§-Math Methods Q&A
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Math Methods Q&A by dot point
A short Q&A bank for every SA Math Methods syllabus dot point. Each question and answer is drawn directly from our worked dot-point page, so you can scan key concepts before opening the long-form answer.
Topic 1: Further Differentiation and Applications
Curve sketching combines intercepts, stationary points, concavity and end behaviour to produce an accurate graph.
Optimisation problems use the first derivative to locate the maximum or minimum value of a quantity subject to a constraint.
The product and quotient rules differentiate functions formed by multiplying or dividing two differentiable functions.
The second derivative measures the rate of change of the gradient and determines concavity and points of inflection.
The chain rule differentiates composite functions by multiplying the derivative of the outer function by the derivative of the inner function.
Topic 2: Discrete Random Variables
A discrete random variable assigns a numerical value to each outcome, and its probability distribution lists every value together with its probability.
The expected value is the long-run mean of a discrete random variable; the variance and standard deviation measure how spread out its values are.
The Bernoulli distribution models a single success/failure trial, and the binomial distribution counts successes across n independent trials with constant probability.
Topic 3: Integral Calculus
Antidifferentiation reverses differentiation to find the family of functions whose derivative is a given function, always including a constant of integration.
The area between a curve and the x-axis equals the definite integral, with regions below the axis requiring a sign adjustment.
The area between two curves is the integral of the upper function minus the lower function over the interval where they enclose a region.
The Fundamental Theorem of Calculus evaluates a definite integral as the difference of an antiderivative at the two limits.
Topic 4: Logarithmic Functions
The derivative of e^x is itself, the derivative of ln x is 1/x, and the chain rule extends both to composite functions.
The logarithmic graph is the reflection of the exponential graph in the line y = x, with a vertical asymptote and a characteristic slow growth.
Exponential equations are solved by taking logarithms of both sides and applying the power law to bring the unknown exponent down.
The logarithm laws turn products into sums, quotients into differences, and powers into multipliers, mirroring the index laws.
Topic 5: Continuous Random Variables and the Normal Distribution
A continuous random variable is described by a probability density function, where probability is the area under the curve found by integration.
The normal distribution is a symmetric bell-shaped density defined by its mean and standard deviation, with predictable probabilities given by the empirical rule.
A z-score standardises a normal value to the standard normal distribution, enabling exact probabilities to be found by technology or tables.
Topic 6: Sampling and Confidence Intervals
A confidence interval gives a range of plausible values for the population mean, built from the sample mean plus or minus a critical z-value times the standard error.
The margin of error sets the precision of an estimate; rearranging it determines the sample size needed for a target margin.
The sampling distribution of the sample mean is approximately normal, centred on the population mean, with a standard deviation that shrinks as the sample size grows.