§-Unit

Unit 4: Further calculus, trigonometry and statistics

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

Topic 1: Further integration

Apply the definite integral to compute the area between curves (including curves that change relative order), the average value of a function, and kinematics quantities (displacement, distance, position) from velocity and acceleration

Applications of integration. Area between curves, average value of a function, displacement and distance from velocity, position from acceleration with initial conditions, with worked PSMT-style examples.

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Integrate trigonometric functions including

\sin(kx)

\cos(kx)

and

\sec^2(kx)

, and apply the linear reverse-chain rule for integrals of the form

f(ax+b)

Integrating trigonometric functions. Antiderivatives of $\sin(kx)$ , $\cos(kx)$ and $\sec^2(kx)$ with the $1/k$ reverse-chain factor, definite-integral evaluation with exact values at standard angles, and worked PSMT-style applications.

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Topic 3: Continuous random variables and the normal distribution

Define a continuous random variable, its probability density function (pdf), cumulative distribution function (cdf), and compute probabilities, expected value (mean), variance and standard deviation as definite integrals

Continuous random variables. Defines the pdf, cdf, mean, variance and standard deviation as integrals, including the normalisation condition and a worked PSMT-style example.

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Apply the normal distribution

N(\mu, \sigma^2)

and the standardisation

Z = (X - \mu)/\sigma

to compute normal probabilities and inverse probabilities, including the empirical 68-95-99.7 rule

Normal distribution. Standardisation, the empirical rule, normal probability and inverse-normal calculations, and worked PSMT and EA examples.

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Specialist Mathematics content (not a Unit 4 Methods topic)

Apply implicit differentiation to find

\frac{dy}{dx}

from equations relating

x

and

y

that cannot be expressed in the form

y = f(x)

, and apply differentiation to related rates problems

Implicit differentiation and related rates are QCE Specialist Mathematics content (Unit 4), not Mathematical Methods. The four-step procedure for related rates, the chain-rule treatment of $y(x)$ , and PSMT contexts where two or more time-dependent quantities are related geometrically.

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Topics 4 and 5: Sampling and proportions; Interval estimates for proportions

Apply the sampling distribution of the sample proportion

\hat{p}

(mean

p

, standard deviation

\sqrt{p(1-p)/n}

) and construct approximate confidence intervals

\hat{p} \pm z^* \sqrt{\hat{p}(1-\hat{p})/n}

for a population proportion

Sample proportions and confidence intervals. The sampling distribution of $\hat{p}$ , the normal approximation, the CI formula with standard $z^*$ values, and worked Paper 2 / PSMT examples.

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Further differentiation (beyond the Unit 4 topic list)

Apply the product, quotient and chain rules to differentiate composite functions involving exponential, logarithmic, polynomial and trigonometric pieces, including logarithmic differentiation and the differentiation of inverse functions

Further differentiation. Logarithmic differentiation for products and powers, derivatives of inverse functions via $1 / f'(y)$ , and the standard PSMT and EA contexts in which the further rules appear.

9 min read →