HKDSE Maths M1 flashcards
HKDSE Maths M1 · Build recall with a focused review session.
30 cards. Read the term and the definition here. The deck at the top of this page is optional.
- Limit
- 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.
- Continuity
- 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.
- Derivative
- 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.
- Differentiability
- 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.
- Critical Point
- 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.
- Concavity
- 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.
- Definite Integral
- 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.
- Fundamental Theorem of Calculus
- The result linking differentiation and integration: differentiating an accumulated area function recovers the integrand, and a definite integral equals the change in an antiderivative.
- Improper Integral
- 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.
- Taylor Series
- 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.
- Convergence
- 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.
- Partial Derivative
- The derivative of a multivariable function with respect to one variable while holding the others fixed. It measures sensitivity to that single input direction.
- Gradient
- 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.
- Jacobian Matrix
- 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.
- Random Variable
- 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.
- Expectation
- The probability-weighted average value of a random variable, also called its mean. For continuous variables it is computed by integrating value times density.
- Variance
- 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.
- Probability Density Function
- 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.
- Cumulative Distribution Function
- 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.
- Central Limit Theorem
- 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.
- Confidence Interval
- 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.
- Hypothesis Test
- 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.
- Least Squares
- A fitting method choosing parameters that minimize the sum of squared residuals. For linear models the solution follows from normal equations involving matrix transposes.
- Integral
- 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.
- Tangent Line
- 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).
- Mean
- The arithmetic average of a data set, found by summing all values and dividing by the count; sensitive to extreme values or outliers.
- Median
- 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.
- Standard Deviation
- 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.
- Normal Distribution
- 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.
- Probability
- A number between 0 and 1 expressing how likely an event is, computed for equally likely outcomes as favorable outcomes divided by total outcomes.