Glossary
Every technical word in the course and what it means in plain language, with flashcards for practice.
Every technical term in the course, with a plain-language meaning. On every page these words are marked with a dotted underline: hover or tap one to see its meaning. The “Used in” links point to the sections where each term appears.
Ch 1 · SetsCh 2 · Relations and FunctionsCh 3 · Trigonometric FunctionsCh 4 · Complex Numbers and Quadratic EquationsCh 5 · Linear InequalitiesCh 6 · Permutations and CombinationsCh 7 · Binomial TheoremCh 8 · Sequences and SeriesCh 9 · Straight LinesCh 10 · Conic SectionsCh 11 · Introduction to Three Dimensional GeometryCh 12 · Limits and DerivativesCh 13 · StatisticsCh 14 · Probability
Ch 1 · Sets
| Term | Meaning | Used in |
|---|---|---|
| closed interval | $[a,b]=\{x\in\mathbb{R} : a\le x\le b\}$ — both end points included. | 1.2 |
| complement of a set | $A'=U-A$: the elements of the universal set $U$ that are not in $A$. It depends on the choice of $U$. | 1.3 |
| De Morgan's laws | $(A\cup B)'=A'\cap B'$ and $(A\cap B)'=A'\cup B'$: taking a complement turns a union into an intersection and vice versa. | 1.3 14.1 |
| difference of sets | $A-B=\{x : x\in A \text{ and } x\notin B\}$ — the elements of $A$ that are not in $B$. In general $A-B\ne B-A$. | 1.3 |
| disjoint sets | Sets with no element in common: $A\cap B=\phi$. In a Venn diagram their circles do not overlap. | 1.3 6.1 |
| distributive laws for sets | $A\cap(B\cup C)=(A\cap B)\cup(A\cap C)$ and $A\cup(B\cap C)=(A\cup B)\cap(A\cup C)$. | |
| element of a set | One of the objects that make up a set. We write $a\in A$ when $a$ is an element (member) of $A$, and $a\notin A$ when it is not. | 6.3 |
| empty set | The set with no elements, written $\phi$ or $\{\ \}$. Note that $\{0\}$ and $\{\phi\}$ are not empty. | 1.1 1.2 2.1 2.2 |
| equal sets | Sets with exactly the same elements; we write $A=B$. One element in one set but not the other makes them unequal. | 1.1 |
| finite set | A set that is empty or whose elements can be counted with the counting coming to an end; $n(A)$ is its number of distinct elements. | 1.1 1.3 2.1 2.2 |
| infinite set | A set that is not finite, such as $\mathbb{N}$ or the real numbers between $0$ and $1$. | 1.1 2.2 14.1 |
| intersection of sets | $A\cap B=\{x : x\in A \text{ and } x\in B\}$ — the elements common to both sets. | 1.3 2.4 |
| interval | An unbroken stretch of the real line, such as $(a,b)=\{x : a\lt x\lt b\}$ or $[a,b)=\{x : a\le x\lt b\}$. A square bracket includes the end point, a round one excludes it. | 1.1 1.2 1.3 2.1 |
| open interval | $(a,b)=\{x\in\mathbb{R} : a\lt x\lt b\}$ — neither end point included. | 1.2 1.3 2.1 12.1 |
| proper subset | A subset that is not the whole set: $A\subset B$ and $A\ne B$. | 1.2 |
| roster form | Writing a set by listing its elements, each once, inside braces, e.g. $\{2,4,6,8\}$. Order and repetition do not matter. | 1.1 1.3 2.1 2.2 |
| set-builder form | Writing a set by a property its elements share: $\{x : P(x)\}$, read 'the set of all $x$ such that $P(x)$'. | 1.1 1.2 2.2 |
| singleton set | A set with exactly one element, such as $\{5\}$ or $\{\phi\}$. | 1.2 |
| subset | $A\subset B$ when every element of $A$ is also an element of $B$. In this course $A\subset B$ allows $A=B$. | 1.2 1.3 2.1 2.2 |
| superset | If $A$ is a proper subset of $B$, then $B$ is a superset of $A$. | 1.2 |
| union of sets | $A\cup B=\{x : x\in A \text{ or } x\in B\}$ — everything in at least one of the sets, each element written once. | 1.1 1.3 |
| universal set | The background set $U$ of a discussion, containing every set under consideration as a subset; drawn as a rectangle. | 1.2 1.3 14.1 |
| Venn diagram | A picture of sets: the universal set $U$ is a rectangle and each set is a closed region (usually a circle) inside it; overlapping circles show common elements. | 1.2 1.3 14.1 14.2 |
| well-defined | Describes a collection for which every object can be decided, without opinion, to be in it or not. Only well-defined collections are sets. | 1.1 |
Ch 2 · Relations and Functions
| Term | Meaning | Used in |
|---|---|---|
| arrow diagram | A picture of a relation: the two sets drawn as ovals, with an arrow from $a$ to $b$ for every pair $(a,b)$ in the relation. | 2.2 2.3 |
| break-even point | The quantity at which profit $P=R-C$ is zero: revenue exactly covers cost. | 2.4 5.2 |
| Cartesian plane | The plane with each point labelled by an ordered pair of real numbers; as a set it is $\mathbb{R}\times\mathbb{R}$. | 2.1 2.3 |
| Cartesian product | $A\times B=\{(a,b) : a\in A,\ b\in B\}$, the set of all ordered pairs with first component in $A$ and second in $B$; $n(A\times B)=n(A)\cdot n(B)$. | 2.1 2.2 6.1 |
| codomain | For a relation or function from $A$ to $B$, the codomain is the whole set $B$ — the set the arrows aim at. The range is the part of it actually reached. | 2.2 2.3 |
| component of an ordered pair | In $(a,b)$, $a$ is the first component and $b$ the second; in $A\times B$ the first component comes from $A$. | 2.1 2.2 2.3 |
| constant function | $f(x)=c$ for every $x$; range $\{c\}$, graph a horizontal line. | 2.3 2.4 12.1 12.2 |
| domain of a relation | The set of all first components of the pairs in the relation: $\{a : (a,b)\in R\}$. | |
| empty relation | The relation $\phi\subset A\times B$ in which no element is related to any other. | 2.2 |
| greatest integer function | $[x]$ = the greatest integer less than or equal to $x$; e.g. $[2.7]=2$, $[-2.7]=-3$. Range $\mathbb{Z}$; graph a staircase. | 2.3 12.1 |
| identity function | $f(x)=x$ on $\mathbb{R}$; its graph is the line $y=x$. | 2.3 2.4 |
| image and pre-image | If $(a,b)$ is in a relation (or $f(a)=b$ for a function), $b$ is an image of $a$ and $a$ is a pre-image of $b$. | 2.2 |
| modulus function | $f(x)=|x|$, equal to $x$ for $x\ge0$ and $-x$ for $x\lt0$; range $[0,\infty)$, graph a V with corner at the origin. | 2.3 5.2 12.1 |
| ordered pair | A pair $(a,b)$ written in a definite order. $(a,b)=(c,d)$ exactly when $a=c$ and $b=d$, so $(1,2)\ne(2,1)$. | 1.2 2.1 2.2 4.3 |
| ordered triple | A triple $(a,b,c)$ with three components in a definite order; $A\times B\times C$ is the set of all such triples. | 2.1 11.1 |
| pointwise operation | Combining functions value by value: to find $(f+g)(3)$, compute $f(3)$ and $g(3)$ and add them. | 2.4 |
| polynomial function | $f(x)=a_0+a_1x+\cdots+a_nx^n$ with $n$ a non-negative integer and real coefficients; domain $\mathbb{R}$. | 2.3 |
| product of functions | $(fg)(x)=f(x)\,g(x)$, defined on $D_f\cap D_g$. It is not the same as applying one function to the other. | 12.1 |
| quotient of functions | $\left(\dfrac fg\right)(x)=\dfrac{f(x)}{g(x)}$, defined on $D_f\cap D_g$ with the zeros of $g$ removed. | |
| range of a function | The set of all outputs $\{f(x) : x \text{ in the domain}\}$ — a subset of the codomain. | 1.1 |
| range of a relation | The set of all second components of the pairs in the relation: $\{b : (a,b)\in R\}$. It is always a subset of the codomain. | |
| rational function | A quotient $\dfrac{p(x)}{q(x)}$ of polynomials, defined wherever $q(x)\ne0$. | 2.3 12.1 12.2 12.4 |
| real function | A function whose domain and codomain are subsets of $\mathbb{R}$. Given only a formula, its domain is the largest set of real $x$ for which the formula gives a real value. | 2.3 2.4 12.1 12.3 |
| relation | A relation from $A$ to $B$ is any subset $R$ of $A\times B$; $(a,b)\in R$ means $a$ is related to $b$. A relation from $A$ to $A$ is a relation on $A$. | 2.2 2.3 |
| scalar multiple of a function | $(\alpha f)(x)=\alpha\,f(x)$ for a real number $\alpha$; same domain as $f$. Its graph is $f$'s graph stretched vertically by $\alpha$ (and reflected if $\alpha\lt0$). | 2.4 12.1 |
| signum function | $\operatorname{sgn}x=1$ for $x\gt0$, $0$ for $x=0$, $-1$ for $x\lt0$; range $\{-1,0,1\}$. | 2.3 12.1 |
| sum of functions | $(f+g)(x)=f(x)+g(x)$, defined on $D_f\cap D_g$; the difference $f-g$ is defined in the same way. | |
| universal relation | The relation $A\times B$ itself, in which every element of $A$ is related to every element of $B$. | 2.2 |
Ch 3 · Trigonometric Functions
| Term | Meaning | Used in |
|---|---|---|
| allied angles | Angles such as $-x$, $\tfrac{\pi}{2}\pm x$, $\pi\pm x$, $\tfrac{3\pi}{2}\pm x$, $2\pi-x$, whose trigonometric values are $\pm$ those of $x$ (with $\sin\leftrightarrow\cos$ swapped for $\tfrac{\pi}{2}$ and $\tfrac{3\pi}{2}$). | 3.2 3.3 9.1 |
| arc length | The distance along a circle between two points. An arc subtending $\theta$ radians in a circle of radius $r$ has length $s=r\theta$. | 3.1 12.2 |
| ASTC rule | Signs by quadrant: All positive in I, Sine (and cosec) in II, Tangent (and cot) in III, Cosine (and sec) in IV. | 3.2 3.3 |
| beat frequency | When two tones of nearby frequencies $f_1$, $f_2$ sound together, the loudness swells $|f_1-f_2|$ times per second — a direct consequence of the sum-to-product formula. | 3.4 |
| co-function identities | $\sin\left(\tfrac{\pi}{2}-x\right)=\cos x$, $\ \cos\left(\tfrac{\pi}{2}-x\right)=\sin x$, $\ \tan\left(\tfrac{\pi}{2}-x\right)=\cot x$: a function of an angle equals the co-function of its complement. | 12.4 |
| complementary angles | Two angles that add up to $\tfrac{\pi}{2}$ ($90^\circ$), such as $15^\circ$ and $75^\circ$; the sine of one is the cosine of the other. | |
| compound angle | An angle written as a sum or difference of two angles, such as $x+y$ or $45^\circ-30^\circ$; its sine, cosine and tangent are found with the sum and difference formulae. | |
| cosecant | $\operatorname{cosec}x=\dfrac{1}{\sin x}$, defined for $x\ne n\pi$; its range is $(-\infty,-1]\cup[1,\infty)$. | 3.2 |
| cotangent | $\cot x=\dfrac{\cos x}{\sin x}$, defined for $x\ne n\pi$; it has period $\pi$ and range $\mathbb{R}$. | 3.2 3.3 |
| coterminal angles | Angles with the same initial and terminal sides; they differ by a whole number of complete turns, e.g. $30^\circ$, $390^\circ$ and $-330^\circ$. | 3.1 |
| degree measure | Measuring angles in degrees: one complete turn is $360^\circ$, with $1^\circ=60'$ (minutes) and $1'=60''$ (seconds). | 3.1 |
| double-angle formulae | $\sin2x=2\sin x\cos x$, $\ \cos2x=\cos^2x-\sin^2x=2\cos^2x-1=1-2\sin^2x$, $\ \tan2x=\dfrac{2\tan x}{1-\tan^2x}$. | 3.4 |
| even and odd functions | $f$ is even if $f(-x)=f(x)$ (e.g. $\cos x$) and odd if $f(-x)=-f(x)$ (e.g. $\sin x$, $\tan x$). | 3.2 |
| half-angle formulae | $\sin^2x=\tfrac{1-\cos2x}{2}$, $\ \cos^2x=\tfrac{1+\cos2x}{2}$; so $\sin\tfrac x2=\pm\sqrt{\tfrac{1-\cos x}{2}}$, with the sign set by the quadrant of $\tfrac x2$. | |
| initial side | The starting position of the rotating ray that forms an angle; in standard position it lies along the positive $x$-axis. | 3.1 |
| multiple angle | An angle such as $2x$ or $3x$ that is a whole-number multiple of $x$; its trigonometric values can be written in terms of those of $x$. | 3.4 |
| period of a function | The smallest positive $T$ with $f(x+T)=f(x)$ for all $x$: $2\pi$ for $\sin,\cos,\sec,\operatorname{cosec}$ and $\pi$ for $\tan,\cot$. | 3.2 |
| periodic function | A function with $f(x+T)=f(x)$ for some $T\gt0$ and every $x$ in its domain; its graph repeats every $T$. | 3.2 |
| positive and negative angles | An angle is positive when the rotation is anticlockwise and negative when it is clockwise; e.g. a quarter turn clockwise is $-90^\circ$. | 3.1 3.2 3.3 |
| product-to-sum formulae | $2\sin x\cos y=\sin(x+y)+\sin(x-y)$, $\ 2\cos x\cos y=\cos(x+y)+\cos(x-y)$, $\ 2\sin x\sin y=\cos(x-y)-\cos(x+y)$. | 3.4 |
| quadrantal angle | An angle in standard position whose terminal side lies on a coordinate axis, such as $0^\circ, 90^\circ, 180^\circ, -270^\circ$. | 3.1 |
| radian | The angle subtended at the centre of a circle by an arc equal in length to the radius. $\pi$ radians $=180^\circ$, so $1$ rad $\approx57.3^\circ$. | 3.1 3.2 4.1 12.2 |
| reference angle | The acute angle between the terminal side of an angle and the $x$-axis; trigonometric values at the angle equal $\pm$ those at the reference angle. | 3.2 |
| standard position | An angle is in standard position when its vertex is at the origin and its initial side lies along the positive $x$-axis. | 3.1 3.2 |
| sum and difference formulae | $\sin(x\pm y)=\sin x\cos y\pm\cos x\sin y$, $\ \cos(x\pm y)=\cos x\cos y\mp\sin x\sin y$, $\ \tan(x\pm y)=\dfrac{\tan x\pm\tan y}{1\mp\tan x\tan y}$. | 3.3 3.4 |
| sum-to-product formulae | $\sin x+\sin y=2\sin\tfrac{x+y}{2}\cos\tfrac{x-y}{2}$, $\ \cos x+\cos y=2\cos\tfrac{x+y}{2}\cos\tfrac{x-y}{2}$, $\ \cos x-\cos y=-2\sin\tfrac{x+y}{2}\sin\tfrac{x-y}{2}$. | 3.4 12.2 |
| terminal side | The final position of the rotating ray that forms an angle. Its quadrant decides the signs of the trigonometric functions of the angle. | 3.1 3.2 |
| trigonometric function | Any of $\sin,\cos,\tan,\cot,\sec,\operatorname{cosec}$, defined for real numbers through the unit circle; for acute angles they agree with the right-triangle ratios. | 3.2 12.2 12.3 12.4 |
| triple-angle formulae | $\sin3x=3\sin x-4\sin^3x$, $\ \cos3x=4\cos^3x-3\cos x$, $\ \tan3x=\dfrac{3\tan x-\tan^3x}{1-3\tan^2x}$. | 3.4 |
| unit circle | The circle $x^2+y^2=1$ of radius $1$ centred at the origin. The point reached after travelling a distance $x$ round it from $(1,0)$ is $(\cos x,\sin x)$. | 3.2 3.3 12.2 |
| vertical asymptote | A vertical line $x=a$ that a graph approaches without meeting, because the function grows without bound near $a$ — e.g. $x=\tfrac{\pi}{2}$ for $\tan x$. | 3.2 |
Ch 4 · Complex Numbers and Quadratic Equations
| Term | Meaning | Used in |
|---|---|---|
| additive inverse | The number $-z$ with $z+(-z)=0$; for $z=a+ib$, $-z=-a-ib$. | 4.1 |
| Argand diagram | A drawing of complex numbers as points (or arrows from $O$) in the Argand plane. | 4.3 |
| Argand plane | The plane in which $z=a+ib$ is represented by the point $(a,b)$; also called the complex plane. | 4.3 |
| complex number | A number $z=a+ib$ with $a,b\in\mathbb{R}$ and $i^2=-1$. The set of all complex numbers is $\mathbb{C}$, and $\mathbb{R}\subset\mathbb{C}$. | 4.1 4.2 4.3 |
| conjugate of a complex number | For $z=a+ib$, the number $\bar z=a-ib$: same real part, opposite imaginary part. $z\bar z=|z|^2$ is real. | 4.2 |
| conjugate pair | Two complex numbers $p+iq$ and $p-iq$, each the conjugate of the other; their sum $2p$ and product $p^2+q^2$ are real. | 4.2 |
| equality of complex numbers | $a+ib=c+id$ exactly when $a=c$ and $b=d$ — one complex equation gives two real equations. | 4.1 |
| imaginary axis | The $y$-axis of the Argand plane; its points represent the numbers $0+ib$. | 4.3 |
| imaginary part | For $z=a+ib$ ($a,b$ real), the real number $b$ (not $ib$), written $\operatorname{Im}z$. | 4.1 4.2 4.3 |
| imaginary unit | The number $i$ with $i^2=-1$. For $a\gt0$, $\sqrt{-a}=i\sqrt a$. | 4.1 |
| impedance | In AC circuits, the complex number $Z$ (in ohms) with $V=IZ$; impedances in series add. | 4.1 4.2 |
| modulus of a complex number | For $z=a+ib$, the non-negative real number $|z|=\sqrt{a^2+b^2}$ — its size. $|z_1z_2|=|z_1||z_2|$. | 4.2 |
| multiplicative inverse | For $z\ne0$, the number $z^{-1}=\dfrac{\bar z}{|z|^2}$ with $z\,z^{-1}=1$. | 4.1 4.2 |
| parallelogram law of addition | $z_1+z_2$ is the fourth vertex of the parallelogram with vertices $O$, $z_1$, $z_2$ — arrows added nose to tail. | 4.3 |
| purely imaginary number | A complex number $ib$ with real part $0$ and imaginary part $b\ne0$, e.g. $3i$, $-i\sqrt2$. | 4.1 4.2 |
| quotient of complex numbers | $\dfrac{z_1}{z_2}=z_1z_2^{-1}$ ($z_2\ne0$), found by multiplying numerator and denominator by $\bar z_2$. | |
| real axis | The $x$-axis of the Argand plane; its points represent the real numbers $a+0i$. | 4.3 |
| real part | For $z=a+ib$ ($a,b$ real), the real number $a$, written $\operatorname{Re}z$. | 4.1 4.2 4.3 |
Ch 5 · Linear Inequalities
| Term | Meaning | Used in |
|---|---|---|
| boundary point | For a linear inequality, the value of $x$ where the two sides are equal, $x=-\tfrac ba$ for $ax+b\lt0$. The solution ray starts there; it is included for $\le$, $\ge$ and excluded for $\lt$, $\gt$. | 5.2 |
| break-even distance | The distance at which two options with different fixed and running costs cost the same; beyond it the option with the lower running cost is cheaper. | 5.2 |
| consecutive odd numbers | Odd numbers that follow one another, such as $13$ and $15$: if the smaller is $x$, the next is $x+2$ (the same holds for consecutive even numbers). | 5.2 |
| double inequality | Two inequalities written as one, such as $2\lt x\le5$, meaning $2\lt x$ and $x\le5$. Any rule must be applied to all three parts. | 5.1 5.2 |
| equivalent inequalities | Inequalities with exactly the same solution set, such as $-3x\lt12$ and $x\gt-4$. Each step in solving produces an equivalent inequality. | 5.1 5.2 |
| linear inequality | An inequality that can be written as $ax+b\lt0$, $ax+b\gt0$, $ax+b\le0$ or $ax+b\ge0$ with $a\ne0$ (in one variable $x$). | 5.1 5.2 |
| ray | An interval that is unbounded on one side, such as $(-\infty,4)$ or $[-2,\infty)$. Every linear inequality in one variable with $a\ne0$ has a ray as its solution set. | 5.2 |
| sign flip | Reversing the direction of an inequality ($\lt$ becomes $\gt$, $\le$ becomes $\ge$). Required when both sides are multiplied or divided by a negative number. | 5.1 |
| slack inequality | An inequality using $\le$ or $\ge$, so equality is allowed: $a\le b$ means $a\lt b$ or $a=b$. On the number line its end point is filled. | 5.1 |
| solution set | The set of all values of the variable that make an inequality (or equation) true. It depends on whether the variable is in $\mathbb{N}$, $\mathbb{Z}$ or $\mathbb{R}$. | 1.1 1.2 2.3 5.1 |
| strict inequality | An inequality using $\lt$ or $\gt$, so equality is not allowed: $x\lt4$ excludes $4$. On the number line its end point is hollow. | 5.1 |
| system of linear inequalities | Two or more inequalities in the same variable that must hold at the same time. Its solution set is the intersection of the individual solution sets, and may be empty. | 5.2 |
Ch 6 · Permutations and Combinations
| Term | Meaning | Used in |
|---|---|---|
| addition principle | If every way of doing a job falls into exactly one of several non-overlapping cases, the total number of ways is the sum of the numbers of ways in the cases. | 6.1 6.3 14.1 |
| block method | For objects that must stay together: tie them into one unit, arrange the units, then multiply by the arrangements inside the block. | 6.2 |
| circular permutation | An arrangement round a circle, where only relative positions matter: $n$ different objects give $(n-1)!$ circular arrangements. | 6.2 |
| collinear points | Points that lie on one straight line; $k$ collinear points give one line instead of ${}^kC_2$, and no triangle. | 6.3 11.2 |
| combination | A selection of objects in which order does not matter. ${}^nC_r=\dfrac{n!}{r!\,(n-r)!}$ counts the selections of $r$ objects from $n$ different objects. | 1.1 2.4 6.1 6.3 |
| diagonal of a polygon | A segment joining two vertices that are not neighbours; an $n$-gon has ${}^nC_2-n=\dfrac{n(n-3)}{2}$ diagonals. | |
| dictionary order | Alphabetical order of words; the rank of a word is its position when all arrangements of its letters are listed in this order. | |
| factorial | For a natural number $n$, $n!=1\times2\times\cdots\times n$; also $0!=1$. Key property: $n!=n\times(n-1)!$. | 6.2 |
| fundamental principle of counting | If a job is done in successive stages with $n_1, n_2, \dots, n_k$ ways at each stage (the number not depending on earlier choices), the whole job can be done in $n_1n_2\cdots n_k$ ways. | 2.2 6.1 6.2 6.3 |
| gap method | For objects no two of which may be adjacent: arrange the other objects first, then place the restricted ones in different gaps between and around them. | 6.2 6.3 |
| permutation | An arrangement of objects in a definite order. ${}^nP_r=\dfrac{n!}{(n-r)!}$ counts the permutations of $n$ different objects taken $r$ at a time. | 6.1 6.2 6.3 14.2 |
| slot method | Counting by drawing one box per stage, writing the number of choices in each (most restricted box first) and multiplying. | 6.1 |
| stars and bars | The method that counts ways to put $n$ identical objects into $r$ different boxes as arrangements of $n$ stars and $r-1$ bars: ${}^{n+r-1}C_{r-1}$. | 6.3 |
| tree diagram | A diagram that branches once for each choice at each stage; every path from the start to the end of the tree is one outcome. | 6.1 |
| with or without repetition | In an arrangement with repetition a symbol may be used more than once (like the digits of a PIN); without repetition each symbol is used at most once. | 6.1 6.2 |
Ch 7 · Binomial Theorem
| Term | Meaning | Used in |
|---|---|---|
| binomial approximation | Estimating $(1+x)^n$ by the first few terms of its expansion, such as $(1+x)^n\approx1+nx$; good when $nx$ is small. | |
| binomial coefficient | One of the numbers ${}^nC_r=\dfrac{n!}{r!\,(n-r)!}$ ($r=0,1,\dots,n$) that multiply the terms of $(a+b)^n$; entry $r$ of row $n$ of Pascal's triangle, counting from $0$. | 7.1 7.2 13.2 |
| binomial expansion | The sum of $n+1$ terms obtained by writing out $(a+b)^n$ with the binomial theorem; the indices of $a$ and $b$ in each term add up to $n$. | |
| binomial theorem | For a positive integer $n$, $(a+b)^n=\sum_{r=0}^{n}{}^nC_r\,a^{n-r}b^r={}^nC_0a^n+{}^nC_1a^{n-1}b+\cdots+{}^nC_nb^n$. | 6.3 7.1 7.2 12.3 |
| counting proof | A proof that a formula is true by showing that both sides count the same collection of objects — for example, the coefficient of $a^{n-r}b^r$ counts the ways to choose which $r$ of the $n$ brackets supply $b$. | 6.3 7.1 |
| doubling time | The time needed for a quantity growing at a fixed percentage rate to double; for growth at rate $r$ per year it is the smallest $n$ with $(1+r)^n\ge2$. | 7.2 |
| lower bound | A number that a quantity is known to be at least; for example $1+nx$ is a lower bound for $(1+x)^n$ when $x\ge0$. | 7.2 14.2 |
| Pascal's triangle | The triangle of numbers that starts with a single $1$; every row begins and ends with $1$ and each other entry is the sum of the two entries above it. Row $n$ lists the coefficients of $(a+b)^n$. | 7.1 |
| remainder on division | The whole number $q$ with $0\le q\lt d$ left over when a whole number is written as $d\times(\text{quotient})+q$; never negative. | 7.2 |
| rule of 72 | The bankers' estimate that money growing at $R\%$ per year doubles in about $\tfrac{72}{R}$ years. | 7.2 |
Ch 8 · Sequences and Series
| Term | Meaning | Used in |
|---|---|---|
| AM–GM inequality | For positive $a,b$: $\dfrac{a+b}{2}\ge\sqrt{ab}$, with equality only when $a=b$. Fixed sum ⇒ largest product when equal; fixed product ⇒ smallest sum when equal. | 8.3 |
| arithmetic mean | Of two numbers $a,b$: $A=\dfrac{a+b}{2}$, the number that makes $a,A,b$ an AP. | 3.3 8.3 10.3 |
| arithmetic progression | A sequence in which each term is the previous one plus a fixed common difference $d$: $a_n=a+(n-1)d$, $S_n=\tfrac n2[2a+(n-1)d]$ (Class 10). | 3.1 3.4 8.1 8.2 |
| arithmetico-geometric series | (JEE) A series whose terms are products of matching AP and GP terms, e.g. $1+2x+3x^2+\cdots=\dfrac1{(1-x)^2}$ for $|x|\lt1$. | 8.2 |
| average growth factor | The constant factor that, applied every period, gives the same total change: the geometric mean of the individual growth factors. | 8.3 |
| common ratio | The fixed number $r=\dfrac{a_{n+1}}{a_n}$ by which each term of a GP is multiplied to give the next. | 8.2 8.3 |
| compound interest | Interest added to the amount each period, so the amount is multiplied by $1+\tfrac{R}{100}$ every period: $A=P\left(1+\tfrac{R}{100}\right)^n$, a GP. | 7.1 7.2 8.2 |
| depreciation | Loss of value by a fixed percentage each year, so the values form a GP with ratio $1-\tfrac{R}{100}$. | |
| equality case | The situation in which an inequality becomes an equation — for A.M. $\ge$ G.M., exactly when the two numbers are equal. A bound is a true least/greatest value only if its equality case can occur. | 8.3 |
| Fibonacci sequence | $1,1,2,3,5,8,13,\dots$: $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n\ge3$. Known in India from the study of verse rhythms. | 8.1 |
| geometric mean | Of two positive numbers $a,b$: $G=\sqrt{ab}$, the positive number that makes $a,G,b$ a GP. | 3.3 8.3 10.3 |
| geometric progression | A sequence $a,ar,ar^2,\dots$ ($a\ne0$, $r\ne0$) in which each term is the previous one times a fixed common ratio $r$; $a_n=ar^{n-1}$. | 8.1 8.2 8.3 12.1 |
| infinite geometric series | The series $a+ar+ar^2+\cdots$ with infinitely many terms; it has a sum exactly when $|r|\lt1$. | 8.2 |
| inserting means | Finding $n$ numbers between $a$ and $b$ so that the $n+2$ numbers form an AP ($d=\tfrac{b-a}{n+1}$) or a GP ($r^{n+1}=\tfrac ba$). | 8.3 |
| nth term of a sequence | The term $a_n$ in position $n$ of a sequence, usually given by a formula in $n$ so that any term can be found directly. | |
| partial sum | The sum of the first $n$ terms of a series, $S_n=a_1+\cdots+a_n$. The terms are recovered by $a_1=S_1$, $a_n=S_n-S_{n-1}$. | 7.2 8.1 8.2 |
| recurring decimal | A decimal with a block of digits repeating for ever, e.g. $0.\overline{45}$; it equals the sum to infinity of a GP and hence a fraction. | 8.2 |
| recursive rule | A rule giving each term of a sequence from earlier terms, together with starting terms, e.g. $a_1=3$, $a_n=2a_{n-1}+1$. | 8.1 |
| sequence | A list of numbers $a_1,a_2,a_3,\dots$ in a definite order; formally a function whose domain is $\mathbb{N}$ (infinite sequence) or $\{1,2,\dots,n\}$ (finite sequence). | 6.1 6.2 8.1 8.2 |
| series | The sum $a_1+a_2+a_3+\cdots$ formed from the terms of a sequence; finite or infinite according as the sequence is. | 4.1 4.2 6.1 8.1 |
| sigma notation | Compact notation for sums: $\sum_{k=m}^{n}a_k=a_m+a_{m+1}+\cdots+a_n$, a sum of $n-m+1$ terms; $k$ is the index. | 7.1 8.1 |
| sum to infinity | For a GP with $|r|\lt1$, the number $S_\infty=\dfrac{a}{1-r}$ that the partial sums approach. If $|r|\ge1$ there is none. | 8.2 |
Ch 9 · Straight Lines
| Term | Meaning | Used in |
|---|---|---|
| angle between two lines | Of the two angles $\theta$ and $180^\circ-\theta$ where two lines cross, usually the acute one: $\tan\theta=\left|\dfrac{m_2-m_1}{1+m_1m_2}\right|$ when $1+m_1m_2\ne0$. | 9.1 9.3 |
| angle bisector | A line dividing an angle into two equal parts. Its points are equidistant from the two arms, which gives its equation from the distance formula. | 9.3 |
| distance between parallel lines | The constant gap between two parallel lines, $\dfrac{|C_1-C_2|}{\sqrt{A^2+B^2}}$ for $Ax+By+C_1=0$ and $Ax+By+C_2=0$ (same $A$ and $B$). | 9.3 |
| distance formula | The distance between $(x_1,y_1)$ and $(x_2,y_2)$ in the plane is $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ — Pythagoras' theorem applied to the horizontal and vertical gaps. | 3.3 4.3 9.3 10.1 |
| equation of a line | An equation in $x$ and $y$ satisfied by the coordinates of every point on the line and of no other point, such as $y=2x+1$ or $x=3$. | 9.2 9.3 |
| foot of the perpendicular | The point $N$ where the perpendicular from a point $P$ meets a line; it is the point of the line nearest to $P$. | 9.2 9.3 10.2 11.1 |
| image of a point in a line | The mirror reflection $P'$ of a point $P$ in a line: the line is the perpendicular bisector of $PP'$, so the foot of the perpendicular is the mid-point of $PP'$. | |
| inclination of a line | The angle $\theta$, with $0^\circ\le\theta\lt180^\circ$, measured anticlockwise from the positive $x$-axis to the line. | 9.1 |
| intercept | The $x$-intercept of a line is the number $a$ where it meets the $x$-axis at $(a,0)$; the $y$-intercept is the number $b$ where it meets the $y$-axis at $(0,b)$. Intercepts are signed numbers, not lengths. | 5.2 9.2 9.3 10.4 |
| intercept form | The equation $\dfrac xa+\dfrac yb=1$ of the line with $x$-intercept $a$ and $y$-intercept $b$, both non-zero. | 9.2 9.3 |
| linear model | A description of one quantity as a first-degree function of another, $y=mx+c$, whose graph is a straight line; the slope is the rate of change and $c$ the starting value. | 9.2 |
| mid-point formula | The mid-point of the segment joining $(x_1,y_1)$ and $(x_2,y_2)$ is $\left(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2}\right)$ — the section formula with ratio $1:1$. | 9.1 |
| parallel lines | Lines in the plane that never meet. Two non-vertical lines are parallel (or coincide) exactly when their slopes are equal. | 9.1 9.2 9.3 |
| perpendicular distance | The length of the perpendicular from a point to a line — the shortest distance between them: $\dfrac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}$ for $(x_1,y_1)$ and $Ax+By+C=0$. | 9.3 11.1 |
| perpendicular lines | Lines meeting at a right angle. Two non-vertical lines are perpendicular exactly when $m_1m_2=-1$; a horizontal and a vertical line are also perpendicular. | 9.1 10.4 |
| point-slope form | The equation $y-y_1=m(x-x_1)$ of the line through $(x_1,y_1)$ with slope $m$. | 9.2 12.3 |
| rise and run | For two points on a line, the rise is the change in $y$ and the run is the change in $x$; their quotient is the slope. | |
| section formula | The point dividing the join of $(x_1,y_1)$ and $(x_2,y_2)$ internally in the ratio $p:q$ is $\left(\tfrac{px_2+qx_1}{p+q},\tfrac{py_2+qy_1}{p+q}\right)$. | 9.1 11.2 |
| slope | The steepness of a non-vertical line: $m=\tan\theta=\dfrac{y_2-y_1}{x_2-x_1}$ (rise over run) for any two points on it. Horizontal lines have slope $0$; vertical lines have no slope. | 2.4 3.3 9.1 9.2 |
| slope-intercept form | The equation $y=mx+c$ of the line with slope $m$ and $y$-intercept $c$; every non-vertical line can be written this way. | 9.2 9.3 |
| two-point form | The equation $y-y_1=\dfrac{y_2-y_1}{x_2-x_1}(x-x_1)$ of the line through $(x_1,y_1)$ and $(x_2,y_2)$, $x_1\ne x_2$. | 9.2 |
Ch 10 · Conic Sections
| Term | Meaning | Used in |
|---|---|---|
| asymptotes of a hyperbola | The lines $y=\pm\tfrac bax$ (diagonals of the rectangle with corners $(\pm a,\pm b)$) which the branches of $\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}=1$ approach far from the centre; a sketching guide. | 10.4 |
| axis of a parabola | The line through the focus perpendicular to the directrix; the parabola is symmetric about it. For $y^2=4ax$ it is the $x$-axis. | 10.2 |
| branch of a hyperbola | Either of the two separate pieces of a hyperbola: the points nearer one focus form one branch, those nearer the other focus the other branch. | 10.1 10.4 |
| completing the square | Rewriting $x^2+2gx$ as $(x+g)^2-g^2$; used to find the centre and radius of a circle from its general equation. | 10.1 11.2 |
| confocal conics | Conics with the same foci, for example $\tfrac{x^2}{25}+\tfrac{y^2}{16}=1$ and $\tfrac{x^2}{4}-\tfrac{y^2}{5}=1$, both with foci $(\pm3,0)$. | 10.4 |
| conic section | A curve in which a plane cuts a double cone: a circle, an ellipse, a parabola or a hyperbola (or, through the vertex, a degenerate case). | 10.1 10.2 10.3 10.4 |
| conjugate axis | The segment of length $2b$ through the centre of a hyperbola, perpendicular to the transverse axis, where $b^2=c^2-a^2$. It does not meet the curve. | 10.4 |
| degenerate conic | The section of a double cone by a plane through its vertex: a single point, a single line, or a pair of intersecting lines. | 10.1 |
| directrix | The fixed line in the definition of a parabola: every point of the parabola is as far from it as from the focus. For $y^2=4ax$ it is $x=-a$. | 10.2 10.3 |
| double cone | The surface swept out when a line through a fixed point $V$ of a fixed axis is rotated about the axis at a constant angle $\alpha$; it has two pieces (nappes) meeting at the vertex $V$. | 10.1 |
| eccentricity | The number $e=\tfrac ca$ measuring how stretched a conic is: $0\lt e\lt1$ for an ellipse ($e=0$ for a circle), $e=1$ for a parabola, $e\gt1$ for a hyperbola. | 10.3 10.4 |
| ellipse | The set of all points of a plane the sum of whose distances from two fixed points (the foci) is a constant $2a$ greater than the distance between them; in standard form $\tfrac{x^2}{a^2}+\tfrac{y^2}{b^2}=1$. | 4.3 10.1 10.3 10.4 |
| focal distance | The distance of a point of a conic from a focus. On $y^2=4ax$ the point $(x,y)$ has focal distance $x+a$. | 10.2 10.3 10.4 |
| focus (of a conic) | A fixed point used to define a parabola, ellipse or hyperbola: $(a,0)$ for $y^2=4ax$; $(\pm c,0)$ for $\tfrac{x^2}{a^2}\pm\tfrac{y^2}{b^2}=1$. Plural: foci. | 10.2 10.3 10.4 |
| general equation of a circle | $x^2+y^2+2gx+2fy+c=0$; when $g^2+f^2-c\gt0$ it is the circle with centre $(-g,-f)$ and radius $\sqrt{g^2+f^2-c}$. | 10.1 |
| generator of a cone | Any one of the straight lines through the vertex that lie on the surface of a cone; the cone is made up of all its generators. | 10.1 |
| hyperbola | The set of all points of a plane the difference of whose distances from two fixed points (the foci) has a constant absolute value $2a$, less than the distance between them; in standard form $\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}=1$. | 10.1 10.3 10.4 |
| latus rectum | The chord of a conic through a focus perpendicular to the axis containing the focus. Length $4a$ for $y^2=4ax$, and $\tfrac{2b^2}{a}$ for the standard ellipse and hyperbola. | 10.2 10.3 10.4 |
| major axis | The chord of an ellipse through its foci, of length $2a$; its ends are the vertices. Along the $x$-axis when the larger denominator is under $x^2$. | 10.3 10.4 |
| minor axis | The chord of an ellipse through the centre perpendicular to the major axis, of length $2b$, where $b^2=a^2-c^2$. | 10.3 10.4 |
| nappe | Either of the two pieces of a double cone, one on each side of the vertex. | 10.1 |
| parabola | The set of all points of a plane equidistant from a fixed line (the directrix) and a fixed point not on it (the focus); in standard form $y^2=4ax$. | 2.4 10.1 10.2 10.4 |
| perigee and apogee | The nearest and farthest points of an orbit round the Earth, at distances $a-c$ and $a+c$ from the Earth's centre (the focus). | 10.3 |
| point circle | The case $g^2+f^2-c=0$ of $x^2+y^2+2gx+2fy+c=0$: a 'circle' of radius $0$, consisting of the single point $(-g,-f)$. | 10.1 |
| rectangular hyperbola | A hyperbola with $a=b$, such as $x^2-y^2=a^2$; its eccentricity is $\sqrt2$. | 10.4 |
| reflecting property | Rays parallel to the axis of a parabolic mirror reflect through the focus, and rays from the focus reflect parallel to the axis — used in dish antennas and headlamps. | 10.2 |
| semi-major axis | Half the major axis, the number $a$; for an orbit it is the average of the least and greatest distances from the focus. | 10.3 |
| semi-vertical angle | The fixed angle $\alpha$ between the axis of a cone and each of its generators. | 10.1 |
| transverse axis | The segment of length $2a$ joining the vertices of a hyperbola, on the line through its foci; along the axis of the positive squared term in the standard form. | 10.4 |
| vertex of a parabola | The point where a parabola meets its axis, midway between the focus and the directrix; the origin for the standard forms. | 10.2 |
| vertices of an ellipse | The two ends of the major axis, $(\pm a,0)$ for $\tfrac{x^2}{a^2}+\tfrac{y^2}{b^2}=1$. | 10.3 |
Ch 11 · Introduction to Three Dimensional Geometry
| Term | Meaning | Used in |
|---|---|---|
| box diagonal | A segment joining opposite corners of a rectangular box through its inside; for edges $a,b,c$ its length is $\sqrt{a^2+b^2+c^2}$. | 11.2 |
| coordinate plane | One of the three planes through the origin containing two of the axes: the $XY$-plane ($z=0$), the $YZ$-plane ($x=0$) and the $ZX$-plane ($y=0$). | 11.1 11.2 |
| distance formula in space | The distance between $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ is $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$ — Pythagoras' theorem used twice on a box with the two points as opposite corners. | 11.2 |
| distance from an axis | The distance of $(x,y,z)$ from the $z$-axis is $\sqrt{x^2+y^2}$ (the two coordinates other than $z$); similarly $\sqrt{y^2+z^2}$ from the $x$-axis and $\sqrt{z^2+x^2}$ from the $y$-axis. | 11.2 |
| ellipsoid | An egg-shaped closed surface, e.g. the set of points whose distances from two fixed points add up to a constant; its equation has the form $\tfrac{x^2}{a^2}+\tfrac{y^2}{b^2}+\tfrac{z^2}{c^2}=1$ when centred at $O$ with axes along the coordinate axes. | 11.2 |
| equidistant | At equal distances. The points of space equidistant from two fixed points form a plane, found from $PA^2=PB^2$. | 4.3 9.1 9.3 10.1 |
| octant | One of the eight regions into which the three coordinate planes divide space, fixed by the signs of $x$, $y$, $z$ — e.g. octant I is $x,y,z\gt0$ and octant VII is $x,y,z\lt0$. Points on a coordinate plane lie in no octant. | 11.1 |
| right-handed system | Axes arranged so that the fingers of the right hand, curling from the positive $x$-axis to the positive $y$-axis, leave the thumb pointing along the positive $z$-axis. | 11.1 11.2 |
| sphere | The set of points at a fixed distance $r$ (the radius) from a fixed point $(a,b,c)$ (the centre): $(x-a)^2+(y-b)^2+(z-c)^2=r^2$. | 7.2 11.2 |
| XY-plane | The coordinate plane containing the $x$- and $y$-axes; its equation is $z=0$, and the distance of $(x,y,z)$ from it is $|z|$. | |
| YZ-plane | The coordinate plane containing the $y$- and $z$-axes; its equation is $x=0$, and the distance of $(x,y,z)$ from it is $|x|$. | |
| z-axis | The third coordinate axis in space, perpendicular to both the $x$-axis and the $y$-axis at the origin; with a right-handed system it points 'up' when $x$ comes towards you and $y$ runs to the right. | |
| z-coordinate | The third number of $P(x,y,z)$: the signed distance of $P$ from the $XY$-plane, positive above it and negative below. | |
| ZX-plane | The coordinate plane containing the $z$- and $x$-axes; its equation is $y=0$, and the distance of $(x,y,z)$ from it is $|y|$. |
Ch 12 · Limits and Derivatives
| Term | Meaning | Used in |
|---|---|---|
| algebra of limits | The rules that the limit of a sum, difference, constant multiple, product or quotient is the sum, difference, multiple, product or quotient of the limits — provided the separate limits exist and, for a quotient, the denominator's limit is not $0$. | 12.1 12.2 12.4 |
| average rate of change | For $y=f(x)$ between $x=a$ and $x=a+h$: $\dfrac{f(a+h)-f(a)}{h}$, the change in $y$ per unit change in $x$ — the slope of the chord joining the two points of the graph. | 12.1 12.3 |
| constant-multiple rule | $(cu)'=c\,u'$ for a constant $c$: a constant factor stays, while an added constant differentiates to $0$. | |
| derivative | $f'(x)=\displaystyle\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}$, where the limit exists: the instantaneous rate of change of $f$ at $x$, and the slope of the tangent to $y=f(x)$ there. Also written $\tfrac{dy}{dx}$. | 3.3 12.1 12.2 12.3 |
| difference quotient | $\dfrac{f(a+h)-f(a)}{h}$, $h\ne0$: the average rate of change of $f$ over $[a,a+h]$ and the slope of the secant line through $(a,f(a))$ and $(a+h,f(a+h))$. | 12.3 12.4 |
| differentiable | A function is differentiable at $a$ if $f'(a)=\displaystyle\lim_{h\to0}\tfrac{f(a+h)-f(a)}{h}$ exists. $|x|$ is not differentiable at $0$ (a corner). | 12.3 12.4 |
| direct substitution | Finding $\displaystyle\lim_{x\to a}f(x)$ by simply evaluating $f(a)$. Valid for polynomials, and for rational functions whose denominator is not $0$ at $a$. | 12.2 |
| factor theorem | If $p$ is a polynomial and $p(a)=0$, then $x-a$ is a factor of $p(x)$. It is why a $\tfrac00$ rational limit can be simplified by cancelling $x-a$. | 12.2 |
| first principle | Finding a derivative directly from its definition as the limit of the difference quotient, rather than from rules. | 12.3 12.4 |
| hole in a graph | A single missing point of a graph, drawn as an open circle, where the function is undefined (or has a different value) although the curve approaches a definite height. The height of the hole is the limit there. | 12.1 |
| horizontal tangent | A tangent of slope $0$, at a point where $f'(x)=0$ — for instance at the top of a hill or the bottom of a valley of the graph. | 12.4 |
| indeterminate form | An expression such as $\tfrac00$ obtained by substituting $x=a$. It gives no information about the limit, which must be found by factorising, rationalising or a standard limit. | 12.2 |
| instantaneous rate of change | The derivative $f'(a)$: the limit of average rates of change over smaller and smaller intervals at $a$ — for example, velocity is the instantaneous rate of change of position. | 12.3 |
| instantaneous speed | The speed at a single instant: the limit of the average speeds $\tfrac{s(t+h)-s(t)}{h}$ over shorter and shorter intervals, as $h\to0$. | 12.1 12.3 |
| left-hand limit | $\displaystyle\lim_{x\to a^-}f(x)$: the number $f(x)$ approaches as $x$ tends to $a$ through values less than $a$ (LHL). | 12.1 12.3 |
| Leibniz notation | Writing the derivative of $y=f(x)$ as $\tfrac{dy}{dx}$ or $\tfrac{d}{dx}f(x)$, recalling that it is the limit of $\tfrac{\Delta y}{\Delta x}$. | |
| limit of a function | $\displaystyle\lim_{x\to a}f(x)=l$ means that $f(x)$ can be made as close to $l$ as we like by taking $x$ close enough to $a$, on either side, with $x\ne a$. The value $f(a)$ itself plays no part. | 12.1 |
| marginal cost | The derivative $C'(x)$ of total cost (or $R'(x)$ of revenue) with respect to the number of units — approximately the cost (revenue) of one more unit. | 12.3 12.4 |
| one-sided limit | A left-hand or a right-hand limit. The (two-sided) limit exists exactly when both one-sided limits exist and are equal. | 12.1 12.3 |
| power rule | $\dfrac{d}{dx}x^n=nx^{n-1}$ — proved here for every integer $n$ (with $x\ne0$ when $n\le0$); it also holds for $x^{1/2}$, $x\gt0$. | 12.4 |
| product rule | $(uv)'=u'v+uv'$ when $u$ and $v$ are differentiable — not $u'v'$. | 4.2 12.1 12.2 12.4 |
| quotient rule | $\left(\dfrac uv\right)'=\dfrac{u'v-uv'}{v^2}$ where $v\ne0$: bottom times derivative of top, minus top times derivative of bottom, over bottom squared. | 12.1 12.2 12.4 |
| rationalising | Multiplying the top and bottom of a fraction by a conjugate such as $\sqrt x+\sqrt a$ so that a square-root difference becomes $x-a$, which can then be cancelled. | 3.3 12.2 12.3 |
| right-hand limit | $\displaystyle\lim_{x\to a^+}f(x)$: the number $f(x)$ approaches as $x$ tends to $a$ through values greater than $a$ (RHL). | 12.1 12.3 |
| sandwich theorem | If $f(x)\le g(x)\le h(x)$ near $a$ and $f$ and $h$ both tend to $l$ as $x\to a$, then $g(x)\to l$ too. | 12.2 |
| secant line | A line through two points of a curve. Its slope is the difference quotient; as the points merge it becomes the tangent line. | 12.3 |
| small-angle approximation | For small angles in radians, $\sin\theta\approx\theta$ and $\tan\theta\approx\theta$ — a practical form of $\tfrac{\sin\theta}{\theta}\to1$. | 12.2 |
| standard limit | A limit used as a tool in many others, e.g. $\tfrac{x^n-a^n}{x-a}\to na^{n-1}$ as $x\to a$, and $\tfrac{\sin x}{x}\to1$, $\tfrac{1-\cos x}{x}\to0$ as $x\to0$ ($x$ in radians). | 12.2 12.3 |
| sum rule for derivatives | $(u\pm v)'=u'\pm v'$: a sum or difference is differentiated term by term. | |
| tangent line | The line through $(a,f(a))$ with slope $f'(a)$: $y-f(a)=f'(a)(x-a)$ — the limiting position of secant lines as the second point approaches the first. | 3.2 12.3 |
Ch 13 · Statistics
| Term | Meaning | Used in |
|---|---|---|
| absolute deviation | The distance $|x_i-a|$ of an observation from a central value $a$, ignoring the sign. | 13.1 |
| assumed mean | A convenient value $A$ (usually a central class mark) from which deviations are measured to shorten the arithmetic; the final answer does not depend on the choice. | 13.2 13.3 |
| class mark | The mid-point of a class interval, $\tfrac12(\text{lower limit}+\text{upper limit})$; it stands for every observation in the class. | 13.2 13.3 |
| coded values | The small numbers $u_i=\tfrac{x_i-A}{h}$ used in the step-deviation and shortcut methods. | 13.3 |
| coefficient of variation | (JEE) $\text{C.V.}=\tfrac{\sigma}{\bar x}\times100\%$; compares the variability of series with different means — the smaller C.V. is the more consistent. | 13.3 |
| continuous frequency distribution | A frequency table whose classes are intervals with no gaps between them, such as $0$–$10$, $10$–$20$, …; individual observations are not known. | 13.2 13.3 |
| cumulative frequency | The running total of frequencies: the number of observations less than or equal to a value (or below the upper limit of a class). Used to locate the median. | 13.2 |
| deviation from a central value | The signed difference $x_i-a$ between an observation and a central value $a$. Deviations from the mean always add up to $0$. | 13.1 13.2 13.3 |
| discrete frequency distribution | A frequency table in which the observations take separate values $x_i$ with frequencies $f_i$, e.g. goals per match. | 13.2 13.3 |
| frequency table | A table listing values (or class intervals) together with their frequencies — how many times each occurs. | 13.2 13.3 14.2 |
| mean deviation | The mean of the absolute deviations about a central value $a$: $\text{M.D.}(a)=\tfrac1n\sum|x_i-a|$, usually about the mean $\bar x$ or the median $M$. It is least about the median. | 13.1 13.2 13.3 |
| measure of central tendency | A single value that represents the centre of a data set — the mean, the median or the mode. | 13.1 |
| measure of dispersion | A number describing how widely data are scattered — e.g. the range, the mean deviation, the variance or the standard deviation. It is $0$ when all observations are equal. | 13.1 |
| median class | The first class whose cumulative frequency is at least $\tfrac N2$; the median lies in it. | 13.2 |
| outlier | An observation far from the rest of the data. It can change the range and the mean a lot, but the median hardly at all. | 13.1 |
| range of data | Largest observation minus smallest observation; the simplest measure of dispersion, decided by just two values. | 13.1 |
| shortcut method for variance | Coding the class marks as $u_i=\tfrac{x_i-A}{h}$, finding $\sigma_u$, and using $\sigma_x=h\,\sigma_u$ (the assumed mean $A$ does not affect the answer). | 13.3 |
| standard deviation | The non-negative square root $\sigma$ of the variance; a typical distance of the observations from the mean, in the same unit as the data. | 13.3 |
| step-deviation method | Working with $u_i=\tfrac{x_i-A}{h}$ instead of $x_i$: $\bar x=A+h\cdot\tfrac{\sum f_iu_i}{N}$. Section 13.3 extends it to the variance. | 13.2 13.3 |
| sum of squares of deviations | $\sum(x_i-\bar x)^2$ (weighted by $f_i$ for frequency data); dividing it by $n$ gives the variance. | 13.1 13.3 |
| variance | The mean of the squared deviations from the mean, $\sigma^2=\tfrac1n\sum(x_i-\bar x)^2=\tfrac1n\sum x_i^2-\bar x^{\,2}$; its unit is the square of the data's unit. | 13.1 13.3 |
Ch 14 · Probability
| Term | Meaning | Used in |
|---|---|---|
| addition rule of probability | $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ for any two events; for mutually exclusive events the last term is $0$. | 6.3 7.1 14.1 14.2 |
| axiomatic approach | Defining probability by the properties it must have rather than by a formula: any rule $P$ on the events of $S$ that obeys the three axioms of probability is a probability. | 14.2 |
| axioms of probability | The three rules every probability obeys: $P(E)\ge0$ for every event; $P(S)=1$; and $P(E\cup F)=P(E)+P(F)$ whenever $E$ and $F$ are mutually exclusive. | |
| birthday problem | The probability that among $n$ people at least two share a birthday, $1-\tfrac{{}^{365}P_n}{365^n}$ (all days equally likely). It exceeds $\tfrac12$ once $n\ge23$. | 14.2 |
| complement rule | $P(A')=1-P(A)$: the probability that an event does not happen. Especially useful for "at least one", whose complement is "none". | 14.2 |
| complementary event | For an event $A$, the event $A'=S-A$ ("not $A$"): it occurs exactly when $A$ does not. | 14.1 14.2 |
| compound event | An event with more than one sample point, such as "a doublet" for two dice ($6$ points). | 14.1 |
| equally likely outcomes | Outcomes whose simple events all have the same probability. With $n$ such outcomes each has probability $\tfrac1n$ and $P(E)=\tfrac{n(E)}{n(S)}$. | 7.1 14.2 |
| exhaustive events | Events whose union is the whole sample space, so that on every trial at least one of them occurs. If they are also mutually exclusive, exactly one occurs. | 14.1 |
| impossible event | The empty set $\phi$, regarded as an event: it contains no outcome, so it never occurs. | 14.1 |
| loaded die | A die (or coin) whose outcomes are not equally likely. Its probabilities are any non-negative weights on the faces that add up to $1$. | 14.2 |
| mutually exclusive events | Events with no sample point in common ($A\cap B=\phi$), so that at most one of them can occur on any trial. | 14.1 14.2 |
| random experiment | An experiment that can be repeated under the same conditions, whose possible outcomes are all known in advance but whose actual outcome on any one trial cannot be predicted — tossing a coin, throwing a die, drawing a card. | 14.1 14.2 |
| sample point | One element of a sample space — a single possible outcome of the experiment. | 11.1 14.1 14.2 |
| sample space | The set $S$ of all possible outcomes of a random experiment, listed so that each trial gives exactly one of them. For two dice, $S$ is the $36$ ordered pairs $(r,b)$. | 14.1 14.2 |
| simple event | An event with exactly one sample point, such as $\{(6,6)\}$ for two dice. Also called an elementary event. | 14.1 14.2 |
| sure event | The whole sample space $S$, regarded as an event: it contains every outcome, so it always occurs. | 14.1 |