverses (Week 2)Week 2 of Monash MTH1020 Analysis of Change pairs the principle of mathematical induction with the formal theory of functions.
- Summation Σ and product Π notation; dummy index; telescoping sums via 1/(n(n+1)) = 1/n − 1/(n+1)
- The principle of mathematical induction: base case P(1), induction step P(k) ⇒ P(k+1)
- Canonical induction results: Σ(2i−1) = n², Σ1/(j(j+1)) = n/(n+1), n³−n divisible by 3, 11ⁿ−1 divisible by 10
- Functions f : A → B: domain, co-domain, image, range; the vertical line test and natural domain
- Injective (one-to-one), surjective (onto) and bijective functions; the horizontal line test
- Even (f(−x) = f(x)), odd (f(−x) = −f(x)) and periodic (f(x+T) = f(x)) functions
Vectors: Dot Product & Projections
Vector arithmetic · magnitude · unit vectors · the dot product · scalar & vector projections (Week 3)Week 3 of Monash MTH1020 Analysis of Change introduces vectors: quantities with both magnitude and direction, their component form in ℝ² and ℝ³, and the arithmetic of addition, subtraction and scalar multiplication.
- Vectors vs scalars; equal vectors; the zero vector; representation as arrows (head and tail)
- Component form v = a i + b j + c k (or (a, b, c)); position vectors; AB = (position of B) − (position of A)
- Vector arithmetic: componentwise addition/subtraction, scalar multiplication, the parallelogram law
- Magnitude |v| = √(a²+b²+c²) (Pythagoras) and the unit vector v̂ = v/|v|
- Dot product two ways: a · b = |a||b|cos θ and a · b = a₁b₁ + a₂b₂ + a₃b₃; v · v = |v|²
- Angle between vectors: cos θ = (a · b)/(|a||b|); perpendicularity test a ⊥ b ⇔ a · b = 0
Vectors: Cross Product, Lines & Planes
The cross product · lines & planes in ℝ³ · intersections · angles & distances (Week 4)Week 4 of Monash MTH1020 Analysis of Change extends vectors into three-dimensional geometry.
- Cross product a × b: perpendicular to both a and b, magnitude |a||b|sin θ, right-handed set
- Component formula a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁); anticommutative a × b = −(b × a)
- Area applications: |a × b| = parallelogram area; triangle area = ½|AB × AC|
- Lines: vector equation r(t) = a + td; through two points r(t) = (1−t)a + tb; Cartesian form in ℝ³
- Intersections of lines: parallel/coincident/single-point/skew; test d₁ = k d₂ then solve r₁(t) = r₂(u)
- Planes: normal vector n; vector equation (r − a) · n = 0 ⇔ r · n = a · n; Cartesian n₁x + n₂y + n₃z = k
Complex Numbers: The Argand Plane & Arithmetic
i · the Argand plane · conjugates · arithmetic & division · the conjugate root theorem (Week 5)Week 5 of Monash MTH1020 Analysis of Change introduces the complex numbers ℂ: the imaginary unit i with i² = −1, the number-system hierarchy ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ, and the Argand plane on which z = x + yi is the point (x, y).
- Number systems ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ; motivation x² = −1 ⇒ invent i with i² = −1
- Complex number z = x + yi; real part Re(z) = x, imaginary part Im(z) = y (a real number)
- The Argand (complex) plane: z ↔ (x, y); real axis horizontal, imaginary axis vertical
- Complex conjugate z̄ = x − yi (reflection in the real axis); z + z̄ = 2Re(z); z·z̄ = x² + y² = |z|²
- Addition/subtraction (add real and imaginary parts); multiplication using i² = −1; powers of i cycle mod 4
- Division by realising the denominator: z₁/z₂ = z₁z̄₂/|z₂|²
Complex Numbers: Polar Form & De Moivre
Polar/cis form · exponential/Euler form · De Moivre's theorem · n-th roots · loci · the FTA (Week 6)Week 6 of Monash MTH1020 Analysis of Change re-expresses complex numbers in polar and exponential form: z = r cis θ = r e^(iθ), where r = |z| is the modulus and θ = arg(z) the argument.
- Polar form z = r(cos θ + i sin θ) = r cis θ; modulus r = |z| = √(x²+y²)
- Argument θ = arg(z) in radians; principal argument Arg(z) ∈ (−π, π]; use a diagram, θ ≠ simply arctan(y/x)
- Euler's / exponential form: e^(iθ) = cos θ + i sin θ = cis θ, so z = r e^(iθ); e^(iπ) = −1
- Multiplication z₁z₂ = r₁r₂ cis(θ₁+θ₂) and division z₁/z₂ = (r₁/r₂) cis(θ₁−θ₂): moduli multiply/divide, arguments add/subtract
- De Moivre's theorem: z = r cis θ ⇒ zⁿ = rⁿ cis(nθ), extending to negative integers
- n-th roots: w = ρ cis φ with ρ = r^(1/n), φ = (Arg(z) + 2πk)/n, k = 0, …, n−1 — n roots equally spaced
Differentiation: Limits, Continuity & Rules
Limits · continuity · differentiability · the rules · standard derivatives · implicit & logarithmic differentiation (Week 7)Week 7 of Monash MTH1020 Analysis of Change opens the calculus core with the derivative.
- The derivative as the slope of the tangent: f′(a) = lim(h→0)[f(a+h) − f(a)]/h; Newton and Leibniz notation
- Limits: two-sided limit exists only if both one-sided limits agree; lim f(x) need not equal f(a)
- Continuity: f continuous at a if lim(x→a) f(x) = f(a); continuous classes and combinations
- Differentiable ⇒ continuous, but not conversely (corners |x|, vertical tangents √x, jumps)
- The rules: constant multiple, sum, product (fg)′ = f′g + fg′, quotient, chain (f∘g)′ = f′(g)·g′
- Standard derivatives: xⁿ → n x^(n−1), eˣ → eˣ, sin x → cos x, cos x → −sin x, tan x → sec²x, ln x → 1/x
Applications of Differentiation: Extrema & Concavity
First & second derivative tests · local & absolute extrema · concavity & inflection · the EVT (Week 8)Week 8 of Monash MTH1020 Analysis of Change turns the derivative into a tool for reading the shape of a graph.
- Increasing/decreasing from the first derivative: f′ > 0 ⇒ increasing, f′ < 0 ⇒ decreasing, f′ = 0 ⇒ stationary
- Local maxima and minima; Fermat's theorem: a differentiable local extremum has f′ = 0
- Critical points: where f′ = 0 or f′ does not exist — the only candidates for local extrema
- First derivative test: sign of f′ changing +→− gives a local max, −→+ a local min, no change neither
- Concavity from the second derivative: f″ > 0 ⇒ concave up, f″ < 0 ⇒ concave down
- Points of inflection: where concavity changes (f″ = 0 or undefined, and actually changes sign)
Applications of Differentiation: Sketching, Related Rates & Optimisation
Curve sketching · limits at infinity & asymptotes · the growth hierarchy · related rates · optimisation (Week 9)Week 9 of Monash MTH1020 Analysis of Change completes the differential-calculus applications.
- Curve sketching from the calculus: domain, intercepts, f′ (increasing/decreasing), f″ (concavity), asymptotes
- Limits at infinity: behaviour of xⁿ, 1/xⁿ and eˣ as x → ±∞; horizontal and vertical asymptotes
- The growth hierarchy for a > 1, p > 0: logarithms ≪ powers ≪ exponentials, so lim(x→∞) xᵖ/aˣ = 0
- Related rates: linking dx/dt and dy/dt through an equation and the chain rule
- Related-rates strategy: diagram, name variables, write given/required rates, relate, differentiate, substitute
- Optimisation: identify the quantity, express it as a function of one variable with a domain, find the absolute extremum
Integration: The Definite Integral & the FTC
Riemann sums · the definite integral · signed area · the Fundamental Theorem of Calculus · antiderivatives (Week 10)Week 10 of Monash MTH1020 Analysis of Change introduces integration.
- The definite integral as a limit of Riemann sums: ∫ₐᵇ f dx = lim(n→∞) Σ f(xᵢ*)Δx, Δx = (b−a)/n
- LEFT(n) and RIGHT(n) endpoint sums; the limit is independent of the sample-point choice
- The integral as signed area: regions above the axis count positively, below negatively
- Properties: ∫ₐᵃ f = 0, ∫ₐᵇ f = −∫ᵇₐ f, ∫ₐᵇ f = ∫ₐᶜ f + ∫ᶜᵇ f, linearity
- FTC evaluation form: if F′ = f then ∫ₐᵇ f(x) dx = F(b) − F(a) = [F(x)]ₐᵇ
- FTC derivative form: (d/dx) ∫ₐˣ f(t) dt = f(x)
Integration Techniques
Substitution · partial fractions · integration by parts · trig integrals & substitution (Week 11)Week 11 of Monash MTH1020 Analysis of Change assembles the toolkit for evaluating integrals that the standard antiderivatives cannot handle directly.
- Substitution (reverse chain rule): spot (inner function)·(its derivative), set u = inner, replace dx via du
- Definite integrals by substitution: convert the terminals to the new variable u
- Partial fractions: split a proper rational function; do polynomial division first if improper
- Partial-fraction forms: linear A/(ax+b), repeated (ax+b)², irreducible quadratic (Dx+E)/(ax²+bx+c)
- Integration by parts: ∫ f g′ dx = f g − ∫ f′ g dx (reverse product rule); choosing f and g′
- Repeated by-parts and rearrangement (e.g. ∫eˣcos x dx solved after two applications)
Applications of Integration: Areas, Volumes & Arc Length
Areas between curves · average value · volumes of revolution (discs/shells) · arc length · surface area (Week 12)Week 12 of Monash MTH1020 Analysis of Change closes the unit by turning the definite integral into a measuring tool.
- Area between curves: for f(x) ≥ g(x) on [a, b], Area = ∫ₐᵇ [f(x) − g(x)] dx; find intersection points for limits
- Regions integrated in y when x is a function of y; use ∫|f − g| when the top curve switches
- Average value of a function: (1/(b−a)) ∫ₐᵇ f(x) dx
- Volumes of revolution — method of discs: rotate about the x-axis, V = π ∫ₐᵇ [f(x)]² dx
- Volumes of revolution — cylindrical shells: rotate about the y-axis, V = ∫ₐᵇ 2π x f(x) dx
- Choosing discs vs shells by which axis the region is rotated about and how it touches the axis
Monash frames MTH1020 in two movements: a foundations half — logic and proof, mathematical induction and functions, vectors (the dot and cross product, lines and planes in ℝ³), and complex numbers (the Argand plane, polar and exponential form, and De Moivre's theorem) — and a calculus half that is the analytic heart of the unit, running from limits, continuity and the differentiation rules through the applications of the derivative (extrema, curve sketching, related rates and optimisation) to integration, its techniques, and the applications of integration (areas, volumes of revolution and arc length).
It is a method-and-communication subject: the marks live in choosing the right technique, carrying a clean derivation, and — distinctively at Monash — writing the argument up in full English sentences, because a significant fraction of every task's marks is for mathematical communication (see the Monash Guidelines for Writing in Mathematics).
Assessment for the 2026 offering is published in the Monash Handbook as four components summing to 100%: invigilated applied-class quizzes (15%), three in-seminar mid-semester tests (30% combined), continuous problem-solving activities / online assignments (25%), and an invigilated final examination worth 30% and running 3 hours 10 minutes.
That split — 70% continuous, a 30% capstone exam — rewards steady work across the semester rather than a single SWOTVAC cram, and the MTH1020 result feeds the Weighted Average Mark (WAM) that later Monash mathematics units build on. Note: for the S1-2026 cohort a lecturer stated complex numbers were excluded from the final (still tested in the mid-semester tests) — confirm this for your cohort on Moodle.
Worked example · free
Optimisation: the minimal-surface closed cylinder (height = diameter)
- +1Set up the model with a clear objective and constraint. The total surface area of a closed cylinder is S = 2πr² + 2πrh (two circular ends plus the curved side), and the fixed volume gives the constraint V = πr²h. There are two variables (r, h) and one constraint, so reduce to a single variable.
- +1Eliminate h using the constraint: h = V/(πr²). Substitute into S to get a function of r alone: S(r) = 2πr² + 2πr·(V/(πr²)) = 2πr² + 2V/r, defined on the domain r > 0.
- +1Find the critical point. Differentiate: S′(r) = 4πr − 2V/r². Set S′(r) = 0: 4πr = 2V/r², so 4πr³ = 2V, giving r³ = V/(2π) and therefore r = (V/(2π))^(1/3).
- +1Confirm it is a minimum (second-derivative test). S″(r) = 4π + 4V/r³, which is strictly positive for every r > 0, so S is concave up and the critical point is a local — and, on the domain r > 0, the absolute — minimum.
- +1Show height = diameter. At the optimum r³ = V/(2π), so V = 2πr³. Then h = V/(πr²) = 2πr³/(πr²) = 2r, which is exactly the diameter. The minimal-surface can is therefore as tall as it is wide.
Key terms
- Proof by contradiction
- A proof that assumes the statement is false, derives a logical contradiction, and concludes the statement must be true. Used at Monash for classics such as 'the square root of 2 is irrational'.
- Contrapositive
- The contrapositive of 'if P then Q' is 'if not Q then not P'. A statement is logically equivalent to its contrapositive, so proving the contrapositive proves the original.
- Dot product (a · b)
- The scalar a · b = |a||b|cos θ = a₁b₁ + a₂b₂ + a₃b₃. It gives the angle between vectors (cos θ = a·b/(|a||b|)) and tests perpendicularity (a ⊥ b ⇔ a · b = 0).
- De Moivre's theorem
- If z = r cis θ then zⁿ = rⁿ cis(nθ): the modulus is raised to the power n and the argument is multiplied by n. It is the engine for powers and n-th roots of complex numbers.
- Fundamental Theorem of Calculus (FTC)
- Links the two halves of calculus: if F′ = f then ∫ₐᵇ f(x) dx = F(b) − F(a) (evaluation form), and (d/dx)∫ₐˣ f(t) dt = f(x) (derivative form). The integral of a rate of change is the total change.
- Mathematical communication
- The Monash requirement that a solution be written in full English sentences with defined variables and justified steps. A significant fraction of the marks on every MTH1020 task is awarded for this, not just the final answer.
MTH1020 FAQ
Is MTH1020 hard?
MTH1020 is broad rather than deep: in twelve weeks it moves through proof, induction, vectors, complex numbers and a full run of differential and integral calculus, so the challenge is keeping many techniques and their standard results straight rather than mastering one hard idea.
The algebra is first-year level, but Monash also marks how you write — clear English sentences, defined variables and justified steps — so students who practise both the calculation and the write-up weekly, instead of cramming through SWOTVAC, tend to find it very manageable.
Because MST2 covers Weeks 1–4 and the final is comprehensive, keeping early topics warm matters, and a solid MTH1020 mark feeds the WAM that later Monash mathematics units build on.
How is MTH1020 assessed, and is there a hurdle?
The 2026 Monash Handbook publishes four components summing to 100%: invigilated applied-class quizzes (15%, best 4 of 7 count), three in-seminar mid-semester tests (30% combined, with MST3 = 15%), continuous problem-solving activities / online assignments (25%), and a 30% invigilated final examination running 3 hours 10 minutes.
The 2026 Handbook lists no pass hurdle for MTH1020 (all four components show a null hurdle) — an earlier ≥40%-exam / ≥50%-overall threshold appears only in historic handbook years, so do not assume it applies; confirm the current rules and any competency thresholds on Moodle and your unit guide.
Can AI help me with MTH1020?
Yes — Sia is an AI tutor trained on how MTH1020 is actually taught and assessed at Monash. It can walk you through a proof by contradiction, a De Moivre calculation, an implicit-differentiation step or a volume-of-revolution set-up one line at a time, and it checks your reasoning and unit work rather than just handing over a result.
Bring your own applied-class or revision question and ask Sia to explain each step; it does not do graded assessment for you, and Monash University academic-integrity rules still apply.
Where can I find past exam papers / practice for MTH1020?
Start on the unit's Moodle page, where MTH1020 posts its official practice material, applied-class problem sets and any released past papers, and search the Monash University Library exam-paper collection for earlier MTH1020 finals. Your weekly applied-class problem sets and the mid-semester-test revision sheets are the closest match to how the exam is marked.
This guide also includes a re-authored practice exam that mirrors the final's shape (proof, vectors, calculus and integration) with fresh numbers, and you can ask Sia to generate extra practice in the same style and explain each step. Treat any third-party 'model answers' with caution and confirm what is officially provided on Moodle.
Are complex numbers on the MTH1020 final exam?
For the S1-2026 cohort a lecturer stated that complex numbers were NOT on the final exam, although they were still examined in the mid-semester tests. This is cohort-specific, is not stated in the Monash Handbook, and may change for the S2-2026 offering — so treat it as a planning note, not a rule, and confirm the exam scope for your cohort on Moodle and in your unit guide.
Either way, learn the complex-numbers material: the Argand plane, polar form and De Moivre are examinable somewhere in the unit and underpin later mathematics.
Questions students ask in each MTH1020 chapter
- What's the difference between the converse and the contrapositive?
- The converse of 'if P then Q' swaps the two parts to give 'if Q then P', and it need not have the same truth value — proving a statement tells you nothing about its converse. The contrapositive flips and negates both parts to give 'if not Q then not P', and it is logically equivalent to the original, which is why 'proof by contrapositive' is a valid strategy. A common Monash exam slip is to 'prove' a statement by accidentally proving its converse. Chapter 01 · Logic & Proof: Foundations
- Why isn't checking lots of cases the same as a proof by induction?
- Checking n = 1, 2, 3, … only verifies finitely many cases, but a 'for all natural n' statement has infinitely many. Induction closes that gap: the induction step proves that truth at any k forces truth at k+1, so once the base case starts the chain, every natural number follows like falling dominoes. That single implication does the work of infinitely many checks. Chapter 02 · Induction & Functions
- What's the difference between the scalar and vector projection?
- The scalar projection comp_a b = (a · b)/|a| is a single signed number — the length of b's shadow along a, negative if that shadow points backwards. The vector projection proj_a b = ((a · b)/|a|²) a is that shadow as an actual vector pointing along a. The link is proj_a b = (comp_a b) â, where â is the unit vector along a. Watch the denominators: divide by |a| for the scalar version and by |a|² for the vector version. Chapter 03 · Vectors: Dot Product & Projections
- When do I use the dot product versus the cross product?
- Use the dot product when you want a number or an angle — a · b = |a||b|cos θ gives angles, perpendicularity (a · b = 0) and projections. Use the cross product when you want a vector perpendicular to two others, or an area — a × b has magnitude |a||b|sin θ, points normal to both, and gives parallelogram and triangle areas. A useful memory hook: dot involves cos and stays in any dimension; cross involves sin, needs 3-D, and produces a new vector. Chapter 04 · Vectors: Cross Product, Lines & Planes
- How do you divide two complex numbers?
- Multiply the numerator and denominator by the conjugate of the denominator. That turns the denominator into z₂·z̄₂ = |z₂|², a positive real number, so the quotient becomes z₁z̄₂/|z₂|², which you can then split into real and imaginary parts. For example, (3 + 2i)/(1 − i) becomes (3 + 2i)(1 + i)/((1 − i)(1 + i)) = (1 + 5i)/2 = 1/2 + (5/2)i. The whole trick is 'realising' the denominator. Chapter 05 · Complex Numbers: The Argand Plane & Arithmetic
- Why is the argument not just arctan(y/x)?
- Because arctan only returns angles in (−π/2, π/2), it cannot distinguish the four quadrants. For z = −1 − i the ratio y/x = 1 gives arctan(1) = π/4, but the point sits in the third quadrant, so the true argument is −3π/4 (or 5π/4). Always plot z on the Argand plane, use tan θ = y/x for the reference angle, then adjust for the correct quadrant. This quadrant check is a frequent MTH1020 exam trap. Chapter 06 · Complex Numbers: Polar Form & De Moivre
- Does continuous mean the same as differentiable?
- No — differentiable is the stronger condition. Every differentiable function is continuous, but not every continuous function is differentiable: |x| is continuous everywhere yet has no derivative at 0 because of the sharp corner, and √x has a vertical tangent at 0. So a graph can be unbroken but still fail to have a well-defined tangent slope at certain points. Getting this implication the right way round is a common exam question. Chapter 07 · Differentiation: Limits, Continuity & Rules
- What's the difference between the first and second derivative tests?
- Both classify a critical point, but they use different information. The first derivative test checks the sign of f′ on each side of the point: it always works, even when f′ does not exist there. The second derivative test evaluates f″ at the point: f″ > 0 means a minimum, f″ < 0 a maximum — quick when f″ is easy to compute, but inconclusive when f″(c) = 0, in which case you fall back on the first derivative test. Chapter 08 · Applications of Differentiation: Extrema & Concavity
- What's the general strategy for a related-rates problem?
- Draw a diagram and name the variables, write the given and required quantities as derivatives with respect to time, find an equation relating the variables (often a geometric formula), then differentiate that equation with respect to t using the chain rule. Only at the final step do you substitute the specific instantaneous values and solve for the unknown rate. The order matters: differentiate the general relation first, substitute numbers last. Chapter 09 · Applications of Differentiation: Sketching, Related Rates & Optimisation
- What's the difference between a definite and an indefinite integral?
- A definite integral ∫ₐᵇ f(x) dx has limits and evaluates to a number — the signed area under the graph between a and b. An indefinite integral ∫ f(x) dx has no limits and represents a whole family of antiderivatives, F(x) + C, where the constant of integration C accounts for the fact that any two antiderivatives differ by a constant. The FTC connects them: you evaluate the definite integral by taking an antiderivative and subtracting its values at the two limits. Chapter 10 · Integration: The Definite Integral & the FTC
- How do I know which integration technique to use?
- Read the structure of the integrand. If you see a composite function multiplied by (something close to) the derivative of its inner part, use substitution. If you see a product of two unlike functions — a polynomial times an exponential or trig, say — use integration by parts. If you have a rational function (a ratio of polynomials), use partial fractions, after polynomial division if it is improper. Roots like √(a²−x²) point to a trig substitution. Recognising the type is itself examined, so practise classifying before integrating. Chapter 11 · Integration Techniques
- How do I decide between discs and shells for a volume of revolution?
- Look at which axis the region is rotated about and how the region meets that axis. When a region sitting against the x-axis is rotated about the x-axis, each cross-section perpendicular to the axis is a disc of radius f(x), so V = π∫[f(x)]² dx. When a region under y = f(x) is rotated about the y-axis, slicing parallel to that axis gives thin cylindrical shells of radius x and height f(x), so V = ∫2πx f(x) dx. Sketching the region and one representative slice tells you which one applies far more reliably than memorising rules. Chapter 12 · Applications of Integration: Areas, Volumes & Arc Length
How to study for the exam
Treat MTH1020 as a catalogue of recurring techniques and rehearse them weekly on Moodle material rather than cramming through SWOTVAC.
Each week, work one full problem of that week's method end to end — state the technique in words, define your variables, carry one clean derivation, then interpret the result and cross-check it a second way (for example, verify an antiderivative by differentiating it back, or check a vector projection with the dot product).
Because MST2 covers Weeks 1–4 and MST3 sits in the Week-10 applied class, early topics get examined more than once, so keep old test material warm instead of filing it away. Practise the Monash write-up deliberately: full English sentences, no ∀/∃/⇒ used as prose shorthand, labelled diagrams, correct units, and a final sentence that answers the question — a significant fraction of every mark is for communication.
For the comprehensive 30% final (3 hours 10 minutes, ~November 2026 — confirm the date, room and open/closed-book status on Moodle), prioritise breadth first: make sure you can start a proof, a vectors question, a differentiation problem and an integration problem before deepening the ones you find hardest.
When a step won't click, ask Sia to re-explain that single step a different way and set you a fresh practice question in the same style — it teaches the method and checks your reasoning, never substituting for your own graded work.
Your AI Mathematics tutor for MTH1020
Stuck on a hard MTH1020 question? Sia is AskSia’s AI Mathematics tutor — ask any MTH1020 Analysis of Change question and get a clear, step-by-step explanation grounded in how the course is actually taught and assessed. Read this whole study guide free, then take your hardest questions to Sia.
