§-Unit

Unit 4

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

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

WACE Year 12 Mathematics Methods Unit 3 the Fundamental Theorem of Calculus: both forms linking differentiation and integration, evaluating definite integrals as F(b) minus F(a), differentiating an integral function, with worked SCSA-style examples.

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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

WACE Year 12 Mathematics Methods Unit 4 margin of error and sample size: the margin of error formula, how confidence level and sample size affect width, and solving for the sample size needed for a required precision, with worked examples.

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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

WACE Year 12 Mathematics Methods Unit 4 the normal distribution: the bell shape, symmetry about the mean, the role of mu and sigma, and the empirical 68-95-99.7 rule, with worked SCSA-style examples.

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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

WACE Year 12 Mathematics Methods Unit 4 the sample proportion: random sampling, the sampling distribution of p-hat, its mean p, the standard error, and approximate normality for large samples, with worked SCSA-style examples.

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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

WACE Year 12 Mathematics Methods Unit 4 probability density functions: the validity conditions, probability as area, finding an unknown constant, the median, and mean and variance by integration, with worked SCSA-style examples.

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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

WACE Year 12 Mathematics Methods Unit 3 integration in kinematics: recovering velocity from acceleration and displacement from velocity, using initial conditions, and finding distance travelled by splitting where velocity is zero, with worked examples.

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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

WACE Year 12 Mathematics Methods Unit 3 area between curves: finding intersection points, integrating upper minus lower, splitting where curves cross, with worked SCSA-style examples and the common sign mistake.

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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

WACE Year 12 Mathematics Methods Unit 4 standardisation: converting to a z-score, computing normal probabilities, symmetry of the standard normal, and inverse problems finding a value from a probability, with worked SCSA-style examples.

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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

WACE Year 12 Mathematics Methods Unit 3 the definite integral: the limit of Riemann sums, signed area above and below the axis, properties of the definite integral, and total area, with worked SCSA-style examples.

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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

WACE Year 12 Mathematics Methods Unit 3 total change from a rate: integrating a rate of change to find net change, recovering a quantity from its rate plus an initial value, with worked SCSA-style examples in real contexts.

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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

WACE Year 12 Mathematics Methods Unit 4 continuous random variables: probability density functions, mean and variance by integration, the normal distribution, standardisation with z-scores and the 68-95-99.7 rule, with worked SCSA-style examples.

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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

WACE Year 12 Mathematics Methods Unit 4 confidence intervals for proportions: the sampling distribution of the sample proportion, the standard error, the approximate 95 percent confidence interval, and correct interpretation, with worked SCSA-style examples.

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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

WACE Year 12 Mathematics Methods Unit 3 integration: antiderivatives, the definite integral, the Fundamental Theorem of Calculus, area under and between curves, and kinematics, with worked SCSA-style examples.

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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

WACE Year 12 Mathematics Methods Unit 3 antidifferentiation: reversing differentiation, the constant of integration, antiderivatives of powers, exponentials, trig and one over x, and finding a particular solution from a condition, with worked examples.

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