European Baccalaureate Mathematics (5-period course) key terms

A straight line that the graph of a function approaches ever more closely as the input or output grows without bound, without necessarily ever meeting it.

The value that the outputs of a function approach as the input approaches a given point or grows without bound, whether or not the function is defined there.

A property of a function at a point where the limit exists, the function is defined there, and the two agree; informally, the graph has no break or jump.

The instantaneous rate of change of a function, defined as the limit of the difference quotient; geometrically it gives the slope of the tangent line at a point.

The property that a function has a finite derivative at a point; it implies continuity there, but continuity does not guarantee it, as sharp corners show.

The straight line through a point of a curve whose slope equals the derivative there, giving the best linear approximation to the curve near that point.

An interior point of the domain where the derivative is zero or fails to exist; candidates for local maxima and minima of a differentiable function.

A point where a function takes a value at least as large, or at most as small, as at all nearby points; found by testing critical points with the sign of the derivative.

The derivative of the derivative; it measures how the slope itself changes and determines concavity, so a positive value at a critical point signals a local minimum.

A point where a curve changes from concave to convex or the reverse; for twice differentiable functions it occurs where the second derivative changes sign.

The way a curve bends: concave up when the slope increases and the graph lies above its tangents, concave down when the slope decreases and the graph lies below them.

The family of all antiderivatives of a function, written with an arbitrary additive constant; differentiating any member returns the original function.

The limit of Riemann sums over an interval, interpreted as the signed area between the graph and the horizontal axis; evaluated using antiderivatives via the fundamental theorem.

The result linking the two branches of calculus: the definite integral of a continuous function over an interval equals the change in any of its antiderivatives across that interval.

A technique derived from the product rule that rewrites the integral of a product as a boundary term minus another integral, useful when one factor simplifies on differentiation.

An integration technique that replaces a composite expression by a new variable, converting the integral into a simpler standard form; the limits must be transformed for definite integrals.

The differentiation rule for composite functions: the derivative of f of g of x equals f prime at g of x multiplied by g prime of x.

A method for limits of indeterminate forms such as zero over zero, replacing the ratio of two functions by the ratio of their derivatives when the required conditions hold.

A function given by one polynomial divided by another; its graph may have vertical asymptotes where the denominator vanishes and a horizontal or oblique asymptote at infinity.

A function formed by adding terms of the form a times x to a whole-number power, with real coefficients; its degree is the highest exponent appearing.

A polynomial has a factor x minus a exactly when substituting a gives zero; repeated use breaks a polynomial into linear and irreducible quadratic factors.

Dividing a polynomial by x minus a leaves a remainder equal to the value of the polynomial at a, which links division to evaluation and to the factor theorem.

The quantity b squared minus four a c in a quadratic equation; its sign tells whether there are two, one, or no real roots.

A number of the form a plus b i, where i squared equals minus one; represented as a point in the plane with modulus and argument, and closed under all four operations.

The polar description of a complex number: its distance from the origin and the angle its position vector makes with the positive real axis, used for multiplication and powers.

A rule stating that raising a complex number in polar form to a power raises its modulus to that power and multiplies its argument by it; it also yields roots.

A sequence in which each term after the first is obtained by adding a fixed constant; the nth term is linear in n and the sum is quadratic in n.

A sequence in which each term after the first is a fixed multiple of the previous one; it converges only when the modulus of the ratio is less than one.

The property that the partial sums of an infinite series approach a finite limit; for geometric series this happens exactly when the ratio lies strictly between minus one and one.

A compact way of writing a sum using the capital Greek letter sigma, with an index, a lower limit, and an upper limit indicating which terms are added.

A proof technique for statements about natural numbers: establish the base case, assume the statement for n, then deduce it for n plus one.

A rule combining two vectors to give a number, equal to the product of their lengths and the cosine of the angle between them; zero exactly when they are perpendicular.

An operation on two vectors in space producing a vector perpendicular to both, whose length equals the area of the parallelogram they span; it vanishes for parallel vectors.

A description of a line as a fixed position vector plus a real parameter times a direction vector; it works in the plane and in three-dimensional space alike.

A description of a plane using a point and a normal vector, expressed by requiring the scalar product of the normal with any displacement in the plane to be zero.

The size of an angle measured as the length of the arc it cuts on a unit circle; a full turn is two pi radians, and it makes derivative formulas for sine and cosine simplest.

An equation in sine, cosine, or tangent that holds for all angles, such as the Pythagorean identity or the double-angle formulas, used to simplify expressions and solve equations.

Relations in any triangle: the sine rule equates side-to-sine ratios, while the cosine rule generalises Pythagoras to give a side from two sides and the included angle.

The probability of an event given that another event has occurred, computed as the probability of both divided by the probability of the condition, provided that is not zero.

A relationship between two events where knowing one occurred does not change the probability of the other; equivalently the probability of both equals the product of the probabilities.

A formula that reverses a conditional probability, expressing the probability of a cause given evidence in terms of the prior probability and the likelihood of the evidence.

The probability model for the number of successes in a fixed number of independent trials, each with the same success probability; described by two parameters.

A continuous symmetric bell-shaped probability model fixed by its mean and standard deviation; standardisation converts any such variable to the standard normal for table lookup.

The probability-weighted average of the values of a random variable, computed by summing value times probability in the discrete case; a measure of long-run centre.

The square root of the variance, expressing the typical distance of observations from their mean in the original units of the data or random variable.

Replacing a curve near a point by its tangent line, so that the function is approximated by its value plus the derivative times the displacement from that point.