Secondary 5 Mathematics Extended Part

The number a function approaches as its variable approaches a given value, written lim x→a f(x).

The process of a variable or function value getting arbitrarily close to a number without necessarily reaching it.

Finding a limit by substituting the value into the function; if the result is a definite number, that number is the limit.

A form such as zero over zero obtained by substitution, where the limit is not yet determined and simplification is needed.

Writing a polynomial as a product of factors so that common factors can be cancelled to remove zero over zero.

Multiplying numerator or denominator by a conjugate surd to remove the square root, using the difference of squares.

Theorems for limits of sums, differences, products, quotients, scalar multiples and composite functions, usable without proof.

The behaviour of a function as x grows without bound; for rational functions, judged by comparing the highest degrees of numerator and denominator.

lim θ→0 (sin θ/θ) = 1 and lim x→0 ((eˣ−1)/x) = 1, the keys to limits of trigonometric and exponential functions.

The change in function value divided by the change in the variable, [f(b)−f(a)]/(b−a), equal to the slope of the secant line.

The instantaneous rate of change of a function, written f'(x) or dy/dx, numerically the slope of the tangent at that point.

The process of finding a derivative of a function; the official name in Module 2 of the curriculum.

Finding a derivative from the definition f'(x) = lim h→0 [f(x+h)−f(x)]/h.

d(uv)/dx = v(du/dx) + u(dv/dx); the derivative of a product is not the product of the derivatives.

d(u/v)/dx = [v(du/dx) − u(dv/dx)]/v²; the order of terms in the numerator must not be swapped.

dy/dx = (dy/du)(du/dx); for composite functions, the outer derivative times the inner derivative.

Differentiating both sides of an equation in x and y, attaching dy/dx to y-terms, then solving for dy/dx.

The derivative of the derivative, written f''(x) or d²y/dx², used to classify turning points and determine concavity.

A local maximum or minimum within a neighbourhood; the curriculum requires handling both global and local extrema.

Whether a curve is concave upward (f'' > 0) or concave downward (f'' < 0), decided by the sign of the second derivative.

The family of all antiderivatives of f(x), written ∫f(x)dx, with the constant of integration C.

A function F(x) whose derivative is f(x), i.e. F'(x) = f(x).

An arbitrary constant C representing the family of antiderivative curves shifted vertically.

The function f(x) being integrated in ∫f(x)dx.

∫xⁿdx = xⁿ⁺¹/(n+1) + C (n ≠ −1): raise the power by one, then divide by the new power.

∫kf(x)dx = k∫f(x)dx and ∫[f(x)±g(x)]dx = ∫f(x)dx ± ∫g(x)dx.

Letting the inner function be u, converting dx into du, integrating the simpler form and substituting back; integration by parts is outside the curriculum.

The curriculum's characterisation of indefinite integration: it reverses differentiation, each undoing the other.

A given function value (such as a point on the curve) used to pin down the constant of integration C.

The logarithm to base e, written ln x; ∫(1/x)dx = ln x + C.

A random variable taking any value in an interval, such as height or lifetime; probabilities are areas under a curve.

The bell-shaped continuous probability distribution symmetric about its mean, written X ~ N(μ, σ²).

The probability curve of a normal distribution: bell-shaped, symmetric, with total area one underneath.

The location parameter of the normal distribution; the mean, mode and median are all equal, on the axis of symmetry.

The spread parameter controlling the flatness of the curve; the larger the σ, the wider and flatter the curve.

Converting X into Z = (X−μ)/σ, counting how many standard deviations x lies from the mean.

The normal distribution with mean 0 and variance 1, Z ~ N(0, 1), read from its own table.

A table listing P(Z ≤ z); combined with symmetry it yields all interval probabilities.

The standard normal curve is symmetric about zero: P(Z < 0) = 0.5 and P(Z < −a) = P(Z > a).

About 68%, 95% and 99.7% of data lie within 1, 2 and 3 standard deviations of the mean.

A fixed constant describing the population, such as the mean μ or variance σ²; usually unknown and estimated from samples.

A quantity computed from a sample, such as x̄ or s²; it varies between samples and estimates a parameter.

The probability distribution of a statistic over all possible samples, e.g. the distribution of x̄.

The standard deviation of x̄, equal to σ/√n; it measures the precision of estimating μ by x̄.

For n large enough, the distribution of x̄ is approximately normal regardless of the population distribution.

Estimating a parameter by a single value: x̄ for μ and s² for σ².

An estimator whose expectation equals the parameter it estimates; E(x̄) = μ, and s² with divisor n−1 has expectation σ².

s² = Σ(xᵢ − x̄)²/(n−1); the divisor n−1 guarantees unbiasedness.

An interval estimate of a parameter centred on a statistic with a stated confidence level, e.g. x̄ ± 1.96σ/√n.

The long-run success rate of the interval procedure; 95% corresponds to the critical value 1.96 and 99% to 2.575.

The three equivalent notations y', f'(x) and dy/dx; evaluation at a point is written as dy/dx at x = x₀.

Understanding the derivative through lim h→0 [f(x+h)−f(x)]/h; M1 does not require lengthy computations with it.

f'(x₀) is the slope of the tangent to the curve at x = x₀ — the geometric meaning of the derivative.

dy/dx = (dy/du)(du/dx); for composite functions, outer derivative times inner derivative.

At a stationary point, f'' > 0 indicates a local minimum and f'' < 0 a local maximum.

The curve is concave upward when f'' > 0 and concave downward when f'' < 0.

The absolute maximum and minimum over the whole domain, versus relative extrema in a neighbourhood; applied questions must address both.

The derivative as instantaneous rate of change; speed, cost changes and inflation rates in applied questions.

The reverse of differentiation, ∫f(x)dx = F(x) + C; M1 requires only the basic formulas and substitution.

Simplify the integrand with u, integrate, then substitute back; integration by parts is outside M1.