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 · Relations and FunctionsCh 2 · Inverse Trigonometric FunctionsCh 3 · MatricesCh 4 · DeterminantsCh 5 · Continuity and DifferentiabilityCh 6 · Application of DerivativesCh 7 · IntegralsCh 8 · Application of IntegralsCh 9 · Differential EquationsCh 10 · Vector AlgebraCh 11 · Three Dimensional GeometryCh 12 · Linear ProgrammingCh 13 · Probability
Ch 1 · Relations and Functions
| Term | Meaning | Used in |
|---|---|---|
| bijective function | A function that is both one-one and onto, so every element of the codomain is the image of exactly one element of the domain. | 1.2 |
| codomain | The set $B$ named in $f:A\to B$, in which the images are allowed to lie. The range $f(A)$ is a subset of it; $f$ is onto when they are equal. | 1.2 2.1 |
| empty relation | The relation on $A$ with no ordered pairs at all, $R=\varnothing\subseteq A\times A$: no element is related to any element. | 1.1 |
| equivalence class | For an equivalence relation $R$ on $A$ and $a\in A$, the set $[a]=\{x\in A:(x,a)\in R\}$ of all elements related to $a$. | 1.1 |
| equivalence relation | A relation that is reflexive, symmetric and transitive, such as equality or “leaves the same remainder on division by $n$”. | 1.1 |
| horizontal line test | A real function is one-one if every horizontal line meets its graph at most once, and onto $B$ if every line $y=c$ with $c\in B$ meets it at least once. | 1.2 2.1 |
| many-one function | A function that is not one-one: at least two different inputs share the same image, e.g. $x^2$ on $\mathbb R$. | 1.2 |
| one-one function | A function $f:A\to B$ in which different inputs always have different images: $f(x_1)=f(x_2)\Rightarrow x_1=x_2$. Also called injective. | 1.2 2.1 2.2 5.3 |
| onto function | A function $f:A\to B$ whose range is the whole codomain: every $y\in B$ equals $f(x)$ for some $x\in A$. Also called surjective. | 1.2 2.1 |
| partition | A collection of non-empty, pairwise disjoint subsets of a set whose union is the whole set, so each element lies in exactly one piece. | 1.1 13.3 |
| pigeonhole principle | If more than $n$ objects are put into $n$ boxes, some box receives at least two objects; so no function from a larger finite set to a smaller one is one-one. | 1.2 |
| reflexive relation | A relation $R$ on $A$ with $(a,a)\in R$ for every $a\in A$ — every element is related to itself. | 1.1 |
| symmetric relation | A relation $R$ such that $(a,b)\in R$ always implies $(b,a)\in R$ — every pair comes with its mirror. | 1.1 |
| transitive relation | A relation $R$ such that $(a,b)\in R$ and $(b,c)\in R$ always imply $(a,c)\in R$ — every chain has its shortcut. | 1.1 |
| universal relation | The relation on $A$ containing every ordered pair, $R=A\times A$: every element is related to every element. | 1.1 |
Ch 2 · Inverse Trigonometric Functions
| Term | Meaning | Used in |
|---|---|---|
| allied angles | Angles such as $-\theta$, $\pi\pm\theta$, $2\pi-\theta$ whose trigonometric values equal those of $\theta$ up to sign; used to move an angle into a principal value branch. | 2.1 2.2 |
| horizontal asymptote | A line $y=c$ that a graph approaches as $x\to\infty$ or $x\to-\infty$, e.g. $y=\pm\tfrac\pi2$ for $y=\tan^{-1}x$. | 2.2 |
| inverse cosecant | $\operatorname{cosec}^{-1}:\mathbb R-(-1,1)\to\left[-\tfrac\pi2,\tfrac\pi2\right]-\{0\}$; the angle in that set whose cosecant is $x$. | 2.1 |
| inverse cosine | $\cos^{-1}:[-1,1]\to[0,\pi]$; $\cos^{-1}x$ is the angle between $0$ and $\pi$ whose cosine is $x$. | 2.1 |
| inverse cotangent | $\cot^{-1}:\mathbb R\to(0,\pi)$; $\cot^{-1}x$ is the angle strictly between $0$ and $\pi$ whose cotangent is $x$. | 2.1 |
| inverse function | For a one-one, onto function $f:X\to Y$, the function $f^{-1}:Y\to X$ that sends each $y$ back to the unique $x$ with $f(x)=y$. Only functions that are both one-one and onto have inverses. | 2.1 2.2 5.2 |
| inverse secant | $\sec^{-1}:\mathbb R-(-1,1)\to[0,\pi]-\left\{\tfrac\pi2\right\}$; the angle in that set whose secant is $x$. | 2.1 |
| inverse sine | $\sin^{-1}:[-1,1]\to\left[-\tfrac\pi2,\tfrac\pi2\right]$; $\sin^{-1}x$ is the angle between $-\tfrac\pi2$ and $\tfrac\pi2$ whose sine is $x$. It is not $\tfrac1{\sin x}$. | 2.1 7.2 7.3 |
| inverse tangent | $\tan^{-1}:\mathbb R\to\left(-\tfrac\pi2,\tfrac\pi2\right)$; $\tan^{-1}x$ is the angle strictly between $-\tfrac\pi2$ and $\tfrac\pi2$ whose tangent is $x$. | 2.1 7.2 |
| inverse trigonometric function | Any of $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$, $\operatorname{cosec}^{-1}$, $\sec^{-1}$, $\cot^{-1}$: the inverse of a trigonometric function restricted to an interval on which it is one-one and onto. Its value is an angle. | 2.1 5.2 5.3 |
| mirror image in y = x | The reflection of a point or curve in the line $y=x$; the point $(a,b)$ goes to $(b,a)$. The graph of an inverse function is the mirror image of the graph of the function. | |
| principal value | The value of an inverse trigonometric function at a point: the one angle in the principal value branch with the given sine, cosine, … — e.g. $\cos^{-1}\!\left(-\tfrac12\right)=\tfrac{2\pi}3$. | 2.1 2.2 5.2 6.2 |
| principal value branch | The agreed interval of angles on which a trigonometric function is restricted before it is inverted, e.g. $\left[-\tfrac\pi2,\tfrac\pi2\right]$ for sine and $[0,\pi]$ for cosine. It is the range of the inverse function. | 2.1 2.2 5.2 6.2 |
| reference angle | The acute (or zero, or right) angle $\alpha\in\left[0,\tfrac\pi2\right]$ whose sine, cosine, … equals the absolute value of the given number; the principal value is then $\alpha$, $-\alpha$ or $\pi-\alpha$ depending on the sign and the branch. | 2.1 2.2 |
| sawtooth graph | A zig-zag graph made of straight pieces, like $y=\sin^{-1}(\sin x)$ or $y=\cos^{-1}(\cos x)$; it agrees with $y=x$ only on the principal value branch. | 2.2 |
| vertical asymptote | A line $x=c$ near which a graph shoots off to $\pm\infty$, e.g. $x=\pm\tfrac\pi2$ for $y=\tan x$. Reflection in $y=x$ turns it into a horizontal asymptote. | 2.2 |
Ch 3 · Matrices
| Term | Meaning | Used in |
|---|---|---|
| column matrix | A matrix with exactly one column, of order $m\times1$. | 3.1 3.2 3.3 |
| commuting matrices | Square matrices $A,B$ with $AB=BA$. In general matrices do not commute; scalar matrices commute with every matrix of their order. | 3.2 |
| diagonal matrix | A square matrix whose entries off the main diagonal are all zero: $a_{ij}=0$ whenever $i\ne j$. Diagonal entries may be any numbers, including $0$. | 3.1 3.3 3.4 |
| equal matrices | Two matrices are equal when they have the same order and every pair of corresponding entries is equal: $a_{ij}=b_{ij}$ for all $i,j$. | 3.1 3.4 |
| identity matrix | The square matrix $I_n$ with $1$ in every position of the main diagonal and $0$ elsewhere; the scalar matrix with $k=1$. It satisfies $AI=IA=A$. | 3.1 3.3 4.2 |
| inverse of a matrix | For an invertible matrix $A$, the unique matrix $A^{-1}$ with $AA^{-1}=A^{-1}A=I$. It is not obtained by inverting each entry. | |
| invertible matrix | A square matrix $A$ for which some square matrix $B$ of the same order satisfies $AB=BA=I$. Matrices that are not square, or have a row or column of zeros, are not invertible. | 3.4 4.2 |
| involutory matrix | A square matrix with $A^2=I$, so that $A^{-1}=A$ — for example a reflection such as $\begin{bmatrix}0&1\\1&0\end{bmatrix}$. | 3.4 |
| key matrix | In a matrix code, the invertible matrix $K$ used to encode a message $P$ as $C=KP$; the receiver decodes with $P=K^{-1}C$. | 3.4 |
| main diagonal | In a square matrix, the entries $a_{11},a_{22},\dots,a_{nn}$ (those with $i=j$), running from the top-left to the bottom-right corner. | 3.1 3.3 4.1 |
| negative of a matrix | $-A=(-1)A=[-a_{ij}]$, the matrix with $A+(-A)=O$. | |
| order of a matrix | The pair $m\times n$ giving the number of rows $m$ and the number of columns $n$ of a matrix (rows first). An $m\times n$ matrix has $mn$ entries. | 3.1 4.1 |
| power of a matrix | For a square matrix, $A^2=AA$, $A^3=A^2A$, … and $A^0=I$. Not defined for non-square matrices. | 3.2 |
| product of matrices | For $A$ of order $m\times n$ and $B$ of order $n\times p$, $AB$ is the $m\times p$ matrix with $(i,k)$ entry $\sum_j a_{ij}b_{jk}$ (row $i$ of $A$ times column $k$ of $B$). Defined only when the inner numbers match. | 3.2 4.2 |
| reversal law for transposes | $(AB)'=B'A'$ whenever $AB$ is defined: transposing a product reverses the order of the factors. Likewise $(ABC)'=C'B'A'$, and for invertible matrices $(AB)^{-1}=B^{-1}A^{-1}$. | 3.3 3.4 |
| row matrix | A matrix with exactly one row, of order $1\times n$, such as $\begin{bmatrix}3&-1&0\end{bmatrix}$. | 3.1 3.2 3.3 3.4 |
| scalar matrix | A diagonal matrix whose diagonal entries are all equal to the same number $k$, such as $\begin{bmatrix}2&0\\0&2\end{bmatrix}$. | 3.1 3.2 |
| scalar multiple of a matrix | For a real number $k$, $kA=[k\,a_{ij}]$: every entry of $A$ is multiplied by $k$. | 3.2 |
| skew-symmetric matrix | A square matrix with $A'=-A$, i.e. $a_{ji}=-a_{ij}$ for all $i,j$. Its diagonal entries are all $0$. | 3.1 3.3 4.1 |
| square matrix | A matrix with as many rows as columns, of order $n\times n$ (a square matrix of order $n$). | 3.1 3.2 3.3 3.4 |
| sum of matrices | For matrices of the same order, $A+B=[a_{ij}+b_{ij}]$: add corresponding entries. Not defined for different orders. | 3.2 |
| symmetric matrix | A square matrix with $A'=A$, i.e. $a_{ij}=a_{ji}$ for all $i,j$: each entry equals its mirror image in the main diagonal. | 3.1 3.3 3.4 4.2 |
| symmetric part | For a square matrix $A$, the symmetric matrix $\tfrac12(A+A')$; together with the skew-symmetric part $\tfrac12(A-A')$ it adds up to $A$. | 3.3 |
| transpose | The transpose $A'$ (also $A^T$) of an $m\times n$ matrix $A=[a_{ij}]$ is the $n\times m$ matrix $[a_{ji}]$: the rows of $A$ written as columns. | 3.1 3.3 3.4 4.2 |
| zero matrix | A matrix of any order all of whose entries are $0$, written $O$. Zero matrices of different orders are different matrices. | 3.1 3.2 3.3 4.2 |
Ch 4 · Determinants
| Term | Meaning | Used in |
|---|---|---|
| adjoint | The adjoint $\operatorname{adj}A$ of a square matrix is the transpose of its matrix of cofactors: the cofactors of row $i$ fill column $i$. It satisfies $A(\operatorname{adj}A)=(\operatorname{adj}A)A=|A|\,I$. | 4.2 4.3 |
| chessboard sign pattern | The pattern of signs $(-1)^{i+j}$ used when expanding a determinant: $+$ in the top-left corner, then alternating $+,-$ along every row and column. | 4.1 4.2 |
| coefficient matrix | In a system written as $AX=B$, the square matrix $A$ of the coefficients of the unknowns; row $i$ holds the coefficients of equation $i$, column $j$ those of the $j$-th unknown. | 4.3 |
| cofactor | The cofactor of $a_{ij}$ is $A_{ij}=(-1)^{i+j}M_{ij}$: the minor with the chessboard sign attached. Entries of a row times their cofactors add up to $|A|$. | 4.2 4.3 |
| collinear points | Points that lie on one straight line. Three points $(x_i,y_i)$ are collinear exactly when $\begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{vmatrix}=0$. | 4.1 10.1 10.2 10.3 |
| consistent system | A system of equations with at least one solution. It may have exactly one solution or infinitely many. | 4.3 |
| determinant | The number $|A|$ (or $\det A$) associated with a square matrix $A$. For order $2$, $\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc$; for order $3$ it is found by expanding along a row or column. It can be negative — the bars are not absolute-value bars. | 3.1 3.4 4.1 4.2 |
| expansion along a row or column | Evaluating a $3\times3$ determinant by taking each entry $a_{ij}$ of one chosen row or column, multiplying it by $(-1)^{i+j}$ and by the $2\times2$ determinant left after deleting row $i$ and column $j$, and adding. Every row and column gives the same value. | 4.1 |
| homogeneous system | A system $AX=O$ whose constants are all zero. It always has the solution $X=O$, which is the only one when $|A|\ne0$. | 4.3 |
| inconsistent system | A system of equations with no solution at all — for two equations in two unknowns, two distinct parallel lines. | 4.3 |
| matrix method | Solving $AX=B$ with $|A|\ne0$ as $X=A^{-1}B=\tfrac1{|A|}(\operatorname{adj}A)B$. | 4.3 |
| minor | The minor $M_{ij}$ of the entry $a_{ij}$ of a square matrix is the determinant left after deleting row $i$ and column $j$. It carries no sign. | 4.2 4.3 8.2 |
| non-singular matrix | A square matrix with non-zero determinant. Exactly the non-singular matrices are invertible, with $A^{-1}=\tfrac1{|A|}\operatorname{adj}A$. | 4.2 |
| order of a determinant | The order $n$ of the square matrix the determinant comes from: an $n\times n$ matrix gives a determinant of order $n$. | |
| signed area | An area with a sign attached: the determinant $\begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{vmatrix}$ is $+2\times$ area when the vertices go anticlockwise and $-2\times$ area when they go clockwise. | 4.1 7.4 |
| singular matrix | A square matrix with determinant $0$. A singular matrix has no inverse. | 4.2 |
Ch 5 · Continuity and Differentiability
| Term | Meaning | Used in |
|---|---|---|
| algebra of continuous functions | If $f,g$ are continuous at $c$, so are $f\pm g$, $fg$, $\lambda f$, and $\tfrac fg$ provided $g(c)\ne0$. | 5.1 |
| chain rule | The rule for differentiating a composite: $\dfrac{d}{dx}v(u(x))=v'(u(x))\,u'(x)$, or $\dfrac{dy}{dx}=\dfrac{dy}{dt}\cdot\dfrac{dt}{dx}$ — rates multiply through the layers. | 5.2 5.3 5.4 6.1 |
| change of base | The identities $\log_bx=\dfrac{\log x}{\log b}$ and $b^x=e^{x\log b}$, which turn any base into base $e$. | 5.3 |
| composite function | The function $f\circ g$, $(f\circ g)(x)=f(g(x))$: apply $g$ first, then $f$. Example: $\sin(x^2)$ is $\sin$ composed with $x^2$. | 5.1 |
| concave down | Bending downward like a cap: the slope decreases as $x$ increases, which happens where $f''(x)\lt0$. | 5.4 7.3 |
| concave up | Bending upward like a cup: the slope increases as $x$ increases, which happens where $f''(x)\gt0$. | 5.4 |
| continuity at a point | $f$ is continuous at a point $c$ of its domain when $\lim_{x\to c}f(x)$ exists and equals $f(c)$: LHL $=$ RHL $=f(c)$. | 5.1 |
| continuous function | A function that is continuous at every point of its domain. Points outside the domain do not count, so $\tfrac1x$ is a continuous function. | 5.1 5.2 6.3 6.4 |
| cusp | A sharp point of a curve, such as where a cycloid touches the line it rolls on; in parametric form both $\tfrac{dx}{dt}$ and $\tfrac{dy}{dt}$ are $0$ there. | 5.2 5.4 6.3 |
| cycloid | The curve $x=a(t-\sin t)$, $y=a(1-\cos t)$ traced by a point on the rim of a wheel of radius $a$ rolling along a straight line. | 5.4 |
| differentiable function | A function that is differentiable at every point of its domain: at each point the left-hand and right-hand derivatives exist and are equal. | 5.2 5.4 6.1 6.3 |
| explicit function | A function given by a formula $y=f(x)$ that expresses $y$ directly in terms of $x$, such as $y=x^3-2x$. | 9.1 |
| exponential function | A function $f(x)=b^x$ with a fixed base $b\gt0$, $b\ne1$; domain $\mathbb R$, range $(0,\infty)$. The most important one is $e^x$. | 5.3 |
| implicit differentiation | Finding $\dfrac{dy}{dx}$ by differentiating both sides of an equation in $x$ and $y$, treating $y$ as a function of $x$ (so $\tfrac{d}{dx}y^2=2y\tfrac{dy}{dx}$), then solving for $\dfrac{dy}{dx}$. | 5.2 5.3 |
| implicit function | A function $y$ of $x$ defined by an equation linking $x$ and $y$, such as $x^2+y^2=25$, without a formula $y=f(x)$ being given. | |
| infinite discontinuity | A point of the domain near which $f(x)$ grows without bound on at least one side, e.g. $\tfrac{1}{(x-1)^2}$ with $f(1)$ given a value. | |
| jump discontinuity | A point where the left-hand and right-hand limits are both finite but different, so the graph steps up or down, as $[x]$ does at every integer. | 5.1 |
| left-hand derivative | $Lf'(c)=\lim_{h\to0^-}\dfrac{f(c+h)-f(c)}{h}$, the limit of chord slopes from points to the left of $c$. | 5.2 |
| logarithmic differentiation | Differentiating $y$ by first taking $\log y$ (to bring down exponents or split products), differentiating implicitly to get $\tfrac{y'}{y}$, then multiplying by $y$. Used for $x^x$, $(\sin x)^{\cos x}$ and long products. | 5.3 |
| logarithmic function | The inverse of $b^x$: $\log_bx=y\iff b^y=x$, defined for $x\gt0$. Its graph is the mirror image of $y=b^x$ in the line $y=x$. | 5.3 |
| natural logarithm | The logarithm to base $e$. In this course $\log x$ always means $\log_ex$, and $\tfrac{d}{dx}\log x=\tfrac1x$ for $x\gt0$. | 5.3 6.2 9.2 |
| parametric form | A description of a curve by $x=f(t)$, $y=g(t)$, where each value of the parameter $t$ gives one point; its slope is $\tfrac{dy}{dx}=\tfrac{dy/dt}{dx/dt}$ where $\tfrac{dx}{dt}\ne0$. | 5.4 11.2 |
| point of discontinuity | A point $c$ in the domain of $f$ at which $f$ is not continuous. A point where $f$ is not defined is never a point of discontinuity. | 5.1 |
| removable discontinuity | A point where $\lim_{x\to c}f(x)$ exists but differs from $f(c)$; redefining the single value $f(c)$ makes $f$ continuous there. | 5.1 |
| right-hand derivative | $Rf'(c)=\lim_{h\to0^+}\dfrac{f(c+h)-f(c)}{h}$, the limit of chord slopes from points to the right of $c$. | 5.2 |
| second-order derivative | The derivative of the derivative, $\dfrac{d^2y}{dx^2}=\dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)$, also written $f''(x)$ or $y_2$; for a position it is the acceleration. | 5.4 6.3 6.4 9.1 |
| the number e | The irrational number $e=2.71828\ldots$, the limit of $\left(1+\tfrac1n\right)^n$; the one base for which $\lim_{h\to0}\tfrac{e^h-1}{h}=1$, so that $\tfrac{d}{dx}e^x=e^x$. | 5.3 |
| vertical tangent | A tangent line parallel to the $y$-axis, as for $x^{1/3}$ at $0$; it has no slope, so the function has no derivative there. | 5.2 5.4 |
Ch 6 · Application of Derivatives
| Term | Meaning | Used in |
|---|---|---|
| absolute extreme value | An absolute maximum or absolute minimum value. For $f$ continuous on $[a,b]$ both exist and occur at critical points in $(a,b)$ or at the end points. | 6.4 |
| absolute maximum | The greatest value of $f$ on a set $I$: $f(c)$ with $f(c)\ge f(x)$ for every $x\in I$. Also called the greatest value or global maximum. | 6.4 |
| absolute minimum | The least value of $f$ on a set $I$: $f(c)$ with $f(c)\le f(x)$ for every $x\in I$. Also called the least value or global minimum. | 6.4 |
| average rate of change | The change in $y$ divided by the change in $x$ over an interval: $\dfrac{f(x_0+h)-f(x_0)}{h}$ — the slope of the chord joining two points of the graph. | 6.1 |
| critical point | A point $c$ of the domain where $f'(c)=0$ or where $f$ is not differentiable. Every local extremum at an interior point is a critical point, but not conversely. | 6.3 6.4 |
| decreasing function | A function $f$ is decreasing on an interval $I$ if $x_1\lt x_2$ in $I$ implies $f(x_1)\ge f(x_2)$: moving right never takes the graph up. | 6.2 |
| extreme value theorem | A function continuous on a closed, bounded interval $[a,b]$ attains an absolute maximum and an absolute minimum there (Theorem 6.6). It can fail on open or unbounded intervals. | |
| first derivative test | Classify a critical point $c$ by the sign of $f'$ on either side: $+\to-$ local maximum; $-\to+$ local minimum; no change of sign, neither. | 6.3 6.4 |
| increasing function | A function $f$ is increasing on an interval $I$ if $x_1\lt x_2$ in $I$ implies $f(x_1)\le f(x_2)$: moving right never takes the graph down (flat stretches allowed). | 6.2 7.1 |
| instantaneous rate of change | The limit of the average rate of change as the interval shrinks to a point: the derivative $f'(x_0)$. | 6.1 |
| linked rates | Rates of change of quantities tied together by an equation; differentiating the equation with respect to time $t$ (chain rule) links them, e.g. $\dfrac{dV}{dt}=4\pi r^2\dfrac{dr}{dt}$ for a sphere. | 6.1 |
| local extremum | A local maximum or local minimum (plural: local extrema). At an interior point where $f$ is differentiable, a local extremum forces $f'(c)=0$. | 6.3 |
| local maximum | $f$ has a local maximum at $c$ if $f(c)\ge f(x)$ for all $x$ in some open interval around $c$: the highest point nearby, not necessarily the highest overall. $f(c)$ is the local maximum value. | 6.3 6.4 |
| local minimum | $f$ has a local minimum at $c$ if $f(c)\le f(x)$ for all $x$ in some open interval around $c$: the lowest point nearby. | 6.3 6.4 |
| marginal cost | The rate of change of total cost with respect to output, $\text{MC}=\dfrac{dC}{dx}$ (rupees per unit); for large outputs it is close to the cost of producing one more unit. | 5.1 6.1 7.1 |
| marginal profit | The rate of change of profit $P=R-C$ with respect to output: $\dfrac{dP}{dx}=\text{MR}-\text{MC}$. | 6.1 |
| marginal revenue | The rate of change of total revenue with respect to the number of units sold, $\text{MR}=\dfrac{dR}{dx}$ (rupees per unit). | 6.1 |
| monotonic function | A function that is increasing on an interval, or decreasing on it, throughout. Its monotonicity is read from the sign of $f'$. | 6.2 6.3 |
| optimisation problem | A problem asking for the largest or smallest possible value of a quantity (area, volume, cost, profit) subject to given conditions, and for the choice that achieves it. | 6.4 12.2 |
| point of inflection | In this course: a point $c$ where $f'(c)=0$ but $f'$ does not change sign, so $f$ has neither a maximum nor a minimum there (e.g. $x^3$ at $0$). | 6.3 |
| rate of change | How fast one quantity changes per unit change of another. The (instantaneous) rate of change of $y=f(x)$ with respect to $x$ at $x_0$ is $f'(x_0)$; its units are units of $y$ per unit of $x$. | 5.2 5.3 5.4 6.1 |
| second derivative test | If $f'(c)=0$: $f''(c)\lt0$ gives a local maximum, $f''(c)\gt0$ a local minimum; if $f''(c)=0$ the test fails and the first derivative test must be used. | 6.3 6.4 |
| sign chart | A number line marked with the zeros of $f'$ (and points where $f'$ does not exist), showing the sign $+$ or $-$ of $f'$ on each interval between them; it shows at a glance where $f$ rises and falls. | 4.1 6.2 6.3 |
| stationary point | A point where $f'(c)=0$: the graph is momentarily level. It may be a local maximum, a local minimum or a point of inflection. | 6.3 12.2 |
| strictly decreasing | $f$ is strictly decreasing on $I$ if $x_1\lt x_2$ in $I$ implies $f(x_1)\gt f(x_2)$. Guaranteed when $f$ is continuous on $I$ and $f'\lt0$ at its interior points. | 2.1 6.2 6.3 |
| strictly increasing | $f$ is strictly increasing on $I$ if $x_1\lt x_2$ in $I$ implies $f(x_1)\lt f(x_2)$. Guaranteed when $f$ is continuous on $I$ and $f'\gt0$ at its interior points (isolated zeros of $f'$ allowed). | 1.2 2.1 5.1 6.2 |
Ch 7 · Integrals
| Term | Meaning | Used in |
|---|---|---|
| antiderivative | A function $F$ with $F'(x)=f(x)$ on an interval. Any two antiderivatives of $f$ on an interval differ by a constant. | 7.1 7.2 7.3 7.4 |
| area function | $A(x)=\int_a^xf(t)\,dt$, the net area under $f$ from a fixed $a$ up to a moving end $x$; by the first fundamental theorem $A'(x)=f(x)$. | 7.4 8.1 |
| completing the square | Rewriting $ax^2+bx+c$ as $a\left[\left(x+\frac{b}{2a}\right)^2+\frac{4ac-b^2}{4a^2}\right]$ so that a quadratic integrand matches a special integral. | 7.2 7.3 |
| constant of integration | The arbitrary constant $C$ in $\int f(x)\,dx=F(x)+C$; one extra condition, such as a known value $F(a)$, fixes it. | 7.1 7.4 |
| cover-up rule | For a non-repeated factor $x-a$: hide that factor in the fraction and put $x=a$ in what is left; the result is the constant over $x-a$. | 7.2 |
| cyclic integral | An integral such as $\int e^x\sin x\,dx$ that reappears after integrating by parts twice; call it $I$ and solve the resulting equation for $I$. | 7.3 |
| definite integral | For $f$ continuous on $[a,b]$, the number $\int_a^bf(x)\,dx$: the net area between $y=f(x)$ and the $x$-axis from $x=a$ to $x=b$ (area below the axis counts negatively). It equals $F(b)-F(a)$ for any antiderivative $F$. | 7.4 8.1 |
| dummy variable | The variable of integration in a definite integral; its name does not matter: $\int_a^bf(x)\,dx=\int_a^bf(t)\,dt$ (property P₀). | |
| even function | A function with $f(-x)=f(x)$ (graph symmetric in the $y$-axis); then $\int_{-a}^af(x)\,dx=2\int_0^af(x)\,dx$. | 7.4 |
| fundamental theorem of calculus | For continuous $f$: (first) $\frac{d}{dx}\int_a^xf(t)\,dt=f(x)$; (second) $\int_a^bf(x)\,dx=F(b)-F(a)$ for any antiderivative $F$ of $f$. | 7.4 |
| ILATE rule | The order Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential: in integration by parts, the factor that comes first is taken as the first function $u$. | 7.3 |
| improper rational function | A quotient $\frac{P(x)}{Q(x)}$ with $\deg P\ge\deg Q$; divide first to get a polynomial plus a proper fraction. | 7.2 |
| indefinite integral | The family of all antiderivatives of $f$, written $\int f(x)\,dx=F(x)+C$. | 7.1 |
| integrand | The function being integrated: in $\int f(x)\,dx$ it is $f(x)$. | 7.1 7.2 7.3 7.4 |
| integration by parts | The product rule read backwards: $\int u\,dv=uv-\int v\,du$, or $\int uv\,dx=u\int v\,dx-\int\left(u'\int v\,dx\right)dx$. | 7.3 |
| integration by substitution | Putting $u=g(x)$, $du=g'(x)\,dx$ to turn $\int f(g(x))\,g'(x)\,dx$ into $\int f(u)\,du$ — the chain rule run backwards. | 7.1 |
| irreducible quadratic factor | A quadratic factor with no real roots, such as $x^2+1$ or $x^2+k$ ($k\gt0$); its partial fraction has a linear numerator $Bx+C$. | 7.2 |
| limits of integration | In $\int_a^bf(x)\,dx$, $a$ is the lower limit and $b$ the upper limit; under a substitution $u=g(x)$ they change to $g(a)$ and $g(b)$. | 7.4 |
| net area | Area above the $x$-axis minus area below it, which is what a definite integral measures; the actual area adds the absolute values. | 7.4 8.1 |
| odd function | A function with $f(-x)=-f(x)$ (graph symmetric about the origin); then $\int_{-a}^af(x)\,dx=0$. | 7.4 |
| partial fractions | Simple fractions such as $\frac{A}{x-a}$, $\frac{B}{(x-a)^2}$ or $\frac{Bx+C}{x^2+k}$ whose sum equals a given proper rational function. | 7.2 9.2 9.3 |
| proper rational function | A quotient of polynomials $\frac{P(x)}{Q(x)}$ with $\deg P\lt\deg Q$; only these are split directly into partial fractions. | 7.2 |
| reduction formula | A formula expressing an integral with a power $n$ in terms of the same integral with a lower power, e.g. $I_n=-\frac1n\sin^{n-1}x\cos x+\frac{n-1}{n}I_{n-2}$. | 7.3 |
| repeated linear factor | A factor such as $(x-a)^2$ in a denominator; it needs both $\frac{A}{x-a}$ and $\frac{B}{(x-a)^2}$ in the partial fractions. | 6.4 7.2 |
| special integrals | The six standard results for $\frac{1}{x^2\pm a^2}$, $\frac{1}{a^2-x^2}$, $\frac{1}{\sqrt{x^2\pm a^2}}$ and $\frac{1}{\sqrt{a^2-x^2}}$, e.g. $\int\frac{dx}{x^2+a^2}=\frac1a\tan^{-1}\frac xa+C$. | 7.2 7.3 7.4 |
| standard integral | An integral read straight off a known derivative, e.g. $\int\cos x\,dx=\sin x+C$ or $\int\frac{dx}{1+x^2}=\tan^{-1}x+C$. | 7.1 8.2 9.3 |
| variable of integration | The variable named by $dx$ (or $dt$, $du$…) in an integral; the integrand is treated as a function of it. | 7.1 |
Ch 8 · Application of Integrals
| Term | Meaning | Used in |
|---|---|---|
| area between two curves | For $f(x)\ge g(x)$ on $[a,b]$, the area of the region between them is $\displaystyle\int_a^b[f(x)-g(x)]\,dx$ (upper minus lower). Beyond the Board syllabus; asked in JEE. | 8.2 |
| area of an ellipse | The ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ encloses area $\pi ab$; with $b=a$ this is the circle's $\pi a^2$. | 8.2 |
| area under a curve | The area of the region bounded by $y=f(x)\ge0$, the $x$-axis and the lines $x=a$, $x=b$; it equals $\displaystyle\int_a^b f(x)\,dx$. | 8.1 |
| elementary strip | A very thin rectangle used to build up the area of a region; adding the areas of all such strips gives a definite integral. | 8.1 |
| horizontal strip | An elementary strip lying along the $y$-axis direction at height $y$, of thickness $dy$ and length $x$; its area is $dA=x\,dy$. Used for curves of the form $x=g(y)$. | 8.1 8.2 12.3 |
| latus rectum | The chord of a parabola through its focus, perpendicular to its axis. For $y^2=4ax$ it is the segment of $x=a$ from $(a,-2a)$ to $(a,2a)$, of length $4a$. | 8.2 |
| parabolic segment | The region bounded by a parabola and a chord; for $y^2=4ax$ and the line $x=c$ its area is $\tfrac83\sqrt a\,c^{3/2}$, two-thirds of the bounding rectangle. | 8.2 |
| segment of a circle | The region between a circle and a chord. The smaller piece is the minor segment, the larger the major segment; its area is sector minus triangle, or an integral of $\sqrt{a^2-x^2}$. | 8.2 |
| symmetric region | A region that is its own mirror image in a line (an axis); its area is twice the area of the part on one side of that line. | |
| total distance travelled | For a body with velocity $v(t)$, the total distance travelled is $\displaystyle\int|v|\,dt$ — the area between the velocity graph and the $t$-axis — while the displacement is $\displaystyle\int v\,dt$, the net area. | 8.1 |
| two-thirds rule | Archimedes' result that a parabolic segment cut off by a chord perpendicular to the axis has $\tfrac23$ of the area of its bounding rectangle. | 8.2 |
| vertical strip | An elementary strip standing on the $x$-axis at position $x$, of width $dx$ and height $y$; its area is $dA=y\,dx$. | 8.1 8.2 |
Ch 9 · Differential Equations
| Term | Meaning | Used in |
|---|---|---|
| continuous compounding | Interest added at every instant, so that a principal obeys $\dfrac{dP}{dt}=rP$ and grows to $P_0e^{rt}$ at annual rate $r$. | 5.3 9.2 9.4 |
| degree of a differential equation | When the equation is a polynomial in its derivatives, the highest power of the highest-order derivative. It is not defined if a derivative appears inside, e.g., $\sin$ or $e^{(\ )}$. | 9.1 |
| degree of a homogeneous function | The number $n$ in $F(\lambda x,\lambda y)=\lambda^nF(x,y)$ — for a polynomial, the common total degree of all its terms. | 9.3 |
| differential equation | An equation involving an unknown function and one or more of its derivatives, such as $\dfrac{d^2y}{dx^2}+y=0$. With a single independent variable it is an ordinary differential equation. | 5.4 9.1 9.2 9.3 |
| doubling time | For exponential growth $N=N_0e^{kt}$, the time $\dfrac{\log2}{k}$ in which the quantity doubles, whatever its size. | 5.3 9.2 |
| equilibrium value | For $\dfrac{dA}{dt}+kA=r$ ($k\gt0$, $r$ constant), the value $\dfrac rk$ that every solution approaches as $t\to\infty$. | |
| Euler's method | A step-by-step numerical method for $\dfrac{dy}{dx}=f(x,y)$: move a small step $h$ along the current slope, $y_{n+1}=y_n+h\,f(x_n,y_n)$. | 9.1 |
| general solution | A solution of a differential equation containing as many independent arbitrary constants as the order of the equation, such as $y=A\cos2x+B\sin2x$ for $y''+4y=0$. | 9.1 9.2 9.3 9.4 |
| half-life | For exponential decay $N=N_0e^{-\lambda t}$, the time $T_{1/2}=\dfrac{\log2}{\lambda}$ in which the amount halves; it does not depend on the starting amount. | 5.3 9.2 |
| homogeneous differential equation | A first-order equation $\dfrac{dy}{dx}=F(x,y)$ with $F$ homogeneous of degree $0$, so that $\dfrac{dy}{dx}=g\!\left(\dfrac yx\right)$; solved by the substitution $y=vx$. | 9.3 |
| homogeneous function | A function $F(x,y)$ with $F(\lambda x,\lambda y)=\lambda^nF(x,y)$ for every non-zero $\lambda$; $n$ is its degree. Example: $x^2-3xy+5y^2$ (degree $2$). | 9.3 |
| initial condition | A condition such as $y(x_0)=y_0$ or $y'(x_0)=y_1$, given at one value of the independent variable, used to find the constants in a general solution. | 9.1 9.2 9.3 9.4 |
| integrating factor | For $\dfrac{dy}{dx}+Py=Q$, the multiplier $\text{I.F.}=e^{\int P\,dx}$ that turns the left side into $\dfrac{d}{dx}(y\cdot\text{I.F.})$, giving $y\cdot\text{I.F.}=\int Q\cdot\text{I.F.}\,dx+C$. | 9.4 |
| law of natural growth and decay | The model $\dfrac{dN}{dt}=kN$: a quantity changing at a rate proportional to its size. Its solution is $N=N_0e^{kt}$ — growth if $k\gt0$, decay if $k\lt0$. | 5.3 9.2 |
| linear differential equation | A first-order equation of the form $\dfrac{dy}{dx}+Py=Q$, with $P$, $Q$ constants or functions of $x$ only (or $\dfrac{dx}{dy}+P_1x=Q_1$ with $P_1$, $Q_1$ functions of $y$). | 9.4 |
| Newton's law of cooling | The rate of cooling of a body is proportional to the excess of its temperature over the surroundings: $\dfrac{dT}{dt}=-k(T-T_s)$, so $T=T_s+(T_0-T_s)e^{-kt}$. | 9.2 |
| order of a differential equation | The order of the highest-order derivative of the dependent variable occurring in the equation; $y'''+(y')^7=0$ has order $3$. | 9.1 |
| particular solution | A solution free from arbitrary constants, obtained from the general solution by giving the constants particular values (usually fixed by initial conditions). | 9.1 9.2 9.3 9.4 |
| slope field | A picture of $\dfrac{dy}{dx}=f(x,y)$ made of short dashes of slope $f(x,y)$ drawn at many points; solution curves are tangent to the dashes everywhere. | 7.1 9.1 9.3 |
| standard form of a linear equation | The arrangement $\dfrac{dy}{dx}+Py=Q$ with the coefficient of $\dfrac{dy}{dx}$ equal to $1$, from which $P$ and $Q$ are read off. | |
| substitution y = vx | For a homogeneous equation put $y=vx$, so $\dfrac{dy}{dx}=v+x\dfrac{dv}{dx}$; the equation becomes $x\dfrac{dv}{dx}=g(v)-v$, which has variables separable. | 9.3 |
| time constant | For $\dfrac{dA}{dt}+kA=r$, the time $\dfrac1k$ (for a circuit, $RC$); in each such interval the gap to equilibrium shrinks by the factor $e^{-1}$, so after one time constant about $63\%$ of the change is complete. | 9.4 |
| variables separable | A first-order equation $\dfrac{dy}{dx}=g(x)\,h(y)$ whose right side is a product of a function of $x$ and a function of $y$; solved by $\displaystyle\int\frac{dy}{h(y)}=\int g(x)\,dx+C$. | 9.2 9.3 9.4 |
Ch 10 · Vector Algebra
| Term | Meaning | Used in |
|---|---|---|
| angle between two vectors | The angle $\theta\in[0,\pi]$ between $\vec a$ and $\vec b$ drawn from the same point; $\cos\theta=\dfrac{\vec a\cdot\vec b}{|\vec a|\,|\vec b|}$. | 10.3 11.3 |
| area of a parallelogram (vector form) | The parallelogram with adjacent sides $\vec a,\vec b$ has area $|\vec a\times\vec b|$; with diagonals $\vec d_1,\vec d_2$ its area is $\tfrac12|\vec d_1\times\vec d_2|$. A triangle $ABC$ has area $\tfrac12|\overrightarrow{AB}\times\overrightarrow{AC}|$. | 4.1 10.4 |
| Cauchy–Schwarz inequality | $|\vec a\cdot\vec b|\le|\vec a|\,|\vec b|$ for all vectors, because $|\cos\theta|\le1$; equality exactly when $\vec a$ and $\vec b$ are collinear. | 10.3 |
| centroid | The point where the three medians of a triangle meet; it divides each median in the ratio $2:1$ from the vertex. Its position vector is $\tfrac13(\vec a+\vec b+\vec c)$. | 10.2 11.1 |
| coinitial vectors | Two or more vectors that have the same initial point, such as $\overrightarrow{OA}$, $\overrightarrow{OB}$ and $\overrightarrow{OC}$. | 10.1 |
| collinear vectors | Vectors that are parallel to the same line, whatever their magnitudes and senses. Non-zero $\vec a$ and $\vec b$ are collinear exactly when $\vec b=\lambda\vec a$ for some scalar $\lambda$. | 10.1 10.2 10.4 11.2 |
| components of a vector | In $\vec a=a_1\hat i+a_2\hat j+a_3\hat k$ the numbers $a_1,a_2,a_3$ are the scalar components of $\vec a$, and $a_1\hat i$, $a_2\hat j$, $a_3\hat k$ are its vector components along the axes. | 10.1 11.1 |
| cross product | Another name for the vector product $\vec a\times\vec b$, computed as the determinant with rows $\hat i,\hat j,\hat k$; $a_1,a_2,a_3$; $b_1,b_2,b_3$. It is perpendicular to both vectors and $\vec b\times\vec a=-(\vec a\times\vec b)$. | 10.4 11.3 |
| direction cosines | The cosines $l=\cos\alpha$, $m=\cos\beta$, $n=\cos\gamma$ of the angles a non-zero vector makes with the positive $x$-, $y$- and $z$-axes. For $\vec a=a_1\hat i+a_2\hat j+a_3\hat k$ they are $\tfrac{a_1}{|\vec a|},\tfrac{a_2}{|\vec a|},\tfrac{a_3}{|\vec a|}$, and always $l^2+m^2+n^2=1$. | 10.1 10.2 10.3 11.1 |
| direction ratios | Any three numbers $a,b,c$ proportional to the direction cosines $l,m,n$ of a vector or line. They are not unique: $1,2,2$ and $2,4,4$ describe the same direction. Divide by $\sqrt{a^2+b^2+c^2}$ to get direction cosines. | 10.1 11.1 11.2 11.3 |
| dot product | Another name for the scalar product $\vec a\cdot\vec b=|\vec a|\,|\vec b|\cos\theta=a_1b_1+a_2b_2+a_3b_3$ — a number, not a vector. | 10.3 10.4 |
| equal vectors | Vectors with the same magnitude and the same direction, whatever their initial points. In components, $\vec a=\vec b$ exactly when $a_1=b_1$, $a_2=b_2$ and $a_3=b_3$. | 10.1 10.2 |
| external division | A point $R$ divides $AB$ externally in the ratio $m:n$ ($m\ne n$) when $R$ lies on line $AB$ outside the segment and $AR:RB=m:n$. | 10.2 |
| internal division | A point $R$ divides $AB$ internally in the ratio $m:n$ when $R$ lies between $A$ and $B$ and $AR:RB=m:n$. | |
| Lagrange's identity | $|\vec a\times\vec b|^2+(\vec a\cdot\vec b)^2=|\vec a|^2\,|\vec b|^2$ — the vector form of $\sin^2\theta+\cos^2\theta=1$. | 10.4 |
| moment of a force | The turning effect about a point $O$ of a force $\vec F$ acting at the point with position vector $\vec r$: the vector $\vec r\times\vec F$. | 10.4 |
| negative of a vector | The vector $-\vec a$ with the same magnitude as $\vec a$ and the opposite direction; for example $\overrightarrow{BA}=-\overrightarrow{AB}$. | 10.1 |
| normal vector | A vector perpendicular to a surface or to two given directions; for a flat panel with edge vectors $\vec u,\vec v$, the cross product $\vec u\times\vec v$ is a normal vector. | 10.4 |
| parallelogram law of vector addition | If $\vec a=\overrightarrow{OA}$ and $\vec b=\overrightarrow{OB}$ start at the same point, their sum is the diagonal $\overrightarrow{OC}$ of the parallelogram $OACB$. The other diagonal $\overrightarrow{BA}$ is $\vec a-\vec b$. | 10.2 |
| perpendicular vectors | Non-zero vectors at right angles. The test is $\vec a\cdot\vec b=0$. | 10.3 10.4 |
| position vector | The position vector of a point $P(x,y,z)$ is $\overrightarrow{OP}=x\hat i+y\hat j+z\hat k$, the arrow from the origin $O$ to $P$. Its magnitude is $\sqrt{x^2+y^2+z^2}$. | 10.1 10.2 10.4 11.2 |
| projection of a vector | The projection of $\vec a$ on $\vec b$ is the signed length $|\vec a|\cos\theta=\dfrac{\vec a\cdot\vec b}{|\vec b|}$ of the shadow of $\vec a$ on the line of $\vec b$. | 10.3 |
| projection vector | The vector $\dfrac{\vec a\cdot\vec b}{|\vec b|^2}\,\vec b$ along $\vec b$ whose signed length is the projection of $\vec a$ on $\vec b$; what is left, $\vec a$ minus it, is perpendicular to $\vec b$. | 10.3 |
| right-hand rule | Curl the fingers of your right hand from $\vec a$ towards $\vec b$; your thumb points along $\vec a\times\vec b$. It fixes which of the two perpendicular directions the cross product takes. | 10.4 |
| right-handed system | Three vectors (or axes) $\vec a,\vec b,\vec c$ such that curling the right hand from $\vec a$ to $\vec b$ makes the thumb point to the side of $\vec c$; the axes $\hat i,\hat j,\hat k$ form one, so $\hat i\times\hat j=\hat k$. | 10.1 10.4 |
| scalar multiple of a vector | $\lambda\vec a$ has magnitude $|\lambda|\,|\vec a|$ and the direction of $\vec a$ when $\lambda\gt0$, the opposite direction when $\lambda\lt0$; in components $\lambda\vec a=\lambda a_1\hat i+\lambda a_2\hat j+\lambda a_3\hat k$. | |
| scalar product | The number $\vec a\cdot\vec b=|\vec a|\,|\vec b|\cos\theta$, where $\theta$ is the angle between $\vec a$ and $\vec b$. In components it is $a_1b_1+a_2b_2+a_3b_3$. Also called the dot product. | 10.1 10.3 11.2 11.3 |
| section formula | The point dividing the join of $\vec a$ and $\vec b$ in the ratio $m:n$ has position vector $\dfrac{m\vec b+n\vec a}{m+n}$ (internally) or $\dfrac{m\vec b-n\vec a}{m-n}$ (externally, $m\ne n$). | 10.2 |
| triangle inequality | $|\vec a+\vec b|\le|\vec a|+|\vec b|$: one side of a triangle is never longer than the other two together. Equality holds only when $\vec a$ and $\vec b$ point the same way. | 10.3 |
| triangle law of vector addition | To add $\vec b$ to $\vec a$, start $\vec b$ at the tip of $\vec a$; the sum runs from the tail of $\vec a$ to the tip of $\vec b$: $\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$. | 10.2 |
| unit vector | A vector of magnitude $1$. The unit vector in the direction of a non-zero vector $\vec a$ is $\hat a=\dfrac{\vec a}{|\vec a|}$; $\hat i,\hat j,\hat k$ are the unit vectors along the axes. | 10.1 10.2 10.3 10.4 |
| vector joining two points | For $P_1(x_1,y_1,z_1)$ and $P_2(x_2,y_2,z_2)$: $\overrightarrow{P_1P_2}=(x_2-x_1)\hat i+(y_2-y_1)\hat j+(z_2-z_1)\hat k$ — position vector of the terminal point minus that of the initial point. | 10.2 |
| vector product | For non-collinear $\vec a,\vec b$ at angle $\theta$: $\vec a\times\vec b=|\vec a|\,|\vec b|\sin\theta\,\hat n$, where $\hat n$ is the unit vector perpendicular to both given by the right-hand rule; $\vec 0$ if $\vec a,\vec b$ are collinear. Also called the cross product. | 10.4 11.3 |
| work done by a force | A constant force $\vec F$ acting through a displacement $\vec d$ does work $W=\vec F\cdot\vec d=|\vec F|\,|\vec d|\cos\theta$ (joules, for newtons and metres). | 10.1 10.3 |
| zero vector | The vector $\vec 0$ whose initial and terminal points coincide. It has magnitude $0$ and no definite direction (it may be regarded as having any direction). | 10.1 10.3 10.4 11.2 |
Ch 11 · Three Dimensional Geometry
| Term | Meaning | Used in |
|---|---|---|
| angle between two lines | The non-obtuse angle $\theta$ between lines through one point parallel to the two lines: $\cos\theta=\dfrac{|\vec b_1\cdot\vec b_2|}{|\vec b_1|\,|\vec b_2|}$, where $\vec b_1,\vec b_2$ are direction vectors. | 11.3 |
| body diagonal | A segment joining two opposite corners of a cube or cuboid through its interior, such as $O$ to $(s,s,s)$ in a cube of side $s$. It makes an angle $\cos^{-1}\tfrac1{\sqrt3}\approx54.74^\circ$ with each edge through $O$. | 10.3 11.1 11.3 |
| Cartesian equation of a line | $\dfrac{x-x_1}{a}=\dfrac{y-y_1}{b}=\dfrac{z-z_1}{c}$ for the line through $(x_1,y_1,z_1)$ with direction ratios $a,b,c$; the common value of the fractions is the parameter $\lambda$. A zero denominator means that numerator is $0$. | 11.2 11.3 |
| common perpendicular | The line that meets two skew lines and is perpendicular to both. The segment of it between the two lines is the shortest segment joining them; its direction is that of $\vec b_1\times\vec b_2$. | 11.3 |
| directed line | A line together with a choice of one of its two directions. Reversing the direction changes every direction cosine $l,m,n$ to $-l,-m,-n$. | 10.1 11.1 |
| direction vector of a line | Any non-zero vector parallel to the line. Its components are direction ratios of the line, and every non-zero multiple of it is again a direction vector. | 11.2 |
| face diagonal | A diagonal of one face of a cube or cuboid, such as $O$ to $(s,s,0)$; its direction cosines are $\left(\tfrac1{\sqrt2},\tfrac1{\sqrt2},0\right)$. | 11.1 11.3 |
| foot of the perpendicular | The point $F$ of a line where the perpendicular from a point $P$ meets it. For $\vec r=\vec a+\lambda\vec b$ it is found from $\overrightarrow{PF}\cdot\vec b=0$. | 10.1 11.1 11.2 11.3 |
| image of a point in a line | The reflection $P'$ of $P$ in a line: the foot of the perpendicular $F$ is the mid-point of $PP'$, so $P'=2F-P$. | |
| intersecting lines | Two lines with exactly one common point. Non-parallel lines intersect exactly when $(\vec b_1\times\vec b_2)\cdot(\vec a_2-\vec a_1)=0$. | 11.3 |
| line through two points | Exactly one line passes through two different points $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$; its direction ratios are $x_2-x_1,\ y_2-y_1,\ z_2-z_1$. | 11.1 11.2 11.3 |
| parallel lines in space | Lines with proportional direction ratios, $a_1:b_1:c_1=a_2:b_2:c_2$. Two different parallel lines never meet and are a constant distance apart. | 4.3 11.1 11.2 11.3 |
| perpendicular distance from a point to a line | The length $PF$, where $F$ is the foot of the perpendicular from $P$ to the line; it is the shortest distance from $P$ to any point of the line. | 11.2 11.3 |
| perpendicular lines in space | Lines whose direction ratios satisfy $a_1a_2+b_1b_2+c_1c_2=0$. In space perpendicular lines need not meet — they may be skew. | 11.3 |
| shortest distance between two lines | The least distance between a point of one line and a point of the other. For non-parallel lines it is $\left|\dfrac{(\vec b_1\times\vec b_2)\cdot(\vec a_2-\vec a_1)}{|\vec b_1\times\vec b_2|}\right|$; for parallel lines $\dfrac{|\vec b\times(\vec a_2-\vec a_1)|}{|\vec b|}$. | 10.4 11.3 |
| skew lines | Two lines in space that are neither parallel nor intersecting. They do not lie together in any flat sheet, and the shortest distance between them is positive. | 11.3 |
| vector equation of a line | $\vec r=\vec a+\lambda\vec b$, $\lambda\in\mathbb R$: the points with position vector $\vec r$ on the line through the point with position vector $\vec a$, parallel to the non-zero vector $\vec b$. | 11.2 |
Ch 12 · Linear Programming
| Term | Meaning | Used in |
|---|---|---|
| bounded region | A feasible region that can be enclosed in a circle. On a non-empty bounded region the objective function has both a maximum and a minimum, at corner points. | 8.1 12.2 12.3 |
| constraint | In a linear programming problem, a linear inequality or equation that the decision variables must satisfy, such as $3x+y\le30$ for a limited resource. | 12.1 12.2 12.3 |
| corner point | A vertex of a feasible region: a point of the region where two of its boundary lines cross. Optimal values of a linear objective function occur at corner points. | 12.1 12.2 12.3 |
| corner point method | Solving a linear programming problem by finding all corner points of the feasible region, evaluating $Z$ at each, and choosing the largest or smallest value (with an extra test when the region is unbounded). | 12.2 |
| decision variables | The quantities to be decided in a linear programming problem, such as the numbers $x$ and $y$ of two products to make. | 12.1 |
| empty feasible region | A feasible region with no points, which happens when the half-planes of the constraints have nothing in common. | 12.3 |
| feasible region | The set of all points satisfying every constraint of a linear programming problem, non-negative constraints included — the common part of all the half-planes. | 12.1 12.2 12.3 |
| feasible solution | Any point of the feasible region of a linear programming problem: a plan that satisfies every constraint (not necessarily the best one). | 12.1 12.2 12.3 |
| half-plane | Either of the two parts into which a straight line divides the coordinate plane. It is closed if the line is included ($\le$, $\ge$; solid line) and open if not ($\lt$, $\gt$; dashed line). | 12.1 12.2 12.3 |
| infeasible problem | A linear programming problem whose constraints have no common solution: its feasible region is empty, so it has no feasible and no optimal solution. | 12.3 |
| infeasible solution | A point that violates at least one constraint of a linear programming problem, so it lies outside the feasible region. | 12.1 |
| iso-cost line | A line $ax+by=k$ joining all plans with the same cost $Z=k$ in a minimisation problem; the minimum is where the sliding line first meets the feasible region. | 12.2 |
| iso-profit line | A line $ax+by=k$ joining all plans with the same profit $Z=k$. Iso-profit lines are parallel; increasing $k$ slides them in the direction $(a,b)$. | 12.2 |
| linear inequality in two variables | An inequality $ax+by\lt c$, $ax+by\le c$, $ax+by\gt c$ or $ax+by\ge c$ with $a,b$ not both zero. Its solutions form a half-plane. | 12.1 |
| linear programming problem | A problem (LPP) of finding the maximum or minimum of a linear objective function $Z=ax+by$ when the decision variables satisfy a set of linear constraints. | 12.1 12.2 12.3 |
| multiple optimal solutions | When $Z$ takes its optimal value at two points (for example two adjacent corners), it takes it at every point of the segment joining them — infinitely many optimal solutions. | 12.2 |
| non-negative constraints | The constraints $x\ge0,\ y\ge0$ of a linear programming problem, saying that the decision variables cannot be negative. | 12.1 12.2 |
| objective function | The linear function $Z=ax+by$ (profit, cost, time…) that a linear programming problem maximises or minimises. | 12.1 12.2 12.3 |
| open half-plane test | For an unbounded feasible region $R$ with largest corner value $M$: if the open half-plane $ax+by\gt M$ has no point in common with $R$, then $M$ is the maximum of $Z$; otherwise there is no maximum. Similarly with $ax+by\lt m$ for the minimum. | 12.3 |
| optimal solution | A feasible solution of a linear programming problem at which the objective function takes its maximum (or minimum) value over the whole feasible region. | 12.2 12.3 |
| optimal value | The maximum (or minimum) value of the objective function of a linear programming problem, attained at an optimal solution. | 12.2 12.3 |
| unbounded feasible region | A feasible region that no circle can enclose — it stretches out without end. On it the objective function may or may not have a maximum or a minimum; the open half-plane test decides. | 12.2 12.3 |
Ch 13 · Probability
| Term | Meaning | Used in |
|---|---|---|
| base rate | How common a cause is in the group being studied — the prior probability. Ignoring it (the base-rate fallacy) makes positive results for rare conditions look far more conclusive than they are. | 13.3 |
| Bayes' theorem | For a partition $E_1,\dots,E_n$ and $P(A)\ne0$: $P(E_i\mid A)=\dfrac{P(E_i)P(A\mid E_i)}{\sum_j P(E_j)P(A\mid E_j)}$. It turns the probability of the evidence given a cause into the probability of the cause given the evidence. | 13.1 13.3 |
| Bernoulli trials | Independent repetitions of an experiment with two results, success (probability $p$, the same each time) and failure. Exactly $k$ successes in $n$ trials has probability $^nC_k\,p^kq^{n-k}$ (JEE extension). | 13.2 |
| conditional probability | The probability $P(E\mid F)=\dfrac{P(E\cap F)}{P(F)}$ that $E$ happens, given that $F$ has happened ($P(F)\ne0$). Read the bar as "given". | 13.1 13.2 13.3 |
| conditioning event | The event $F$ after the bar in $P(E\mid F)$ — the information we are given. It becomes the new, reduced sample space. | 13.1 |
| dependent events | Events that are not independent: $P(E\cap F)\ne P(E)P(F)$, so information about one changes the probability of the other. | |
| expectation | Another name for the mean of a random variable, $E(X)=\sum x_ip_i$. It need not be a value that $X$ can take. | 13.4 |
| fair game | A game in which the player's expected gain is $0$: the expected prize equals the cost of playing. | |
| false positive | A positive test result for someone who does not have the condition. The false-positive rate is $P(+\mid\text{no condition})$. | 13.3 |
| independent events | Events with $P(E\cap F)=P(E)P(F)$: knowing that one happened does not change the probability of the other ($P(F\mid E)=P(F)$ when $P(E)\ne0$). | 13.2 |
| mean of a random variable | $E(X)=\mu=\sum x_ip_i$: the probability-weighted average of the values; the long-run average and the balance point of the distribution. | |
| multiplication theorem on probability | $P(E\cap F)=P(E)\,P(F\mid E)$ when $P(E)\ne0$; for three events $P(E\cap F\cap G)=P(E)P(F\mid E)P(G\mid E\cap F)$. It finds the chance that several things all happen, stage by stage. | 13.2 13.3 13.4 |
| mutually independent events | Events $A,B,C$ for which every pair satisfies the product rule and also $P(A\cap B\cap C)=P(A)P(B)P(C)$. | 13.2 |
| pairwise independent | Three or more events such that every pair satisfies the product rule. This does not imply that the events are mutually independent. | 13.2 |
| partition of a sample space | Events $E_1,\dots,E_n$ that are pairwise disjoint, together make up the whole sample space $S$, and each have non-zero probability: exactly one of them happens. | 13.3 |
| posterior probability | $P(E_i\mid A)$ in Bayes' theorem: how likely the cause $E_i$ is after the evidence $A$ has been observed. | 13.3 |
| prior probability | $P(E_i)$ in Bayes' theorem: how likely the cause $E_i$ is before the evidence is taken into account. | 13.3 |
| probability distribution | The list of the values $x_1,\dots,x_n$ of a random variable with their probabilities $p_i=P(X=x_i)$, where $p_i\gt0$ and $\sum p_i=1$. | 13.4 |
| random variable | A real-valued function on the sample space of a random experiment: it attaches a number (number of heads, a sum, a gain) to every outcome. | 13.4 |
| reduced sample space | The set of outcomes still possible once an event $F$ is known to have happened: $F$ itself. For equally likely outcomes $P(E\mid F)=\dfrac{n(E\cap F)}{n(F)}$. | 13.1 13.2 |
| sensitivity of a test | The probability $P(+\mid\text{condition})$ that a test is positive for someone who really has the condition. | |
| theorem of total probability | For a partition $E_1,\dots,E_n$ and any event $A$: $P(A)=\sum_j P(E_j)P(A\mid E_j)$ — multiply along each branch of the tree and add. | 13.1 13.3 |
| tree diagram | A diagram with one level of branches per stage of an experiment, each labelled with a conditional probability. Multiply along a path; add across paths. | 13.2 13.3 |
| two-way table | A table of counts classified by two characteristics at once (rows and columns, with totals). A row or a column total is the denominator of a conditional probability. | 13.1 |
| variance of a random variable | $\operatorname{Var}(X)=\sum(x_i-\mu)^2p_i=E(X^2)-[E(X)]^2$, a measure of spread; its square root is the standard deviation (JEE extension). | 13.4 |
| without replacement | Drawing objects one after another without putting them back, so the contents (and the probabilities) change after each draw. Draws with replacement are independent; draws without replacement are not. | 13.2 13.4 |