IB Mathematics: Analysis and Approaches HL key terms

A sequence in which each term after the first comes from adding a fixed constant, the common difference, to the previous term; the nth term is u₁ + (n − 1)d.

A sequence in which each term after the first is found by multiplying the previous term by a fixed non-zero constant, the common ratio; the nth term is u₁r^(n−1).

For a geometric series whose common ratio r satisfies |r| < 1, the total of all infinitely many terms, given by u₁/(1 − r); if |r| ≥ 1 the sum does not exist.

The exponent to which a base must be raised to produce a given number: log_b x = y means b^y = x, where the base b is positive and not 1 and x is positive.

Rules that rewrite products, quotients and powers as sums, differences and multiples: log(ab) = log a + log b, log(a/b) = log a − log b, and log(a^n) = n log a, all for the same base.

The count of ways to choose r objects from n distinct objects, written nCr or as a binomial coefficient, equal to n!/(r!(n−r)!), and appearing in the expansion of (a + b)^n.

The expression b² − 4ac for the quadratic ax² + bx + c; its sign shows whether the equation has two distinct real roots, one repeated real root, or no real roots.

For a complex number z = a + bi, the conjugate is a − bi, written z*; multiplying a number by its conjugate gives the real value a² + b², which helps simplify division.

The set of all input values for which a function is defined; for example, f(x) = √x has domain x ≥ 0 unless the question states a restriction.

The set of all output values a function actually produces as its input runs through the domain; it can be read from a graph or found by analysing the function.

A function built by applying one function to the output of another, written (f ∘ g)(x) = f(g(x)); its domain excludes any x for which g(x) or f(g(x)) is undefined.

The function that undoes another, written f⁻¹, so that f⁻¹(f(x)) = x; it exists only when f is one-to-one, and its graph is the reflection of f in the line y = x.

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

A function of the form f(x) = a·b^x where b is positive and not 1 and the variable sits in the exponent; it models growth or decay at a constant percentage rate.

The logarithm to base e, written ln x; it is the inverse of e^x and is used to solve equations in which the unknown appears in an exponent.

A unit of angle measure in which one radian is the angle at the centre of a circle subtended by an arc equal in length to the radius; π radians = 180°.

The length along part of a circle's circumference; for a circle of radius r and a central angle θ measured in radians, it is rθ.

The area of the region enclosed by two radii and an arc of a circle; with radius r and central angle θ in radians it equals ½r²θ.

In any triangle, a/sin A = b/sin B = c/sin C; it finds a side or angle when a side and its opposite angle are known, but may give the ambiguous case.

In any triangle, c² = a² + b² − 2ab cos C; it finds the third side from two sides and the included angle, or an angle from all three sides.

The situation where the sine rule produces two possible triangles from two given sides and a non-included angle, because an angle and its supplement have the same sine.

The identity sin²θ + cos²θ = 1, which holds for every angle and comes from the unit circle; it converts between sine and cosine and simplifies expressions.

An identity giving a trigonometric ratio of 2θ in terms of ratios of θ, especially sin 2θ = 2 sin θ cos θ and cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ.

A measure of how spread out data values are, giving a typical distance from the mean; it is the square root of the variance and uses the same units as the data.

The difference between the upper quartile and lower quartile, Q₃ − Q₁; it describes the spread of the middle half of the data and is not affected by extreme values.

A number r between −1 and 1 that measures how closely two variables follow a straight line; its sign gives the direction and its size the strength of the linear relationship.

The line that makes the sum of the squared vertical distances from the data points as small as possible; it is used to predict one variable from the other.

The probability that event A happens given that event B has already happened, written P(A|B) and equal to P(A ∩ B)/P(B) provided P(B) > 0.

Two events whose probabilities do not influence each other; they satisfy P(A ∩ B) = P(A) × P(B), and knowing one occurred does not change the probability of the other.

The distribution of the number of successes in n independent trials that each have the same success probability p; individual probabilities use nCr p^r (1 − p)^(n−r).

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

The number of standard deviations a value lies above or below the mean, z = (x − μ)/σ; it converts any normal distribution to the standard normal for comparison and probability.

The instantaneous rate of change of a function at a point, defined as the limit of a chord's gradient as the two points merge; it equals the gradient of the tangent.

Finding a derivative directly from the limit definition f'(x) = lim(h→0) [f(x + h) − f(x)]/h instead of using a shortcut rule.

The rule for differentiating a composite function: dy/dx = dy/du × du/dx, used when one expression is nested inside another, as in (3x + 1)^5.

A straight line that touches a curve at a point and has the same gradient as the curve there; its gradient is the derivative at that point.

A point on a graph where the derivative is zero and the tangent is horizontal; it may be a local maximum, a local minimum or a point of inflection.

A point where a curve changes from concave up to concave down or the reverse; the second derivative is zero or undefined there, but that alone does not guarantee one.

A function whose derivative is the given function; because constants differentiate to zero, antiderivatives of the same function differ only by a constant, written + C.

The limit of a sum of rectangle areas between two x-values; it gives the signed area between a curve and the x-axis, with area below the axis counted as negative.