§-Math Methods syllabus

WA · SCSA← Math Methods

Math Methods syllabus, dot point by dot point

Every dot point in the WA Math Methods syllabus, in syllabus order, with a focused answer for each. Open any dot point for the answer, exam-style questions and links to related points.

Unit 3

Unit overview →

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Unit 4

Unit overview →

How does the Fundamental Theorem of Calculus link differentiation and integration into inverse processes?

State and apply the Fundamental Theorem of Calculus to evaluate definite integrals and to differentiate integral functions

What controls the width of a confidence interval, and how large must a sample be to achieve a required precision?

Relate the margin of error to confidence level and sample size, and determine the sample size needed for a required margin of error

What are the defining properties of the normal distribution and the empirical 68-95-99.7 rule?

Describe the normal distribution, its symmetry and parameters, and apply the empirical 68-95-99.7 rule

How does the sample proportion behave across repeated random samples, and why is it approximately normal?

Describe the sampling distribution of the sample proportion, including its mean, standard error and approximate normality for large samples

How does a probability density function describe a continuous random variable, and how do we find probabilities from it?

Use probability density functions to find probabilities as areas, determine an unknown constant, and compute the mean and variance by integration

How do we use integration to recover velocity and displacement from acceleration, and find distance travelled?

Apply integration to straight-line motion, recovering velocity from acceleration and displacement from velocity, and distinguishing displacement from distance travelled

How do we find the area enclosed between two curves, including where they cross?

Calculate the area enclosed between two curves by integrating the difference of the upper and lower functions over the correct interval

How do we standardise a normal variable and calculate normal probabilities, including inverse problems?

Standardise a normal variable to a z-score and calculate normal probabilities, including finding a value from a given probability

How is the definite integral defined as a limit of sums, and why does it give signed area under a curve?

Interpret the definite integral as a limit of Riemann sums and as the signed area between a curve and the x-axis

How do we recover the total change in a quantity from its rate of change?

Use the definite integral of a rate of change to find the total or net change in a quantity over an interval

How do we model continuous data using probability density functions and the normal distribution?

Use probability density functions to find probabilities, mean and variance for continuous random variables, and apply the normal distribution with standardisation

How do we use a sample proportion to estimate a population proportion with a confidence interval?

Use the distribution of the sample proportion to construct and interpret approximate confidence intervals for a population proportion

How do we reverse differentiation to find areas, total change and other quantities through integration?

Find antiderivatives, evaluate definite integrals using the Fundamental Theorem of Calculus, and apply integration to areas, total change and kinematics

What does it mean to reverse differentiation, and how do we find the family of antiderivatives of a function?

Find antiderivatives of standard functions, include the constant of integration, and determine a particular antiderivative from an initial condition