School Study Skills

Estimate the limit of (a^h − 1)/h as h approaches 0 using technology and identify the defining property of e.

Analyse the intercepts, asymptotes, domains, ranges and inverse relationship of y = e^x and y = ln(x).

Differentiate functions of the form e^{f(x)} using the chain rule.

Differentiate functions of the form ln(f(x)), stating any relevant domain restrictions.

Solve equations involving exponential and logarithmic functions with base e without technology and verify the admissibility of each solution.

Solve and interpret rate-of-change problems involving exponential and logarithmic models, selecting calculator-free or calculator-use methods appropriately.

Differentiate composite sine and cosine functions in radian measure.

Apply the chain rule to composite functions involving exponential, logarithmic and trigonometric functions.

Apply the product rule to functions involving exponential, logarithmic and trigonometric factors.

Apply the quotient rule to functions involving exponential, logarithmic and trigonometric expressions.

Differentiate functions requiring combinations of the chain, product and quotient rules, expressing the result in a useful simplified or factorised form.

Solve calculator-free and calculator-use tangent, rate and stationary-point problems while checking angle mode and domain restrictions.

Calculate and interpret the second derivative of a function.

Determine displacement, velocity and acceleration relationships using first and second derivatives.

Determine intervals of concavity from the sign of the second derivative.

Locate and verify points of inflection using sign changes in concavity.

Classify stationary points using the second derivative test.

Sketch a curve from first- and second-derivative information and verify its key features using graphing technology.

Formulate an objective function and feasible domain from an optimisation context.

Find all stationary-point and endpoint candidates for an optimum using calculator-free or calculator-use methods.

Classify and justify local or global extrema using derivative evidence and domain constraints.

Solve an unfamiliar applied optimisation problem using first and second derivatives.

Evaluate an optimisation result using constraints, units and reasonableness checks, and correct any identified error.

Determine indefinite integrals of power functions using linearity and the power rule.

Determine indefinite integrals involving e^x, 1/x, sin(x) and cos(x).

Integrate functions of the form f(ax + b) covered by the syllabus.

Determine a function from its derivative and an initial condition.

Determine displacement from a velocity function and an initial position.

Determine velocity and displacement from acceleration and initial conditions, then verify the result by differentiation.

Construct or validate a probability function for a discrete random variable using stated conditions or relative-frequency data.

Calculate the expected value of a discrete random variable.

Calculate the variance and standard deviation of a discrete random variable.

Compare discrete distributions using their expected values and standard deviations in context.

Solve calculator-free and calculator-use problems involving uniform and non-uniform discrete random variables.

Determine whether a context can be modelled by a Bernoulli variable or a binomial distribution and justify the choice.

Calculate an exact binomial probability algebraically for a specified number of successes.

Translate phrases such as “at least”, “at most” and “between” into correct binomial probability statements.

Calculate cumulative and interval binomial probabilities using graphing technology.

Calculate and interpret the mean and variance of Bernoulli and binomial distributions.

Solve contextual binomial-distribution problems and check the reasonableness of the result.

Estimate area under a curve using a finite sum of rectangle areas.

Explain how a definite integral represents a limit of sums and a signed area.

Evaluate definite integrals using the fundamental theorem of calculus.

Calculate the total area enclosed by a curve and the x-axis, splitting at zeros when necessary.

Calculate the area between two curves after determining their intersection points.

Apply the trapezoidal rule and assess whether an approximation is plausible.

Calculate total change from a rate function, including displacement and distance interpretations in motion problems.

Solve triangles using the sine rule, including identifying and resolving the ambiguous case.

Solve triangles using the cosine rule.

Calculate the area of a triangle using one-half ab sin(C).

Solve two-dimensional problems involving bearings, directions, and angles of elevation or depression.

Solve three-dimensional trigonometric problems and check the geometric plausibility of the result.

Interpret probability density functions, cumulative distribution functions and interval probabilities for continuous random variables.

Determine an unknown parameter in a probability density function by using the total-area condition.

Calculate the expected value of a continuous random variable from its probability density function.

Calculate the variance and standard deviation of a continuous random variable from its probability density function.

Interpret the parameters and graphical features of a normal distribution written as X ~ N(μ, σ²).

Standardise normal observations using z-scores and compare observations from different distributions.

Use graphing technology to calculate normal probabilities and quantiles, then interpret the result in context.

Distinguish populations, samples, parameters, statistics and sample proportions in a statistical context.

Identify sources of sampling bias in a proposed data-collection method.

Select and justify a procedure that produces a random sample.

Calculate and interpret the mean and standard deviation of the sampling distribution of a sample proportion.

Determine when a normal approximation for a sample proportion is appropriate and standardise a sample proportion.

Use simulated or graphical sampling distributions to explain the effects of population proportion and sample size.

Construct an approximate confidence interval for a population proportion using a given confidence level.

Interpret a confidence interval for a population proportion without claiming that it gives the probability of a fixed parameter.

Calculate and interpret the margin of error for an estimated population proportion.

Determine how confidence level and sample size affect margin of error.

Determine a minimum sample size required to achieve a specified approximate margin of error.

Explain why confidence intervals vary between samples and why the chosen confidence level does not guarantee coverage for every sample.

Solve calculator-free and calculator-use contextual problems involving interval estimates for proportions.

Present sufficient mathematical working, notation, units and rounding in a calculator-use response, then apply a structured checking routine to a timed mixed Units 3–4 question set.