§-Unit

Unit 3

17 dot points, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

What is a discrete random variable, and what conditions make a table of probabilities a valid probability distribution?

Define discrete random variables and construct and verify discrete probability distributions, including finding unknown probabilities

WACE Year 12 Mathematics Methods Unit 3 discrete probability distributions: defining a discrete random variable, the two validity conditions, finding an unknown probability, and uniform distributions, with worked SCSA-style examples.

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How do we calculate exact and cumulative binomial probabilities, including at-least and at-most events?

Recognise binomial conditions and calculate exact and cumulative binomial probabilities, using complements for at-least and at-most events

WACE Year 12 Mathematics Methods Unit 3 binomial probabilities: recognising the four conditions, computing exact probabilities, cumulative at-least and at-most events using the complement, with worked SCSA-style examples.

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How do we combine intercepts, stationary points, concavity and asymptotes into a complete and accurate sketch of a curve?

Sketch curves using intercepts, stationary points, their nature, points of inflection, asymptotes and end behaviour

WACE Year 12 Mathematics Methods Unit 3 curve sketching with calculus: a systematic method using intercepts, stationary points and their nature, points of inflection, asymptotes and end behaviour, with a fully worked SCSA-style sketch.

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How do we model and analyse discrete random variables and the binomial distribution?

Construct discrete probability distributions, calculate expected value and variance, and apply the binomial distribution to repeated independent trials

WACE Year 12 Mathematics Methods Unit 3 discrete random variables: probability distributions, expected value, variance and the binomial distribution with mean np and variance np(1-p), shown through worked SCSA-style examples.

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How do we differentiate, integrate and apply exponential and logarithmic functions?

Differentiate and integrate exponential and natural logarithm functions and apply them to growth, decay and other modelling contexts

WACE Year 12 Mathematics Methods Units 3 and 4 exponential and logarithmic functions: derivatives and integrals of e^x and ln x, the chain rule with exponentials, and growth and decay modelling with worked SCSA-style examples.

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How do we use differentiation to find the maximum or minimum value of a quantity in a real context?

Solve optimisation problems by modelling a quantity as a function of one variable and using the derivative to find and justify extreme values

WACE Year 12 Mathematics Methods Unit 3 optimisation: modelling a quantity with one variable using a constraint, solving f-prime equals zero, justifying the extremum, checking domain endpoints, with a worked SCSA-style problem.

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How do we model a single trial with exactly two outcomes, and what are its mean and variance?

Use the Bernoulli distribution to model a single two-outcome trial and find its mean and variance

WACE Year 12 Mathematics Methods Unit 3 the Bernoulli distribution: a single success-or-failure trial, its probability function, mean p and variance p times one minus p, and its role as the building block of the binomial, with a worked example.

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What are the mean and variance of a binomial distribution, and how do the parameters n and p shape it?

Find and apply the mean np and variance np(1-p) of a binomial distribution and describe the effect of n and p on its shape

WACE Year 12 Mathematics Methods Unit 3 mean and variance of the binomial distribution: the formulas np and np(1-p), the standard deviation, the effect of n and p on shape and skew, with worked SCSA-style examples.

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How do we differentiate sine, cosine and tangent functions, including composite trigonometric expressions?

Establish and use the derivatives of sine, cosine and tangent functions, including the chain rule for trigonometric functions of a function of x

WACE Year 12 Mathematics Methods Unit 3 derivatives of trigonometric functions: differentiating sin, cos and tan, the chain rule for sin and cos of a function, the radian requirement, and worked SCSA-style examples.

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What does the second derivative tell us about the shape of a curve and the nature of its stationary points?

Find and interpret the second derivative, determine concavity and points of inflection, and apply the second derivative test

WACE Year 12 Mathematics Methods Unit 3 the second derivative: concavity, points of inflection where the second derivative changes sign, the second derivative test for stationary points, and worked SCSA-style examples.

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How do we differentiate a function that is a product or a quotient of two simpler functions?

Establish and apply the product rule and the quotient rule to differentiate products and quotients of functions

WACE Year 12 Mathematics Methods Unit 3 the product and quotient rules: their statements, when to use each, applying them to polynomial, exponential and trigonometric factors, with worked SCSA-style examples and the numerator order trap.

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What is the expected value of a discrete random variable, and how do we use it to make decisions?

Calculate and interpret the expected value (mean) of a discrete random variable and apply it to decision contexts such as fair games

WACE Year 12 Mathematics Methods Unit 3 expected value: the mean as a probability-weighted average, its interpretation as a long-run average, expected value of a linear function, and fair-game decisions, with worked SCSA-style examples.

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How do we measure the spread of a discrete random variable around its mean?

Calculate and interpret the variance and standard deviation of a discrete random variable, including the effect of a linear transformation

WACE Year 12 Mathematics Methods Unit 3 variance and standard deviation of a discrete random variable: the definition, the E of X squared minus mean squared shortcut, the linear transformation rule, with worked SCSA-style examples.

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Why is the natural exponential function its own derivative, and how do we differentiate exponential functions in general?

Establish and use the derivative of the exponential function, including the chain rule for e raised to a function of x

WACE Year 12 Mathematics Methods Unit 3 derivatives of exponential functions: why e to the x is its own derivative, the chain rule for e to a function of x, base-a exponentials, and worked SCSA-style differentiation.

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How does the derivative describe how fast a quantity changes, and how do we link the rates of related quantities?

Interpret the derivative as an instantaneous rate of change and use the chain rule to relate the rates of change of connected quantities

WACE Year 12 Mathematics Methods Unit 3 rates of change: the derivative as instantaneous rate, average versus instantaneous rate, related rates through the chain rule, with worked SCSA-style examples in context.

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How do we differentiate the natural logarithm function and logarithms of more complex expressions?

Establish and use the derivative of the natural logarithm function, including the chain rule for the logarithm of a function of x

WACE Year 12 Mathematics Methods Unit 4 derivatives of logarithmic functions: the derivative of natural log x, the chain rule giving f-prime over f, using log laws to simplify before differentiating, with worked SCSA-style examples.

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How do the product, quotient and chain rules let us differentiate and apply more complex functions?

Apply the product, quotient and chain rules to differentiate functions, and use the derivative in optimisation, rates of change and curve sketching

WACE Year 12 Mathematics Methods Unit 3 further differentiation: the product, quotient and chain rules combined, optimisation, rates of change and the second derivative for curve sketching, with worked SCSA-style examples.

9 min read →