Unit 3: Calculus & Probability

VCE Mathematical Methods Study Design 2023-2027

Units 3 & 4 (Exam-Assessed) 73 Skills • 3 Levels Each

Unit 3 Skill Map

Advanced functions & graphs, algebra, differential calculus for all function types, and discrete probability.

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Functions & Graphs (Advanced)

24 skills

Identify key features of polynomial functions (degree, end behaviour, multiplicity)20200 Sketch polynomial functions using intercepts and end behaviour20201 Sketch exponential functions with transformations20202 Sketch logarithmic functions with transformations20203 Sketch circular functions with combined transformations20204 Determine the rule of a function from its graph20205 Find the domain and range of transformed functions20206 Form composite functions f(g(x)) and determine domain20207 Form sum and difference functions f + g, f - g20208 Form product functions f * g20209 Apply multiple transformations in correct order (dilations, reflections, translations)20210 Write the rule for a transformed function y = A*f(b(x - h)) + k20211 Find inverse functions and state domain restrictions20212 Sketch a function and its inverse on the same axes20213 Verify that f(f^(-1)(x)) = x20214 Use parameters to explore function families (interactive)20215 Model practical situations with appropriate function types20216 Match rule to graph (all function types mixed)20217 Sketch transformed hyperbolas y = a/(x-h) + k20218 Sketch hyperbolas with intercepts20219 Sketch applied/modelling functions (projectile, cooling, tides)20231 Addition of ordinates for combining functions20236 Power function transformations y = a(x-h)^n + k20237 Mapping notation for transformations20238

Algebra (Advanced)

16 skills

Solve exponential equations analytically20220 Solve logarithmic equations analytically20221 Solve trigonometric equations over specified domains20222 Solve trigonometric equations with transformations20223 Solve equations involving composite functions20224 Solve literal equations (rearrange for a given variable)20225 Solve systems of simultaneous equations (linear and non-linear)20226 Solve equations graphically and numerically20227 Apply Newton's method to find approximate roots20228 Trace Newton's method pseudocode20229 Determine number of solutions using graphical analysis20230 Solve equations graphically (intersection points)20232 Polynomial division and factorisation20233 General solution of trigonometric equations20234 Trigonometric identities and symmetry properties20235 Function notation and identities20239

Differential Calculus (All Function Types)

20 skills

Differentiate e^(ax+b)20240 Differentiate ln(ax+b)20241 Differentiate sin(ax+b) and cos(ax+b)20242 Differentiate tan(x)20243 Apply the chain rule to composite functions20244 Apply the product rule20245 Apply the quotient rule20246 Differentiate combinations requiring multiple rules20247 Find equations of tangent and normal lines (all function types)20248 Determine where a function is increasing/decreasing (all types)20249 Find and classify stationary points (all function types)20250 Find absolute maximum and minimum on a closed interval20251 Sketch curves using derivatives (all function types)20252 Solve optimisation problems in context20253 Rates of change in context (growth, decay, motion)20254 Relate graphs of f, f', and f''20255 Determine f from information about f'20256 Limits and continuity (advanced)20257 Conditions for differentiability20258 Kinematics — velocity and acceleration from position function20259

Probability (Discrete Random Variables & Binomial)

13 skills

Construct probability distribution tables (advanced)20270 Calculate E(X), Var(X), sd(X) for discrete distributions20271 Properties of expected value: E(aX + b) = aE(X) + b20272 Properties of variance: Var(aX + b) = a^2 Var(X)20273 Identify and set up binomial distributions X ~ Bi(n, p)20274 Calculate binomial probabilities (by hand for small n)20275 Calculate binomial probabilities (CAS for larger n)20276 Find the expected value and variance of Bi(n, p)20277 Determine sample size n for a binomial problem20278 Solve problems involving at least/at most in binomial contexts20279 Graph binomial probability distributions20280 Apply binomial distribution to real-world problems20281 Proofs for E(aX+b) and Var(aX+b)20282