UPMSP Intermediate Mathematics key terms

A set of ordered pairs linking elements of one set to elements of another, or of a set to itself; the first elements form the domain and the second elements form the range.

A relation in which every element of the domain is paired with exactly one element of the codomain, so no input has two different outputs.

A function where different inputs always give different outputs, that is, f(x1) = f(x2) forces x1 = x2.

A function whose range equals its codomain, meaning every element of the codomain is the image of at least one element of the domain.

A function that is both one-one and onto; such functions are exactly the ones that possess an inverse function.

The function f⁻¹ that reverses f, defined only when f is bijective, so that f⁻¹(y) = x whenever f(x) = y.

A relation that is reflexive, symmetric and transitive; it splits the set into disjoint classes whose members are mutually related.

A rule that combines any two elements of a set to produce another element of the same set, such as addition on integers.

The unique value chosen for an inverse trigonometric function within its restricted range, for example sin⁻¹x lies between −π/2 and π/2.

Limiting a trigonometric function to an interval where it is one-one, which makes its inverse well defined and single-valued.

A scalar computed from a square matrix that measures whether the matrix is invertible; a zero determinant means no inverse exists.

The minor of an entry is the determinant left after deleting its row and column; the cofactor is that minor multiplied by (−1) raised to the row plus column index.

The transpose of the cofactor matrix; multiplying a matrix by its adjoint gives the determinant times the identity matrix.

A square matrix whose determinant is zero, so it has no multiplicative inverse and its rows or columns are linearly dependent.

A symmetric matrix equals its transpose; a skew-symmetric matrix equals the negative of its transpose, so all its diagonal entries are zero.

The matrix obtained by interchanging rows and columns; the transpose of a product reverses the order of the factors.

A function is continuous at a point when its limit there equals its value there; informally, the graph has no break or jump at that point.

A function is differentiable at a point when its derivative exists there; differentiability implies continuity, but continuity does not imply differentiability.

The instantaneous rate of change of a function with respect to its variable, defined as the limit of the difference quotient.

A rule for differentiating a composite function: the derivative of f(g(x)) is f′(g(x)) multiplied by g′(x).

A technique for finding dy/dx when the relation between x and y is not solved for y, by differentiating both sides and treating y as a function of x.

If a function is continuous on a closed interval, differentiable inside it, and equal at both endpoints, then its derivative vanishes somewhere inside the interval.

If a function is continuous on a closed interval and differentiable inside, some interior point has derivative equal to the average rate of change over the whole interval.

An interior point where the derivative is zero or does not exist; local maxima and minima can occur only at such points or at endpoints.

Classifying a critical point by the sign change of the derivative: positive to negative gives a maximum, negative to positive gives a minimum.

The way a curve bends, determined by the sign of the second derivative: positive means concave up, negative means concave down.

A point where the curve changes concavity, typically where the second derivative changes sign.

The family of all antiderivatives of a function, written with an arbitrary constant of integration because derivatives of constants are zero.

The limit of a sum of areas over a partition of an interval; it gives a signed area and is evaluated using the fundamental theorem of calculus.

A technique derived from the product rule: the integral of u dv equals uv minus the integral of v du, useful for products of unlike functions.

Breaking a rational function into a sum of simpler fractions so that each piece can be integrated using standard formulas.

The region bounded by a curve, the x-axis and two vertical lines, found by integrating the function between the limits; regions below the axis count as negative.

The order of the highest derivative appearing in the equation; degree is the power of that highest derivative after the equation is made polynomial in derivatives.

A solution of a differential equation containing as many arbitrary constants as the order of the equation, representing a whole family of curves.

A single solution obtained from the general solution by fixing the arbitrary constants using given initial conditions.

A multiplier, usually a function of x, that turns a non-exact first-order linear differential equation into one whose left side is the derivative of a product.

The dot product of one vector with the cross product of the other two; its absolute value gives the volume of the parallelepiped they span.

A vector perpendicular to two given vectors, with magnitude equal to the area of the parallelogram they span and direction fixed by the right-hand rule.

The cosines of the angles a line makes with the positive coordinate axes; their squares sum to one.

Two lines in space that are neither parallel nor intersecting; the shortest distance between them is measured along their common perpendicular.

The set of points satisfying all constraints of a linear programming problem simultaneously, usually a convex polygon in the first quadrant.

The linear expression, in profit or cost, that a linear programming problem seeks to maximise or minimise subject to the given constraints.

Solving a linear programming problem by evaluating the objective function at each vertex of the feasible region and choosing the best value.

The probability of event A given that event B has occurred, written P(A|B) and equal to P(A ∩ B) divided by P(B).

Events whose joint probability equals the product of their individual probabilities, so knowing one occurred does not change the chance of the other.

A formula that reverses a conditional probability, updating the probability of a cause given observed evidence using prior probabilities.

The probability distribution of the number of successes in n independent trials with constant success probability p, given by nCr p^r q^(n−r).

A real-valued function defined on the sample space of an experiment, assigning a number to each outcome.

The tangent of the angle a line makes with the positive x-axis, equal to the change in y divided by the change in x between two points.

A number describing how far a conic departs from a circle: zero for a circle, below one for an ellipse, one for a parabola, above one for a hyperbola.

The chord of a conic through its focus perpendicular to the axis of symmetry; its length is 4a for a parabola and 2b²/a for an ellipse or hyperbola.

A permutation counts ordered arrangements of chosen items, while a combination counts unordered selections; the combination count equals the permutation count divided by r factorial.

A formula expanding (a + b)^n as a sum of terms nCr a^(n−r) b^r, with the general term useful for finding a specific coefficient.

The value a function approaches as its input approaches a point, whether or not the function is defined at that point.