IB Mathematics: Analysis and Approaches HL flashcards

40 cards. Read the term and the definition here. The deck at the top of this page is optional.

Arithmetic sequence

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.

Geometric sequence

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).

Sum to infinity

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.

Logarithm

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.

Laws of logarithms

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.

Binomial coefficient

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.

Discriminant

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.

Complex conjugate

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.

Domain

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.

Range

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.

Composite 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.

Inverse function

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.

Asymptote

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

Exponential function

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.

Natural logarithm

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.

Radian

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°.

Arc length

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.

Sector area

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.

Sine rule

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.

Cosine rule

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.

Ambiguous case

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.

Pythagorean identity

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.

Double-angle identity

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.

Standard deviation

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.

Interquartile range

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.

Correlation coefficient

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.

Least squares regression line

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.

Conditional probability

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.

Independent events

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.

Binomial distribution

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).

Normal distribution

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.

Z-score

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.

Derivative

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.

Differentiation from first principles

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.

Chain 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.

Tangent

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.

Stationary 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.

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.

Antiderivative

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.

Definite integral

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.