HKDSE Maths M1 key terms

The value a function approaches as its input approaches a chosen point, whether or not the function is defined there. It is the foundation for continuity, derivatives, and integrals.

A property of a function whose value at a point equals the limit of the function there. Informally, the graph has no break, jump, or hole at that point.

The instantaneous rate of change of a function with respect to its variable, defined as the limit of a difference quotient. Geometrically it gives the slope of the tangent line.

The property that a function has a well-defined derivative at a point. Differentiability at a point implies continuity there, but continuity does not imply differentiability.

An interior point of a function's domain where the derivative is zero or fails to exist. Such points are candidates for local maxima, minima, or inflection behavior.

The way a curve bends, determined by the sign of the second derivative. Positive second derivative means concave up; negative means concave down; a sign change marks an inflection point.

The signed area between a function's graph and the horizontal axis over an interval, defined as a limit of Riemann sums. It yields a number, unlike the indefinite integral.

The result linking differentiation and integration: differentiating an accumulated area function recovers the integrand, and a definite integral equals the change in an antiderivative.

An integral over an unbounded interval or of a function with an infinite discontinuity. It is defined by a limit and either converges to a finite value or diverges.

An infinite power series expansion of a smooth function about a point, built from the function's derivatives there. Truncating it gives polynomial approximations with controllable error.

The property of a sequence or series whose terms approach a finite limit or whose partial sums approach a finite sum. Divergence means no such finite limit exists.

The derivative of a multivariable function with respect to one variable while holding the others fixed. It measures sensitivity to that single input direction.

The vector of all first partial derivatives of a scalar function. It points in the direction of steepest increase and is perpendicular to level curves or surfaces.

The matrix of first partial derivatives of a vector-valued function, describing its best linear approximation. Its determinant measures local volume scaling under the transformation.

A function assigning a numerical value to each outcome of a random experiment. It may be discrete, taking countable values, or continuous, described by a density.

The probability-weighted average value of a random variable, also called its mean. For continuous variables it is computed by integrating value times density.

The expected squared deviation of a random variable from its mean, measuring spread. Its square root is the standard deviation, expressed in the original units.

A nonnegative function whose integral over an interval gives the probability that a continuous random variable falls in that interval. It integrates to one over the whole line.

A function giving the probability that a random variable is at most a given value. It is nondecreasing, ranges from zero to one, and its derivative is the density when continuous.

For independent identically distributed variables with finite nonzero variance, the suitably centered and scaled sample mean approaches a standard normal distribution as sample size grows; finite-sample accuracy depends on the population distribution.

A range of plausible values for an unknown parameter, constructed so that a stated proportion of repeated samples would produce intervals containing the true value.

A procedure comparing observed data against a null hypothesis using a test statistic. A small p-value indicates the data are unlikely under the null.

A fitting method choosing parameters that minimize the sum of squared residuals. For linear models the solution follows from normal equations involving matrix transposes.

A quantity representing accumulated change, computed as a limit of sums of areas; the definite integral gives signed area under a curve between two bounds.

A line touching a curve at a point with slope equal to the derivative there; its equation is y - f(a) = f'(a)(x - a).

The arithmetic average of a data set, found by summing all values and dividing by the count; sensitive to extreme values or outliers.

The middle value of an ordered data set, or the average of the two middle values when the count is even; resistant to outliers compared with the mean.

A measure of the typical distance of data values from the mean, equal to the square root of the variance; larger values indicate more spread.

A symmetric, bell-shaped probability distribution defined by its mean and standard deviation; roughly 68% of values fall within one standard deviation of the mean.

A number between 0 and 1 expressing how likely an event is, computed for equally likely outcomes as favorable outcomes divided by total outcomes.