Maths AA terms: C

C

Carrying capacityHL only

The largest population $K$ that an environment can support in the long term; in the logistic model every positive solution approaches $K$ as time increases.

Introduced in 5.18 First-order differential equations

Cartesian equations of a lineHL only · also symmetric form of a line

The equations $\frac{x - x_0}{l} = \frac{y - y_0}{m} = \frac{z - z_0}{n}$ of the line through $(x_0, y_0, z_0)$ with direction vector $\begin{pmatrix} l \\ m \\ n \end{pmatrix}$, found by eliminating the parameter.

Introduced in 3.14 Vector equations of lines

Cartesian form (of a complex number)HL only · also rectangular form, a + bi form

The way of writing a complex number as $z = a + b\mathrm{i}$ with $a, b \in \mathbb{R}$, giving its real and imaginary parts. It corresponds to the point $(a, b)$

Causationalso causal relationship, cause and effect

A relationship in which a change in one variable directly produces a change in another. Correlation alone never shows causation: both variables may depend on a third, or the link may be coincidence.

Introduced in 4.4 Linear correlation and regression

Census

A survey that collects data from every member of the population rather than from a sample.

Introduced in 4.1 Populations, samples and sampling

Chain rule

The rule for differentiating a composite function: if $y = g(u)$ and $u = f(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$

Change of base formula

The rule $\log_a x = \frac{\log_b x}{\log_b a}$

Chordalso chords, chord of a curve, secant line

A straight line segment joining two points on a curve. On a circle the longest chord is a diameter; on the graph of a function, the gradient of a chord is the average rate of change between its two endpoints.

Introduced in 3.4 Radians, arcs and sectors

Circular functionalso trigonometric function, circular functions

One of the functions $\sin x$, $\cos x$ and $\tan x$ of a real number $x$, defined from the coordinates of the point at angle $x$ radians on the unit circle.

Introduced in 3.7 Circular functions

Class intervalalso class, class boundaries, class width

A range of values used to group data, such as $50 \le t < 55$. Its boundaries are the values that separate it from the neighbouring classes; its width is the upper boundary minus the lower boundary.

Introduced in 4.2 Presenting data

Coefficient of determinationalso r²

The square $r^2$ of Pearson's correlation coefficient, with $0 \le r^2 \le 1$: the proportion of the variation in $y$

Coincident linesHL only

Two lines that are the same line: their direction vectors are parallel and they share a point, so every point of one lies on the other.

Introduced in 3.15 Coincident, parallel, intersecting and skew lines

CollinearHL only · also collinear points

Lying on the same straight line. Points $A$, $B$ and $C$ are collinear when $\overrightarrow{AC} = k\overrightarrow{AB}$

CombinationHL only · also selection

A selection of objects in which order does not matter. The number of combinations of $r$ objects chosen from $n$ different objects is ${}^nC_r = \frac{n!}{r!(n-r)!}$

Common difference

The fixed amount $d = u_{n+1} - u_n$ added to each term of an arithmetic sequence to get the next term. It can be positive, negative or zero.

Introduced in 1.2 Arithmetic sequences and series

Common logarithmalso lg, log base 10

A logarithm with base $10$. In IB notation $\log x$ with no base written means $\log_{10} x$, so $\log 1000 = 3$

Common perpendicularHL only

The line segment that joins two skew lines and is perpendicular to both. Its length is the shortest distance between the lines.

Introduced in 3.15 Coincident, parallel, intersecting and skew lines

Common ratio

The fixed number $r = \frac{u_{n+1}}{u_n}$

Complementary anglesHL only

Two angles whose sum is a right angle, $\frac{\pi}{2}$ (or $90^\circ$); the angles $\theta$ and $\frac{\pi}{2} - \theta$

Complementary eventsalso complement, A′

An event $A$ and the event $A'$ ("not $A$"), which contains every outcome in $U$ that is not in $A$. Exactly one of them occurs, so

Completing the square

Rewriting a quadratic $ax^2 + bx + c$ in the form $a(x - h)^2 + k$

Complex conjugateHL only · also conjugate, z*

For $z = a + b\mathrm{i}$, the number $z^* = a - b\mathrm{i}$, with the sign of the imaginary part changed. On an Argand diagram it is the reflection of $z$

Complex numberHL only

A number of the form $z = a + b\mathrm{i}$, where $a, b \in \mathbb{R}$ and $\mathrm{i}^2 = -1$

Component (of a vector)HL only · also components, column vector

One of the numbers $v_1$, $v_2$, $v_3$

Composite functionalso composition, f∘g

The function $f \circ g$ formed by applying $g$ first and then applying $f$ to the result, so that $(f \circ g)(x) = f(g(x))$

Compound angle identityHL only · also addition formulae, compound angle formula

An identity that writes a trigonometric function of a sum or difference of two angles, such as $\sin(A + B)$, in terms of the sines, cosines or tangents of $A$ and $B$ separately.

Introduced in 3.10 Compound angle identities

Compound interest

Interest calculated on the original amount plus all the interest already added, so the balance is multiplied by the same factor each period. The balances form a geometric sequence.

Introduced in 1.4 Financial applications of geometric sequences

Compounding periodalso compounding frequency

The length of time between two consecutive additions of interest, for example one month for monthly compounding. The number of periods per year is written $k$.

Introduced in 1.4 Financial applications of geometric sequences

Concave down

Describes a curve on an interval where its gradient is decreasing, so it bends downwards and lies below its tangents. This happens where $f''(x) < 0$.

Introduced in 5.7 The second derivative

Concave upalso convex

Describes a curve on an interval where its gradient is increasing, so it bends upwards and lies above its tangents. This happens where $f''(x) > 0$.

Introduced in 5.7 The second derivative

Concavity

The direction in which a curve bends on an interval: concave up or concave down. It is decided by the sign of the second derivative.

Introduced in 5.7 The second derivative

Conditional probabilityalso P(A | B)

The probability that $A$ occurs given that $B$ has occurred, $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$

Conjecture

A statement that appears to be true, usually from a pattern in examples, but has not yet been proved. A conjecture becomes a theorem once it is proved and is discarded if a counterexample is found.

Introduced in 1.6 Simple deductive proof

Conjugate root theoremHL only · also conjugate pairs

If a polynomial equation with real coefficients has a root $a + b\mathrm{i}$ with $b \neq 0$, then its conjugate $a - b\mathrm{i}$

Consecutive integersalso consecutive numbers

Integers that follow each other without gaps, each $1$ more than the one before, such as $7, 8, 9$. In algebra they are written $n, n + 1, n + 2, \ldots$

Consistent systemHL only · also inconsistent system, consistent, inconsistent

A system of equations that has at least one solution (a unique solution or infinitely many). A system with no solution is called inconsistent.

Introduced in 1.16 Systems of linear equations

Constant of integrationalso +C

The arbitrary constant $C$ added to an antiderivative to give every function with the same derivative. Its value can be found from a boundary condition, a known point on the curve.

Introduced in 5.5 Introduction to integration

Constant termalso term independent of x

The term of an expression or expansion that does not contain the variable, that is, the term in $x^0$. In $(2x - \frac{1}{x})^6$

Constraintalso constraint equation

A condition linking the variables in an optimisation problem, such as a fixed volume or a fixed length of material. It is used to write the quantity to be optimised in terms of a single variable.

Introduced in 5.8 Stationary points, optimisation and inflexion

ContinuityHL only · also continuous function, continuous

A function $f$ is continuous at $x = a$ if $f(a)$ is defined, $\lim_{x \to a} f(x)$

Continuous compoundingalso compounded continuously

Compound interest in the limit where the number of compounding periods per year increases without bound. A principal $P$ at nominal annual rate $r$ grows to $A = Pe^{rt}$ after $t$

Continuous data

Numerical data that can take any value in an interval and are obtained by measuring, such as height, mass or time. Recorded values are rounded to some degree of accuracy.

Introduced in 4.1 Populations, samples and sampling

Continuous random variableHL only

A random variable that can take any value in an interval of real numbers, such as a time or a length. Its probabilities are areas under a probability density function, and $P(X = a) = 0$ for any single value $a$.

Introduced in 4.14 Continuous random variables

Continuous uniform distributionHL only · also rectangular distribution

The distribution of a continuous random variable whose probability density function is constant, $f(x) = \frac{1}{b - a}$, on an interval $a \le x \le b$

Convenience samplingalso opportunity sampling

A non-random sampling method that uses members of the population who are easy to reach, such as the first people met.

Introduced in 4.1 Populations, samples and sampling

Convergent seriesalso converges, convergence

An infinite series whose partial sums approach a finite limit as the number of terms increases. A geometric series converges exactly when $|r| < 1$.

Introduced in 1.8 Infinite geometric series

CoplanarHL only · also coplanar points, coplanar lines

Lying in the same plane. Any three points are coplanar; four points or two lines may or may not be.

Introduced in 3.17 Vector equations of planes

Correlationalso positive correlation, negative correlation

A relationship between two variables in which one tends to increase (positive correlation) or decrease (negative correlation) as the other increases. Linear correlation is when the points lie close to a straight line.

Introduced in 4.4 Linear correlation and regression

Cosine rule

In any triangle $ABC$, $c^2 = a^2 + b^2 - 2ab\cos C$

CounterexampleHL only · also counter-example

A single specific case that satisfies the conditions of a general statement but for which its conclusion is false. One counterexample is enough to disprove the statement.

Introduced in 1.15 Proof by induction, contradiction and counterexample

Cover-up methodHL only · also substitution method

A quick way to find a partial fraction numerator: substitute the value of $x$ that makes one linear factor zero into the rest of the fraction, with that factor covered up.

Introduced in 1.11 Partial fractions

Critical valueHL only · also critical values

A value of $x$ at which an expression in an inequality is zero or undefined. Critical values split the number line into intervals on which the sign of the expression does not change; a sign diagram records that sign on each interval.

Introduced in 2.15 Solving inequalities

Cumulative frequencyalso ogive, cumulative frequency graph

The running total of the frequencies up to a given value or upper class boundary. Plotted against the upper boundaries, it gives a cumulative frequency graph.

Introduced in 4.2 Presenting data

Cumulative probabilityalso P(X ≤ x), cdf

The probability that a random variable takes a value less than or equal to a given value, $P(X \le x)$. A GDC's binomial cdf function gives it directly.

Introduced in 4.8 The binomial distribution

CuspHL only · also corner

A point where a continuous graph changes direction sharply, so the gradients from the left and the right are different (a corner) or tend to $\pm\infty$ (a cusp, as for $y = x^{2/3}$ at $0$). The function is not differentiable there.

Introduced in 5.12 Continuity, differentiability and first principles

Cycle (of a periodic function)also complete cycle

One complete repetition of the pattern of a periodic graph, covering an interval of $x$ equal to one period.

Introduced in 3.7 Circular functions