BSEB Inter Mathematics key terms

The set of all ordered pairs formed by taking one element from each of two sets in turn. If A has m elements and B has n, then A × B has mn elements.

Any subset of the Cartesian product A × B. It links elements of one set to elements of another, and may be described by listing pairs, a rule, or an arrow diagram.

A relation in which every element of the domain is paired with exactly one element of the codomain. No input may produce two different outputs.

The set of all input values for which a relation or function is defined. For a real function given by a formula, it is the set of real numbers making the formula meaningful.

The set of all actual output values a function produces as its input runs through the domain. It is a subset of the codomain and may be smaller.

A function in which distinct inputs always give distinct outputs, so f(x₁) = f(x₂) forces x₁ = x₂. Also called injective, and it is the condition needed for an inverse.

A function whose range equals its codomain, so every element of the codomain is the image of at least one input. Also called surjective.

The function that reverses another, sending each output back to its unique input. It exists only when the original function is both one-one and onto.

The single agreed output chosen from all possible ones for an inverse trigonometric expression, fixed by the standard restricted range of that inverse function.

The function giving the angle whose sine is a given number, written sin⁻¹x, defined for x between −1 and 1 with output in [−π/2, π/2].

The function giving the angle whose tangent is a given number, written tan⁻¹x, defined for every real x with output strictly between −π/2 and π/2.

A square matrix equal to its own transpose, so the entry in row i, column j equals the entry in row j, column i for every pair of positions.

A square matrix whose transpose is its negative, so entries satisfy aᵢⱼ = −aⱼᵢ. Every diagonal entry must be zero, and only square matrices qualify.

The matrix obtained by rewriting rows as columns and columns as rows. For a product, the transpose reverses the order: (AB)ᵀ = BᵀAᵀ.

A single number computed from a square matrix that tells whether the matrix is invertible. It is zero exactly when the matrix has no inverse.

A square matrix whose determinant is zero, so it has no inverse. Its rows or columns are linearly dependent and it cannot solve a system uniquely.

The transpose of the cofactor matrix of a square matrix. Multiplying a matrix by its adjoint gives the determinant times the identity matrix, which yields the inverse formula.

A property of a function at a point meaning the limit there exists, the value there exists, and the two are equal. Graphically, the curve has no break at that point.

A property meaning the derivative exists at a point, which requires the left-hand and right-hand derivatives to be finite and equal. Differentiability at a point implies continuity there.

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

The rule for differentiating a composite function: the derivative of f(g(x)) is f′(g(x)) multiplied by g′(x). It is applied repeatedly for deeper nestings.

A technique for finding dy/dx when the relation between x and y is not solved for y. Differentiate both sides with respect to x, treating y as a function of x, then solve for dy/dx.

An interior point of a function's domain where the derivative is zero or fails to exist. Local maxima and minima among such points must still be tested.

A value of a function that is at least as large as all nearby values. At such a point the derivative is zero or undefined and the function changes from increasing to decreasing.

A method of classifying critical points by the sign change of the derivative around them: positive to negative means a local maximum, negative to positive means a local minimum.

A straight line touching a curve at a point and having there the same slope as the curve. Its slope equals the derivative at that point.

The straight line perpendicular to the tangent at a point on a curve. Its slope is the negative reciprocal of the derivative at that point.

The family of all antiderivatives of a function, written with an arbitrary constant of integration. Differentiating the result returns the original integrand.

The limit of Riemann sums over an interval, interpreted as signed area between the curve and the axis. It is evaluated using antiderivatives through the fundamental theorem of calculus.

A technique derived from the product rule, stated as ∫u dv = uv − ∫v du. It is used when the integrand is a product of two unlike types of function.

An integration technique that replaces a complicated expression by a new variable so the integral becomes a standard form. It reverses the chain rule.

The region enclosed between a curve and an axis or between two curves, computed as a definite integral. Portions below the axis contribute negatively unless absolute value is taken.

The order of the highest derivative appearing in the equation. It is unrelated to the number of terms or to the degree of any power.

The power of the highest-order derivative after the equation is cleared of radicals and fractions in the derivatives. It is defined only for polynomial-type equations in derivatives.

The solution of a differential equation containing as many arbitrary constants as the order of the equation. Particular solutions come from fixing those constants.

A multiplier that turns a first-order linear differential equation into an exact one. For dy/dx + Py = Q it equals e raised to the integral of P.

The scalar obtained from two vectors as the product of their magnitudes and the cosine of the angle between them. It is zero when the vectors are perpendicular.

The vector obtained from two vectors whose magnitude is the product of their magnitudes and the sine of the angle between them, directed perpendicular to both. It vanishes for parallel vectors.

One of the cosines of the angles a line or vector makes with the positive coordinate axes. Their squares always sum to one.

The dot product of one vector with the cross product of two others, giving a scalar whose absolute value is the volume of the parallelepiped they span. It is zero for coplanar vectors.

The set of points satisfying all constraints of a linear programming problem simultaneously, including non-negativity. The optimum is found among its corner points.

The linear expression in a linear programming problem that is to be maximised or minimised, such as profit or cost. Its value is tested at the corners of the feasible region.

A vertex of the feasible polygon in a linear programming problem. The fundamental theorem guarantees an optimal value of the objective function occurs at such a point.

The probability of an event given that another event has already occurred, written P(A|B) and equal to P(A ∩ B) divided by P(B), provided P(B) is not zero.

Two events whose joint probability equals the product of their separate probabilities. Knowing one occurred does not change the chance of the other.

A formula that revises a prior probability in light of new evidence, expressing P(A|B) in terms of P(B|A), P(A) and P(B). It underlies problems on tests and disease screening.

The probability distribution of the number of successes in a fixed number of independent trials, each with the same success probability. Its mean is np and variance npq.

A real-valued function defined on the sample space of an experiment, assigning a number to each outcome. It may be discrete or continuous.

A number of the form a + bi where a and b are real and i² = −1. It has a real part and an imaginary part and can be drawn as a point in the plane.

The non-negative distance of a complex number from the origin, written |a + bi| and equal to the square root of a² + b².

For a complex number a + bi, the number a − bi obtained by reversing the sign of the imaginary part. A number times its conjugate equals the square of its modulus.

The value a function approaches as its input approaches a specified point, whether or not the function is defined there. It is the foundation of both continuity and the derivative.

If a function is continuous on a closed interval, differentiable inside it, and equal at the endpoints, then its derivative is zero at some interior point.

If a function is continuous on a closed interval and differentiable inside, then at some interior point the derivative equals the average rate of change over the interval.