SACE Stage 2 Mathematical Methods flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Derivative from first principles

The gradient of a curve at a point defined as f'(x) = lim(h→0)[f(x+h) − f(x)]/h. It gives the instantaneous rate of change of the function.

Chain rule

If y = f(g(x)), then dy/dx = f'(g(x))·g'(x). Differentiate the outer function, keeping the inner unchanged, then multiply by the derivative of the inner function.

Product rule

For y = u(x)v(x), the derivative is y' = u'v + uv'. The rule applies whenever two differentiable expressions are multiplied together.

Quotient rule

For y = u/v with v ≠ 0, y' = (u'v − uv')/v². The numerator order matters: subtract the derivative of the denominator times the numerator.

Stationary point test

Set f'(x) = 0 to find stationary points; use the sign of f'(x) either side, or f''(x), to classify each as local maximum, local minimum, or point of inflection.

Definite integral evaluation

For continuous f with antiderivative F, ∫ₐᵇ f(x) dx = F(b) − F(a). It measures signed area between the curve and the x-axis over the interval.

Area between two curves

If f(x) ≥ g(x) on [a, b], the enclosed area is ∫ₐᵇ [f(x) − g(x)] dx. Find intersection points first to determine the limits of integration.

Integration by substitution

Reverse the chain rule: let u = g(x), so du = g'(x) dx, then rewrite the integral entirely in u, integrate, and substitute back. Adjust limits for definite integrals.

Trapezoidal rule approximation

∫ₐᵇ f(x) dx ≈ (h/2)[f(x₀) + 2f(x₁) + … + 2f(xₙ₋₁) + f(xₙ)], where h = (b − a)/n. Accuracy improves as n increases.

Exponential function derivative

If y = e^(kx), then dy/dx = k·e^(kx). More generally, d/dx aˣ = aˣ·ln a, so the exponential function is proportional to its own derivative.

Logarithm derivative

For x > 0, d/dx ln x = 1/x, and d/dx ln(kx) = 1/x. The chain rule extends this to d/dx ln(g(x)) = g'(x)/g(x).

Sine and cosine derivatives

d/dx sin x = cos x and d/dx cos x = −sin x, with x in radians. Chain rule versions: d/dx sin(kx) = k cos(kx) and d/dx cos(kx) = −k sin(kx).

Normal distribution standardising

For X ~ N(μ, σ²), z = (x − μ)/σ converts X to the standard normal Z ~ N(0, 1). Probabilities are then read from the standard normal table or calculator.

Inverse normal calculation

Given a probability, find the z-score from the standard normal table, then convert back using x = μ + zσ. Useful for finding cut-off values, percentiles, and tolerance limits.

Binomial probability formula

For X ~ Bin(n, p), P(X = k) = C(n,k)·pᵏ·(1−p)^(n−k). It requires n independent trials, constant success probability p, and two possible outcomes per trial.

Binomial mean and variance

If X ~ Bin(n, p), then E(X) = np and Var(X) = np(1−p), so the standard deviation is √(np(1−p)). These follow from summing independent Bernoulli trials.

Continuous random variable

A random variable with a probability density function f where f(x) ≥ 0 and ∫f(x) dx = 1 over its range. Probabilities are areas: P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx.

Sample proportion distribution

For large n, the sample proportion p̂ is approximately normal with mean p and standard deviation √(p(1−p)/n). This supports confidence intervals and hypothesis tests.

Confidence interval for proportion

An approximate 95% confidence interval is p̂ ± 1.96√(p̂(1−p̂)/n). The multiplier changes with the confidence level, for example 1.645 for 90%.

Mathematical Investigation purpose

A school-assessed folio task in which students pose a question, build and refine a mathematical model, and justify conclusions using evidence and appropriate mathematical notation.

Skills and Applications Tasks

Supervised in-school assessment tasks worth part of the 70% school grade. They test routine procedures, problem-solving in context, and reasoning against the performance standards.

Performance standards A to E

Descriptions of five achievement levels used to grade school assessment. They describe quality of understanding, fluency, problem-solving, and reasoning rather than a raw score.

Reasoning in written solutions

Showing why a result is true by naming the rule or theorem used, displaying key algebraic steps, and stating a conclusion. Answers alone do not earn reasoning credit.

Fluency in assessment

Accurate and efficient recall and execution of procedures, such as differentiating, integrating, or computing probabilities, with correct units and appropriate rounding.

Modelling assumptions

Statements about simplifications made when building a model, such as constant rate or negligible air resistance. They must be stated and their effect on validity discussed.

Defining variables and units

Before calculating, state what each symbol represents and its unit, for example t = time in seconds. This supports communication marks and correct interpretation of answers.

Checking solutions in context

Verify that an answer satisfies the original equation and is physically sensible, rejecting values outside the domain such as negative lengths or times.

Examination structure

The external paper is three hours long and contributes 30% of the grade. It contains routine, problem-solving, and reasoning questions covering all topics, with a formula sheet provided.

Radians in calculus

Trigonometric derivatives and integrals require angles in radians. Convert degrees to radians by multiplying by π/180 before differentiating or integrating trigonometric functions.

Logarithm laws

ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, and ln(aⁿ) = n ln a, valid for positive a and b. These laws solve exponential equations by linearising them.

Solving exponential equations

Take logarithms of both sides to bring the unknown out of the exponent, then solve the resulting linear equation. Check that the solution lies in the domain.

Limit and continuity check

A function is continuous at x = a when f(a) is defined, lim(x→a) f(x) exists, and the two are equal. Discontinuities break differentiability at that point.