Glossary — English / العربية
Every technical word in the course — its Arabic name and what it means in plain language, with flashcards for practice.
Every technical term in the course, with its Arabic name and a plain-language meaning. On every page these words are marked with a dotted underline: hover or tap one to see this information, and press the ع button (bottom-right) to show Arabic names beside the English words throughout the text. The “Used in” links point to the sections where each term appears.
Ch 5 · Techniques of IntegrationCh 6 · Applications of IntegrationCh 7 · Introduction to Differential EquationsCh 8 · Sequences and SeriesCh 9 · Power SeriesCh 10 · Parametric Equations and Polar CoordinatesCalculus I review
Ch 5 · Techniques of Integration
| English | العربية | Meaning | Used in |
|---|---|---|---|
| absolute error | الخطأ المطلق | The size of the difference between the exact value and the approximation: $|I-A|$. | 5.4 |
| circular segment | القطعة الدائرية | The region of a disc cut off by a chord, such as the cross-section of the liquid in a horizontal cylindrical tank. | 5.2 10.4 |
| comparison test | اختبار المقارنة | If $0\le f\le g$, then convergence of the bigger one ($g$) forces convergence of the smaller ($f$), and divergence of the smaller forces divergence of the bigger. Used for improper integrals and for series. | 5.5 8.3 8.4 8.5 |
| completing the square | إكمال المربع | Rewriting $x^2+bx+c$ as $\left(x+\tfrac b2\right)^2+\left(c-\tfrac{b^2}{4}\right)$ so that a quadratic matches one of the standard forms. | 5.2 5.3 10.3 |
| convergent | متقارب (تقارب) | An improper integral (or a series) converges when the limit that defines it exists and is a finite number. | 5.5 6.1 6.3 6.4 |
| cover-up method | طريقة التغطية (طريقة هيفيسايد) | A shortcut for the constant over a non-repeated linear factor $x-r$: cover that factor and substitute $x=r$ into the rest. It computes $\lim_{x\to r}(x-r)\frac{P(x)}{Q(x)}$. | 5.3 |
| divergent | متباعد (تباعد) | An improper integral (or a series) diverges when the limit that defines it does not exist as a finite number — for example it grows to $\infty$. | 5.5 8.1 8.2 8.3 |
| ellipse | القطع الناقص | The curve $\tfrac{x^2}{a^2}+\tfrac{y^2}{b^2}=1$; it encloses area $\pi ab$. | 5.2 5.4 6.2 10.1 |
| equating coefficients | مساواة المعاملات | Finding unknown constants by clearing denominators, expanding, and matching the coefficients of each power of $x$ on the two sides. | 5.3 |
| error bound | حدّ الخطأ | A guaranteed upper limit on the size of an approximation error, such as $|E_T|\le\frac{K(b-a)^3}{12n^2}$ when $|f''|\le K$. | 5.4 8.3 8.5 9.2 |
| escape velocity | سرعة الإفلات | The launch speed $v=\sqrt{2GM/R}$ needed to escape a planet's gravity, found from the convergent improper integral $\int_R^\infty\frac{GM}{r^2}\,dr$. | 5.5 |
| half-angle identity | متطابقة نصف الزاوية | An identity that lowers a power, e.g. $\sin^2x=\tfrac12(1-\cos2x)$ and $\cos^2x=\tfrac12(1+\cos2x)$. | 5.1 5.2 10.2 10.4 |
| improper integral | التكامل المعتل | An integral over an infinite interval, or of a function that is unbounded somewhere on the interval. It is defined as a limit of ordinary definite integrals, e.g. $\int_1^\infty f=\lim_{b\to\infty}\int_1^b f$. | 5.5 6.3 6.4 8.3 |
| improper rational function | دالة كسرية غير فعلية | A rational function $P/Q$ with $\deg P\ge\deg Q$. Long division turns it into a polynomial plus a proper rational function. | 5.3 |
| infinite discontinuity | انقطاع لانهائي | A point where a function becomes unbounded, typically a vertical asymptote such as $x=0$ for $\frac1x$. Integrating across one makes the integral improper. | 5.5 |
| inverse substitution | التعويض العكسي | A substitution of the form $x=g(\theta)$, which replaces the variable itself rather than naming part of the integrand $u$; $g$ must be one-to-one on the interval used. | |
| irreducible quadratic | مقدار تربيعي غير قابل للتحليل | A quadratic $ax^2+bx+c$ with $b^2-4ac\lt0$: it has no real roots, so it cannot be factored into real linear factors. | 5.3 |
| midpoint rule | قاعدة نقطة المنتصف | $M_n=\Delta x\,[f(m_1)+\cdots+f(m_n)]$: rectangles whose heights are taken at the midpoints $m_i$ of the subintervals. | 5.4 |
| numerical integration | التكامل العددي | Approximating a definite integral by a weighted sum of function values, used when no antiderivative is available or only data are known. | 5.4 6.4 10.2 |
| orthogonality | التعامد | For trigonometric functions: $\int_{-\pi}^{\pi}\sin mx\sin nx\,dx=0$ for positive integers $m\ne n$, so different frequencies do not overlap. | 5.1 |
| p-integral | تكامل p | $\int_1^\infty\frac{dx}{x^p}$ (converges exactly when $p\gt1$) or $\int_0^1\frac{dx}{x^p}$ (converges exactly when $p\lt1$). The standard yardsticks for comparison. | |
| partial fraction | الكسور الجزئية | One of the simple fractions, such as $\frac{A}{x-a}$ or $\frac{Bx+C}{x^2+c^2}$, that a proper rational function can be split into. Each one is easy to integrate. | 5.3 5.5 6.4 7.3 |
| polynomial long division | القسمة المطوّلة لكثيرات الحدود | The division algorithm for polynomials: $P=SQ+R$ with $\deg R\lt\deg Q$, so $\frac PQ=S+\frac RQ$. | 5.3 9.4 |
| probability density function | دالة الكثافة الاحتمالية | A function $f\ge0$ with $\int_{-\infty}^{\infty}f(x)\,dx=1$; probabilities are areas under it, e.g. $P(X\gt a)=\int_a^\infty f(x)\,dx$. | 5.5 |
| product-to-sum identity | متطابقات تحويل الضرب إلى مجموع | An identity rewriting a product such as $\sin A\cos B$ as half a sum of single sines or cosines. | 5.1 |
| proper rational function | دالة كسرية فعلية | A rational function $P/Q$ whose numerator has smaller degree than its denominator: $\deg P\lt\deg Q$. Only proper functions can be decomposed directly. | |
| Pythagorean identity | المتطابقة الفيثاغورية | $\sin^2x+\cos^2x=1$ and its relative $1+\tan^2x=\sec^2x$; used to convert even powers from one function to another. | 5.2 |
| reduction formula | صيغة الاختزال | A formula that expresses an integral containing a power $n$ in terms of the same integral with a smaller power, such as $n-2$. | 5.1 |
| reference triangle | المثلث المرجعي | The right triangle built from a substitution such as $\sin\theta=\tfrac xa$ (opposite $x$, hypotenuse $a$), used to write every trigonometric function of $\theta$ back in terms of $x$. | 5.2 |
| relative error | الخطأ النسبي | The absolute error divided by the size of the exact value, $\frac{|I-A|}{|I|}$; often quoted as a percentage. | 5.4 9.4 |
| repeated linear factor | عامل خطي مكرر | A linear factor that appears to a power, like $(x-2)^3$. It needs one partial fraction for each power: $\frac{A_1}{x-2}+\frac{A_2}{(x-2)^2}+\frac{A_3}{(x-2)^3}$. | 5.3 |
| Simpson's rule | قاعدة سمبسون | $S_n=\frac{\Delta x}{3}[y_0+4y_1+2y_2+\cdots+4y_{n-1}+y_n]$ with $n$ even: fits parabolas through consecutive triples of points. Exact for cubics. | 5.4 6.2 6.3 6.4 |
| trapezoidal rule | قاعدة شبه المنحرف | $T_n=\frac{\Delta x}{2}[f(x_0)+2f(x_1)+\cdots+2f(x_{n-1})+f(x_n)]$: joins neighbouring points by straight lines and adds the trapezoid areas. | 5.4 5.5 6.1 6.2 |
| trigonometric integral | التكامل المثلثي | An integral whose integrand is built from powers or products of trigonometric functions, such as $\int\sin^3x\cos^2x\,dx$. | 5.2 |
| trigonometric substitution | التعويض المثلثي | Replacing $x$ by $a\sin\theta$, $a\tan\theta$ or $a\sec\theta$ so that a Pythagorean identity removes a square root such as $\sqrt{a^2-x^2}$. | 5.2 10.4 |
Ch 6 · Applications of Integration
| English | العربية | Meaning | Used in |
|---|---|---|---|
| arc length | طول القوس | The length of a curve, found as the limit of lengths of inscribed polygons; for a smooth graph $L=\int_a^b\sqrt{1+\left(f'(x)\right)^2}\,dx$. | 5.4 6.4 10.2 10.3 |
| arc-length function | دالة طول القوس | The distance along a curve from a fixed starting point: $s(x)=\int_a^x\sqrt{1+\left(f'(t)\right)^2}\,dt$, so $ds=\sqrt{1+(dy/dx)^2}\,dx$. | 6.4 |
| area between curves | المساحة بين منحنيين | The area of the region enclosed by two graphs, found by integrating (top − bottom) $dx$ or (right − left) $dy$: $A=\int_a^b\left[f(x)-g(x)\right]dx$ when $f\ge g$. | 6.1 |
| axis of revolution | محور الدوران | The line about which a plane region is revolved to produce a solid of revolution. | 6.2 6.3 |
| catenary | المنحنى السلسلي | The curve $y=a\cosh(x/a)$ formed by a flexible chain or cable hanging under its own weight. | 6.4 |
| Cavalieri's principle | مبدأ كافاليري | Two solids whose cross-sections have equal areas at every height have equal volumes. | 6.2 |
| centre of mass | مركز الكتلة | The balance point of a mass distribution, $(\bar x,\bar y)=(M_y/M,\ M_x/M)$; the total moment about it is zero. | 6.5 |
| centroid | المركز الهندسي | The centre of mass of a region of uniform density; it depends only on the shape: $\bar x=\frac1A\int x(f-g)\,dx$, $\bar y=\frac1A\int\frac12\left(f^2-g^2\right)dx$. | 6.5 |
| consumer surplus | فائض المستهلك | The total amount buyers save by paying the market price rather than the most they would pay: $CS=\int_0^{q_e}\left[D(q)-p_e\right]dq$, the area between the demand curve and the price line. | 6.1 |
| cross-section | مقطع عرضي | The flat shape obtained by cutting a solid with a plane; for the slicing method the plane is perpendicular to an axis and its area is $A(x)$. | 5.2 6.2 6.5 10.2 |
| cylindrical shell | القشرة الأسطوانية | The solid between two coaxial cylinders of the same height; with average radius $r$, height $h$ and thickness $\Delta r$ its volume is $2\pi r h\,\Delta r$. | 6.3 |
| disk method | طريقة الأقراص | Volume of a solid of revolution whose slices are solid disks: $V=\int_a^b\pi\left[f(x)\right]^2dx$. | 6.2 |
| frustum | جذع المخروط | The part of a cone between its base and a parallel plane cutting off the tip; its curved surface has area $\pi(r_1+r_2)\ell$, with $\ell$ the slant height. | 6.2 6.4 |
| Gini coefficient | معامل جيني | A measure of inequality between $0$ (perfect equality) and $1$: twice the area between the line $y=x$ and the Lorenz curve, $G=2\int_0^1\left[x-L(x)\right]dx$. | 6.1 |
| Hooke's law | قانون هوك | The force needed to hold a spring stretched or compressed a distance $x$ from its natural length is $F=kx$. | 6.5 |
| hyperbolic cosine | جيب التمام الزائدي | $\cosh x=\tfrac12\left(e^x+e^{-x}\right)$; it satisfies $\cosh^2x-\sinh^2x=1$ and $(\cosh x)'=\sinh x$. | 6.4 |
| hyperbolic sine | الجيب الزائدي | $\sinh x=\tfrac12\left(e^x-e^{-x}\right)$; its derivative is $\cosh x$. | |
| intersection point | نقطة التقاطع | A point where two curves meet; found by setting their equations equal. Intersection points usually give the limits of integration for an area between curves. | 6.1 6.2 10.4 |
| lamina | صفيحة رقيقة | A thin flat plate, modelled as a plane region with mass per unit area $\rho$. | 6.5 |
| lateral area | المساحة الجانبية | The area of the curved side of a cylinder or cone, not counting its ends; for a cylinder of radius $r$ and height $h$ it is $2\pi rh$. | 6.3 6.4 |
| Lorenz curve | منحنى لورنز | A curve $y=L(x)$ giving the fraction of total income earned by the poorest fraction $x$ of a population; it lies below the line of equality $y=x$. | 6.1 |
| market equilibrium | توازن السوق | In a market, the quantity $q_e$ and price $p_e$ at which the demand and supply curves meet: $D(q_e)=S(q_e)=p_e$. | 6.1 |
| method of slicing | طريقة الشرائح | Finding a volume by cutting the solid into thin slices and integrating their cross-sectional areas: $V=\int_a^bA(x)\,dx$. | 6.2 |
| moment | العزم | Mass times signed distance from an axis or point, added over all the mass: $M_y=\sum m_ix_i$ about the $y$-axis, $M_x=\sum m_iy_i$ about the $x$-axis. | 6.5 |
| producer surplus | فائض المنتج | The extra income sellers receive above their minimum acceptable price: $PS=\int_0^{q_e}\left[p_e-S(q)\right]dq$, the area between the price line and the supply curve. | 6.1 |
| shell method | طريقة القشرات الأسطوانية | Finding the volume of a solid of revolution by slicing parallel to the axis into thin cylindrical shells: $V=\int_a^b2\pi\,(\text{radius})(\text{height})\,dx$. | 6.3 |
| solid of revolution | مجسم دوراني | A solid swept out when a plane region is revolved about a line (the axis of revolution) in its plane. | 6.2 6.3 |
| spring constant | ثابت النابض | The constant $k$ (in N/m) in Hooke's law $F=kx$; a stiffer spring has a larger $k$. | 6.5 |
| surface of revolution | سطح دوراني | The surface swept out when a curve is revolved about a line; about the $x$-axis its area is $S=\int_a^b2\pi f(x)\sqrt{1+\left(f'(x)\right)^2}\,dx$. | 6.4 |
| Theorem of Pappus | نظرية بابوس | Revolving a plane region of area $A$ about a line that does not cut it gives volume $V=2\pi dA$, where $d$ is the distance from the region's centroid to the line. | 6.5 |
| torus | الطارة | The doughnut-shaped solid (or surface) formed by revolving a disk (or circle) about a line in its plane that does not cut it. | 6.3 6.5 |
| washer method | طريقة الحلقات | Volume of a solid of revolution whose slices are washers (disks with a hole): $V=\int_a^b\pi\left(R^2-r^2\right)dx$, with $R$ and $r$ the outer and inner radii. | 6.2 |
| work done | الشغل المبذول | Energy transferred by a force acting through a distance; for a variable force along a line $W=\int_a^bF(x)\,dx$, measured in joules (1 J = 1 N·m). | 6.5 |
Ch 7 · Introduction to Differential Equations
| English | العربية | Meaning | Used in |
|---|---|---|---|
| autonomous equation | معادلة تفاضلية ذاتية | A first-order equation $y'=f(y)$ whose right-hand side does not depend explicitly on the independent variable. | 7.2 7.4 |
| carrying capacity | السعة الحاملة | The largest population $K$ that an environment can sustain in the long run; the stable equilibrium of the logistic equation. | 5.3 7.4 |
| direction field | حقل الاتجاهات | A picture of short line segments, one at each grid point $(x,y)$, with slope $f(x,y)$; solution curves of $y'=f(x,y)$ are tangent to it everywhere. | 7.2 7.3 |
| doubling time | زمن التضاعف | The time for an exponentially growing quantity to double: $T_d=\ln 2/k$. | 7.3 |
| equilibrium solution | حل التوازن | A constant solution $y=c$ of $y'=f(y)$, found from $f(c)=0$; its graph is a horizontal line. | 7.1 7.2 7.3 7.4 |
| Euler's method | طريقة أويلر | A numerical method for $y'=f(x,y)$ that follows tangent lines in steps of size $h$: $y_{k+1}=y_k+h\,f(x_k,y_k)$. | 7.2 7.3 |
| explicit solution | حل صريح | A solution written directly as $y=f(x)$. | 7.3 |
| exponential decay | التضاؤل الأسي | Decrease described by $y'=ky$ with $k\lt0$, giving $y=y_0e^{kt}$; the amount halves over every fixed time interval (the half-life). | 7.1 8.2 |
| exponential growth | النمو الأسي | Growth described by $y'=ky$ with $k\gt0$: the rate of increase is proportional to the amount present, giving $y=y_0e^{kt}$. | 7.1 7.3 7.4 |
| family of solutions | عائلة الحلول | The set of all graphs given by the general solution as the arbitrary constant varies. | 7.1 |
| general solution | الحل العام | A formula with arbitrary constants that describes every solution of a differential equation, e.g. $y=x^2+C$ for $y'=2x$. | 7.1 7.2 7.3 |
| half-life | عمر النصف | The time for an exponentially decaying quantity to fall to half its value: $t_{1/2}=\ln 2/|k|$. | 7.1 7.3 8.2 9.1 |
| implicit solution | حل ضمني | A solution given as an equation linking $x$ and $y$, such as $y^3+y=x^2+x+2$, rather than as a formula $y=f(x)$. | 7.3 |
| intrinsic growth rate | معدل النمو الذاتي | The per-capita growth rate $r$ of a population when it is small and resources are plentiful. | 7.4 |
| logistic equation | المعادلة اللوجستية | The differential equation $dP/dt=rP(1-P/K)$, with $r\gt0$ and $K\gt0$: growth that is nearly exponential for small $P$ and slows to zero at $P=K$. | 7.4 |
| logistic function | الدالة اللوجستية | A solution $P(t)=K/(1+Ae^{-rt})$ of the logistic equation, with an S-shaped graph when $0\lt P_0\lt K/2$. | 7.4 |
| maximum sustainable yield | أقصى إنتاج مستدام | The largest constant harvest $H=rK/4$ that a logistic population can support indefinitely, taken at population $K/2$. | 7.4 |
| Newton's law of cooling | قانون نيوتن للتبريد | The rate of change of an object's temperature is proportional to the difference between its temperature and the surroundings: $dT/dt=-k(T-T_a)$ with $k\gt0$. | 7.1 7.3 |
| order of a differential equation | رتبة المعادلة التفاضلية | The order of the highest derivative that appears in the equation; $y''+9y=0$ has order $2$. | 7.1 7.2 7.3 |
| ordinary differential equation | معادلة تفاضلية عادية | A differential equation whose unknown is a function of a single variable, such as $y=y(x)$, so only ordinary (not partial) derivatives appear. | 7.1 |
| particular solution | الحل الخاص | One specific solution, obtained by giving the constants of the general solution definite values, usually from initial conditions. | 7.1 |
| phase line | خط الطور | The $y$-axis of an autonomous equation $dy/dt=f(y)$, marked with its equilibria and with arrows showing whether solutions increase or decrease. | 7.2 7.4 |
| radiocarbon dating | التأريخ بالكربون المشع | Estimating the age of once-living material from the fraction of carbon-14 left, using its half-life of about 5730 years. | 7.3 |
| semi-stable equilibrium | توازن شبه مستقر | An equilibrium approached by solutions on one side but left by solutions on the other side, e.g. $y=1$ for $y'=(y-1)^2$. | 7.2 7.4 |
| separable equation | معادلة تفاضلية قابلة لفصل المتغيرات | A first-order differential equation that can be written as $dy/dx=g(x)\,h(y)$: a function of $x$ times a function of $y$. | 7.3 |
| separation of variables | فصل المتغيرات | Solving $dy/dx=g(x)h(y)$ by rewriting it as $dy/h(y)=g(x)\,dx$ and integrating both sides. Constant solutions where $h(y)=0$ must be found separately. | 7.3 |
| solution curve | منحنى الحل | The graph of one particular solution of a differential equation. | 7.1 7.2 7.4 |
| stable equilibrium | توازن مستقر | An equilibrium that nearby solutions approach, from both sides, as $x\to\infty$; e.g. room temperature for a cooling cup. | 7.2 7.4 |
| step size | طول الخطوة | The horizontal distance $h$ moved in one step of a numerical method; Euler's error is roughly proportional to $h$. | 7.2 |
| unstable equilibrium | توازن غير مستقر | An equilibrium from which nearby solutions on both sides move away. | 7.2 7.4 |
Ch 8 · Sequences and Series
| English | العربية | Meaning | Used in |
|---|---|---|---|
| absolute convergence | التقارب المطلق | $\sum a_n$ converges absolutely when $\sum|a_n|$ converges. Absolute convergence implies convergence, and the sum does not depend on the order of the terms. | 8.5 8.6 |
| alternating harmonic series | المتسلسلة التوافقية المتناوبة | The series $1-\tfrac12+\tfrac13-\tfrac14+\cdots=\sum\frac{(-1)^{n+1}}{n}$. It converges (to $\ln 2$) although the harmonic series diverges. | 8.5 9.1 |
| alternating series | متسلسلة متناوبة | A series whose terms are alternately positive and negative, written $\sum(-1)^{n+1}b_n$ or $\sum(-1)^nb_n$ with every $b_n\gt0$. | 8.5 8.6 9.1 9.2 |
| bounded sequence | متتالية محدودة | A sequence whose terms all lie between two fixed numbers: $m\le a_n\le M$ for every $n$. | 8.1 |
| common ratio | أساس المتسلسلة الهندسية (النسبة المشتركة) | The fixed factor $r$ between consecutive terms of a geometric sequence or series. | 8.2 |
| comparison series | متسلسلة المقارنة | A series of known behaviour (usually geometric or a $p$-series) used to decide the behaviour of another series. | 8.4 8.6 |
| conditional convergence | التقارب المشروط | $\sum a_n$ converges conditionally when it converges but $\sum|a_n|$ diverges, e.g. $\sum\frac{(-1)^{n+1}}{n}$. Its sum can change if the terms are reordered. | 8.5 8.6 |
| convergence test | اختبار التقارب | A rule that decides whether a series converges or diverges by examining its terms, without finding the sum. | 8.3 |
| convergent sequence | متتالية متقاربة | A sequence whose terms approach a single real number $L$: for every $\varepsilon\gt0$, $|a_n-L|\lt\varepsilon$ for all large enough $n$. | 8.1 |
| convergent series | متسلسلة متقاربة | A series whose partial sums approach a finite limit $S$, its sum. | 8.2 8.4 8.5 9.2 |
| direct comparison test | اختبار المقارنة المباشرة | If $0\le a_n\le b_n$ eventually, then $\sum b_n$ convergent implies $\sum a_n$ convergent, and $\sum a_n$ divergent implies $\sum b_n$ divergent. | 8.3 8.4 |
| divergence test | اختبار التباعد (اختبار الحد النوني) | If $\lim a_n\ne0$ (or the limit does not exist), then $\sum a_n$ diverges. If $a_n\to0$ the test says nothing. | 8.3 8.4 8.5 8.6 |
| divergent sequence | متتالية متباعدة | A sequence that does not converge — it grows without bound or keeps oscillating, like $(-1)^n$. | 8.1 |
| divergent series | متسلسلة متباعدة | A series whose partial sums have no finite limit. | 8.2 8.3 8.4 |
| dominant term | الحد المهيمن | The term of an expression that grows fastest as $n\to\infty$; keeping only dominant terms suggests the right comparison series. | 5.5 8.4 8.6 |
| eventually decreasing | متناقصة ابتداءً من حدٍّ ما | Decreasing from some point on, i.e. on an interval $[k,\infty)$ (for a sequence: $b_{n+1}\le b_n$ for all $n\ge N_0$); enough for the integral test and the alternating series test, since finitely many terms never affect convergence. | 8.1 8.3 8.4 8.5 |
| factorial | المضروب | $n!=1\cdot2\cdot3\cdots n$ (with $0!=1$). Useful identity: $(n+1)!=(n+1)\,n!$. | 8.1 8.6 9.3 |
| geometric series | متسلسلة هندسية | A series $\sum ar^{n-1}$ in which each term is $r$ times the previous one; it converges to $\frac{a}{1-r}$ exactly when $|r|\lt1$. | 8.2 8.4 8.5 8.6 |
| harmonic series | المتسلسلة التوافقية | The series $1+\frac12+\frac13+\cdots=\sum\frac1n$; its terms tend to $0$ but it diverges. | 8.2 8.3 8.4 8.5 |
| inconclusive | غير حاسم | Said of a test that gives no decision for a particular series — e.g. the ratio and root tests when $L=1$. Another test must be used. | 5.5 7.2 8.3 8.4 |
| infinite series | متسلسلة لانهائية | An infinite sum $a_1+a_2+a_3+\cdots=\sum a_n$, defined as the limit of its partial sums. | 8.1 8.2 8.5 |
| integral test | اختبار التكامل | If $f$ is positive, continuous and decreasing on $[k,\infty)$ with $a_n=f(n)$, then $\sum_{n\ge k}a_n$ and $\int_k^\infty f(x)\,dx$ both converge or both diverge. | 8.3 8.4 8.5 8.6 |
| limit comparison test | اختبار المقارنة بالنهاية | For positive terms with $a_n/b_n\to L$ and $0\lt L\lt\infty$, the series $\sum a_n$ and $\sum b_n$ both converge or both diverge. | 8.4 |
| Monotone Convergence Theorem | نظرية التقارب الرتيب | Every sequence that is both monotone and bounded converges. | 8.1 8.3 8.4 8.5 |
| monotone sequence | متتالية رتيبة | A sequence that is always increasing ($a_n\le a_{n+1}$) or always decreasing ($a_n\ge a_{n+1}$). | |
| p-series | المتسلسلة-p | A series $\sum 1/n^p$; it converges when $p\gt 1$ and diverges when $p\le1$. | |
| partial sum | مجموع جزئي | The sum of the first $N$ terms of a series, $S_N=a_1+\cdots+a_N$. | 8.1 8.2 8.3 8.4 |
| ratio test | اختبار النسبة | Let $L=\lim\left|\frac{a_{n+1}}{a_n}\right|$. If $L\lt1$ the series $\sum a_n$ converges absolutely; if $L\gt1$ it diverges; if $L=1$ the test gives no information. | 8.3 8.6 9.1 9.3 |
| rearrangement | إعادة الترتيب | A series that uses exactly the same terms as a given series, each exactly once, but in a different order. | 8.5 |
| recursive sequence | متتالية معرّفة تكرارياً (علاقة تكرارية) | A sequence given by starting value(s) and a rule that builds each term from earlier ones, e.g. $a_{n+1}=\sqrt{2+a_n}$. | 8.1 9.4 |
| remainder of a series | باقي المتسلسلة | $R_N=S-S_N=\sum_{n=N+1}^\infty a_n$: the error made when the partial sum $S_N$ is used in place of the sum $S$. | 8.3 8.4 8.5 |
| root test | اختبار الجذر | Let $L=\lim\sqrt[n]{|a_n|}$. If $L\lt1$ the series $\sum a_n$ converges absolutely; if $L\gt1$ it diverges; if $L=1$ the test gives no information. Best when $a_n$ is a whole $n$th power. | 8.5 8.6 9.1 |
| sequence | متتالية | An endless ordered list of numbers $a_1,a_2,a_3,\dots$; formally a function whose domain is the positive integers. | 8.1 8.2 8.3 8.4 |
| Stirling's formula | صيغة ستيرلنغ | The approximation $n!\approx\sqrt{2\pi n}\left(\frac ne\right)^n$ for large $n$; it shows how fast factorials grow compared with powers. | 8.6 |
| telescoping series | متسلسلة تلسكوبية (متداخلة) | A series whose terms are differences $b_n-b_{n+k}$, so that most terms of each partial sum cancel. | 8.2 8.3 |
Ch 9 · Power Series
| English | العربية | Meaning | Used in |
|---|---|---|---|
| Abel's theorem | مبرهنة آبل | If a power series converges at an endpoint of its interval, its sum there equals the one-sided limit of the function it represents — e.g. $\ln 2=1-\frac12+\frac13-\cdots$. | 9.2 |
| analytic function | دالة تحليلية | A function that equals its Taylor series on some open interval around the centre $a$. $e^x$, $\sin x$ and $\cos x$ are analytic everywhere; $e^{-1/x^2}$ is not analytic at $0$. | 9.3 |
| binomial coefficient | معامل ذي الحدين | For real $k$ and integer $n\ge0$: $\binom{k}{n}=\frac{k(k-1)\cdots(k-n+1)}{n!}$, with $\binom{k}{0}=1$. | 9.4 |
| binomial series | متسلسلة ذات الحدين | The expansion $(1+x)^k=\sum_{n=0}^{\infty}\binom{k}{n}x^n$, valid for $|x|\lt1$ and any real exponent $k$; it stops after finitely many terms when $k$ is a non-negative integer. | 8.6 9.4 |
| Cauchy product | ضرب كوشي | The product of two power series, $\left(\sum c_nx^n\right)\left(\sum d_nx^n\right)=\sum e_nx^n$ with $e_n=\sum_{k=0}^{n}c_kd_{n-k}$. | 9.2 |
| centre of a power series | مركز متسلسلة القوى | The number $a$ in $\sum c_n(x-a)^n$. Every power series converges at its centre, to $c_0$. | 5.2 9.1 9.2 9.3 |
| interval of convergence | فترة التقارب | The set of all $x$ at which a power series converges: an interval from $a-R$ to $a+R$, with each endpoint included or not after checking it separately. | 9.1 9.2 |
| Lagrange remainder | باقي لاغرانج | The error term $R_n(x)=f(x)-p_n(x)$ written as $\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$; if $|f^{(n+1)}|\le M$ then $|R_n(x)|\le\frac{M|x-a|^{n+1}}{(n+1)!}$. | 9.3 |
| Leibniz series for π | متسلسلة لايبنتز لحساب π | The series $\frac{\pi}{4}=1-\frac13+\frac15-\frac17+\cdots$, obtained from $\arctan x$ at $x=1$; it converges very slowly. | 8.6 9.2 |
| Maclaurin polynomial | كثيرة حدود ماكلورين | A Taylor polynomial centred at $a=0$. | 9.3 9.4 |
| Maclaurin series | متسلسلة ماكلورين | A Taylor series centred at $a=0$, e.g. $e^x=\sum x^n/n!$. | 9.3 9.4 |
| non-elementary integral | تكامل غير أولي | An integral such as $\int e^{-x^2}dx$ or $\int\frac{\sin x}{x}dx$ whose antiderivative exists but cannot be written with finitely many elementary functions; series give its values. | 9.4 |
| power series | متسلسلة القوى | A series of the form $\sum_{n=0}^{\infty} c_n(x-a)^n$ — an “infinite polynomial” in powers of $x-a$ whose sum depends on $x$. | 8.6 9.1 9.2 9.3 |
| power-series representation | تمثيل الدالة بمتسلسلة قوى | An equation $f(x)=\sum c_n(x-a)^n$ valid on an interval, for example $\frac{1}{1-x}=\sum x^n$ for $|x|\lt 1$. | 9.2 9.3 |
| power-series solution | حل بمتسلسلة القوى | A solution $y=\sum c_nx^n$ of a differential equation found by substituting the series and matching coefficients, which gives a recurrence relation for the $c_n$. | 9.4 |
| radius of convergence | نصف قطر التقارب | The number $R$ such that a power series converges absolutely for $|x-a|\lt R$ and diverges for $|x-a|\gt R$; $R=0$ and $R=\infty$ are possible. | 9.1 9.2 9.3 9.4 |
| recurrence relation | علاقة تكرارية | A rule that gives each term of a sequence from earlier terms, e.g. $c_{m+1}=\frac{2}{m+1}c_{m-1}$. | 9.4 |
| small-angle approximation | تقريب الزاوية الصغيرة | Replacing $\sin\theta$ by $\theta$ (and $\cos\theta$ by $1-\theta^2/2$) for small $\theta$ in radians, with error at most $|\theta|^3/6$ for the sine. | 9.4 |
| Taylor polynomial | كثيرة حدود تايلور | The polynomial $p_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k$ that has the same value and first $n$ derivatives as $f$ at $x=a$. | 9.3 |
| Taylor series | متسلسلة تايلور | The power series $\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$ built from all the derivatives of $f$ at $a$; its partial sums are the Taylor polynomials. | 8.5 9.2 9.3 9.4 |
| Taylor's theorem | مبرهنة تايلور | States that $f(x)=p_n(x)+R_n(x)$ with $R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$ for some $c$ between $a$ and $x$, when $f$ has $n+1$ derivatives on an open interval containing $a$ and $x$. | 9.3 |
| term-by-term differentiation | الاشتقاق حدًّا حدًّا | Differentiating a power series one term at a time: $\frac{d}{dx}\sum c_n(x-a)^n=\sum n c_n(x-a)^{n-1}$, valid inside the interval of convergence with the same radius. | 9.2 9.3 9.4 |
| term-by-term integration | التكامل حدًّا حدًّا | Integrating a power series one term at a time: $\int\sum c_n(x-a)^n dx=C+\sum c_n\frac{(x-a)^{n+1}}{n+1}$, with the same radius of convergence (endpoints may change). | 5.3 9.2 9.4 |
| uniqueness of power series | وحدانية تمثيل متسلسلة القوى | If two power series centred at $a$ agree on an open interval around $a$, they have the same coefficients; in fact $c_n=\frac{f^{(n)}(a)}{n!}$. | 9.2 9.3 |
Ch 10 · Parametric Equations and Polar Coordinates
| English | العربية | Meaning | Used in |
|---|---|---|---|
| cardioid | المنحنى القلبي | The heart-shaped limaçon $r=a(1\pm\cos\theta)$ or $r=a(1\pm\sin\theta)$, with a cusp at the pole. | 10.3 10.4 |
| circular sector | القطاع الدائري | The slice of a disk between two radii; with radius $r$ and angle $\Delta\theta$ (radians) its area is $\tfrac12r^2\Delta\theta$ — the building block of polar area. | 10.4 |
| cusp | نقطة ارتداد | A sharp point where a moving point comes in along a line, stops, and leaves back along the same line, so the curve has a spike; the cycloid has one wherever it touches the road. | 10.2 10.3 |
| cycloid | الدويري (السيكلويد) | The curve traced by a point on the rim of a circle rolling along a straight line: $x=r(\theta-\sin\theta)$, $y=r(1-\cos\theta)$. | 10.1 10.2 |
| eliminating the parameter | حذف الوسيط | Combining $x=f(t)$ and $y=g(t)$ into one equation in $x$ and $y$ only. The result describes the path but loses the orientation, the timing and any restriction on $x$ or $y$ unless these are stated. | 10.1 10.2 |
| horizontal tangent | مماس أفقي | A tangent line with slope $0$. On a parametric curve it occurs where $y^\prime(t)=0$ and $x^\prime(t)\neq0$. | 10.2 10.3 |
| inner loop | الحلقة الداخلية | The small loop of a limaçon $r=a+b\cos\theta$ with $a\lt b$, drawn while $r\lt0$. | 10.3 10.4 |
| lemniscate | المنحنى اللمنسكاتي (منحنى برنولي) | A figure-eight curve through the pole such as $r^2=a^2\cos2\theta$, defined only where the right-hand side is non-negative. | 10.3 10.4 |
| limaçon | منحنى الليماسون (حلزون باسكال) | A curve $r=a+b\cos\theta$ or $r=a+b\sin\theta$; it has an inner loop if $a\lt b$, is a cardioid if $a=b$, dimpled if $b\lt a\lt2b$ and convex if $a\ge2b$ ($a,b\gt0$). | 10.3 10.4 |
| Lissajous figure | منحنى ليساجو | A curve $x=A\sin(at)$, $y=B\sin(bt+\delta)$ produced by two perpendicular oscillations; it closes up when $a/b$ is rational. | 10.1 |
| logarithmic spiral | الحلزون اللوغاريتمي | The curve $r=ae^{b\theta}$; it crosses every ray from the pole at the same angle, and its arc length is $\dfrac{\sqrt{1+b^2}}{b}\left(r_{\text{end}}-r_{\text{start}}\right)$. | 10.3 10.4 |
| orientation | اتجاه المنحنى | The direction in which a parametric curve is traced as the parameter $t$ increases; shown by arrows on the curve. | 10.1 |
| parametrisation | التمثيل الوسيطي | A particular choice of parametric equations for a curve. One curve has many parametrisations, differing in direction, speed or the number of times it is traced. | 10.1 |
| petal | بتلة | One loop of a rose curve, swept out between two consecutive zeros of $r$; for $r=a\cos n\theta$ each petal has area $\tfrac{\pi a^2}{4n}$. | 10.3 10.4 |
| polar axis | المحور القطبي | The fixed ray from the pole from which polar angles are measured, usually drawn along the positive $x$-axis. | 10.3 10.4 |
| polar coordinates | الإحداثيات القطبية | A way of locating a point by its directed distance $r$ from a fixed point (the pole) and the angle $\theta$ from a fixed ray (the polar axis); $x=r\cos\theta$, $y=r\sin\theta$. | 10.3 10.4 |
| polar equation | المعادلة القطبية | An equation in $r$ and $\theta$, often $r=f(\theta)$; its graph is every point having at least one polar name that satisfies it. | 10.3 |
| polar region | منطقة قطبية | A region swept out by the radius arm of a polar curve $r=f(\theta)$ between two rays $\theta=\alpha$ and $\theta=\beta$; its area is $\tfrac12\int_\alpha^\beta r^2\,d\theta$. | |
| pole | القطب | The fixed reference point of the polar coordinate system (the origin); it has coordinates $(0,\theta)$ for every $\theta$. | 5.3 10.3 10.4 |
| projectile motion | حركة المقذوفات | Motion of an object under gravity alone: $x=(v_0\cos\alpha)t$, $y=h+(v_0\sin\alpha)t-\tfrac12gt^2$; unless it is launched straight up, its path is a parabola. | 10.1 10.2 |
| rose curve | منحنى الوردة | A curve $r=a\cos n\theta$ or $r=a\sin n\theta$; it has $n$ petals if $n$ is odd and $2n$ petals if $n$ is even. | 10.3 |
| singular point | نقطة شاذة | A point of a parametric curve where $x^\prime(t)=y^\prime(t)=0$: the moving point momentarily stops, and the curve may have a cusp there. | 10.2 |
| smooth curve | منحنى أملس | A parametric curve whose derivatives $x^\prime$, $y^\prime$ are continuous and never both zero, so it has a tangent direction at every point. | 6.4 7.2 10.2 |
| vertical tangent | مماس رأسي | A tangent line parallel to the $y$-axis (undefined slope). On a parametric curve it occurs where $x^\prime(t)=0$ and $y^\prime(t)\neq0$. | 6.4 10.2 10.3 |
Calculus I review
| English | العربية | Meaning | Used in |
|---|---|---|---|
| absolute maximum | القيمة العظمى المطلقة | The largest value of $f$ on the whole interval. | |
| absolute minimum | القيمة الصغرى المطلقة | The smallest value of $f$ on the whole interval. | |
| absolute value | القيمة المطلقة | The distance of a number from $0$; it is never negative. $|-3|=3$. | 5.1 5.2 5.3 5.4 |
| acceleration | التسارع | Rate of change of velocity, $a(t)=v'(t)=s''(t)$. | 7.1 8.5 10.1 |
| antiderivative | الدالة الأصلية (عكس المشتقة) | A function $F$ whose derivative is $f$: $F'(x)=f(x)$. | 5.1 5.2 5.3 5.4 |
| area | المساحة | The amount of flat space inside a region, in square units. | 5.1 5.2 5.3 5.4 |
| asymptote | خط تقارب | A line the graph gets closer and closer to but (usually) never reaches. | 5.3 5.5 |
| average rate of change | معدل التغيّر المتوسط | Total change $\div$ time (or $x$) elapsed — the slope of a secant line. | |
| average value | القيمة المتوسطة | The average value of $f$ on $[a,b]$: $f_{\text{ave}}=\dfrac{1}{b-a}\displaystyle\int_a^b f(x)\,dx$. | |
| chain rule | قاعدة السلسلة | Derivative of $f(g(x))$ is $f'(g(x))\cdot g'(x)$: outside derivative $\times$ inside derivative. | 7.3 7.4 10.2 |
| closed interval | فترة مغلقة | An interval that INCLUDES its end points, written $[a,b]$. | 9.1 |
| coefficient | معامل | The number multiplying a variable, e.g. $7$ in $7x^2$. | 5.3 5.4 6.1 7.3 |
| composite function | دالة مركّبة | A function inside another, $f(g(x))$: first apply $g$, then apply $f$. | |
| concave down | مقعّرة للأسفل | Curving downward like a cap $\cap$; $f''(x)<0$. | 5.4 7.4 10.2 |
| concave up | مقعّرة للأعلى | Curving upward like a cup $\cup$; $f''(x)>0$. | 5.4 7.2 7.4 10.2 |
| concavity | التقعّر | Which way the graph bends — up ($\cup$) or down ($\cap$). | 10.2 |
| conjugate | المرافق | For $\sqrt{a}-b$, the conjugate is $\sqrt{a}+b$; multiplying them removes the square root. | 8.1 |
| constant | ثابت | A fixed number that does not change, e.g. $5$, $\pi$ or $C$. | 5.1 5.2 5.3 5.4 |
| constant of integration | ثابت التكامل | The $+\,C$ added to an antiderivative, because constants differentiate to $0$. | |
| continuity | الاتصال | The property of being continuous — no holes, jumps or breaks. | 8.1 8.2 |
| continuous | متصلة | A function you can draw without lifting your pencil: $f(a)$ exists, the limit exists, and they are equal. | 5.3 5.4 5.5 6.1 |
| critical point | نقطة حرجة | An interior $x$ where $f'(x)=0$ or $f'(x)$ does not exist — a candidate for a max or min. | |
| decreasing | تناقصية | Going down from left to right; $f'(x)<0$. | 5.4 6.1 7.2 7.4 |
| definite integral | التكامل المحدد | $\displaystyle\int_a^b f(x)\,dx$ — a number: the signed area under $f$ from $a$ to $b$. | 5.1 5.2 5.3 5.4 |
| degree | درجة | For a polynomial: the highest power of $x$. For angles: a unit of measure ($360^\circ$ in a circle). | 5.2 5.3 5.4 9.3 |
| derivative | مشتقة | The instantaneous rate of change of a function — the slope of its tangent line at a point. | 5.1 5.2 5.3 6.4 |
| difference quotient | خارج قسمة الفروق | $\dfrac{f(a+h)-f(a)}{h}$ — the slope of a secant line. | |
| differentiable | قابلة للاشتقاق | Has a derivative there — the graph is smooth, with no corner, cusp or break. | 7.1 7.3 8.1 9.2 |
| differential equation | معادلة تفاضلية | An equation involving an unknown function and its derivatives, e.g. $\dfrac{dy}{dx}=6x^2$. | 7.1 7.2 7.3 7.4 |
| differentiation | الاشتقاق (التفاضل) | The process of finding a derivative. | 5.1 5.3 5.4 7.1 |
| discontinuity | عدم الاتصال | A point where the graph has a hole, jump or break. | 5.5 |
| domain | مجال الدالة | All the $x$-values you are allowed to put into a function. | 5.1 8.1 9.1 |
| equation | معادلة | A statement that two expressions are equal, e.g. $2x+1=7$. | 6.1 6.2 6.3 6.4 |
| exponential function | دالة أُسّية | A function with the variable in the power, e.g. $e^x$ or $2^x$. | |
| Extreme Value Theorem | مبرهنة القيمة القصوى | A continuous function on a closed interval $[a,b]$ always has an absolute max and min. | |
| extremum | قيمة قصوى | A maximum or a minimum. | |
| factor | التحليل إلى عوامل | Writing an expression as a product, e.g. $x^2-4=(x-2)(x+2)$. | 9.1 |
| first derivative test | اختبار المشتقة الأولى | At a critical point: $f'$ changes $+\to-$ gives a max; $-\to+$ gives a min. | |
| function | دالة | A rule that gives exactly one output $y$ for each input $x$. Think of a machine: put $x$ in, get $f(x)$ out. | 5.1 5.2 5.3 5.4 |
| Fundamental Theorem of Calculus | المبرهنة الأساسية في التفاضل والتكامل | Differentiation and integration undo each other; $\displaystyle\int_a^b f\,dx=F(b)-F(a)$. | 5.4 6.4 9.4 |
| graph | التمثيل البياني | The picture of all points $(x, f(x))$ of a function drawn on axes. | 5.1 5.2 5.4 5.5 |
| horizontal asymptote | خط تقارب أفقي | A horizontal line $y=L$ that the graph approaches as $x\to\pm\infty$. | |
| identity | متطابقة | An equation true for every allowed value, e.g. $\sin^2 x+\cos^2 x=1$. | 5.1 5.2 6.4 7.1 |
| implicit differentiation | الاشتقاق الضمني | Differentiating an equation in $x$ and $y$ without solving for $y$ first; every $y$-term gets a $\dfrac{dy}{dx}$. | 7.3 10.3 |
| increasing | تزايدية | Going up from left to right; $f'(x)>0$. | 5.4 6.1 6.4 7.2 |
| indefinite integral | التكامل غير المحدد | $\displaystyle\int f(x)\,dx=F(x)+C$ — the whole family of antiderivatives. | |
| indeterminate form | صيغة غير معيّنة | An expression like $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$ that tells you nothing yet — more work is needed. | |
| infinity | ما لا نهاية | Not a number but the idea of growing without bound ($\infty$). | 5.1 5.2 5.3 5.5 |
| inflection point | نقطة انعطاف | A point where the graph changes from concave up to concave down, or vice versa. | 7.4 |
| initial condition | الشرط الابتدائي | A known value such as $y(x_0)=y_0$ used to find $C$. | 7.1 7.3 9.4 |
| initial-value problem | مسألة القيمة الابتدائية | A differential equation plus a starting condition like $y(0)=5$ that fixes $C$. | 7.1 7.2 7.3 7.4 |
| instantaneous | لحظي | At one exact moment, not averaged over an interval. | 7.1 9.1 |
| integral | تكامل | Either a family of antiderivatives (indefinite) or a signed area (definite). | 5.1 5.2 5.3 5.4 |
| integrand | الدالة المكامَلة | The function being integrated — the $f(x)$ inside $\displaystyle\int \ldots\,dx$. | 5.1 5.2 5.3 5.5 |
| integration | التكامل | The process of finding an integral — the reverse of differentiation. | 5.1 5.2 5.3 5.4 |
| integration by parts | التكامل بالتجزئة | $\displaystyle\int u\,dv=uv-\int v\,du$ — the product rule run backwards. | 5.1 5.2 5.3 |
| interval | فترة | All the numbers between two end points, e.g. $[1,3]$ or $(0,\infty)$. | 5.1 5.2 5.3 5.4 |
| inverse function | دالة عكسية | A function that undoes another: if $f(2)=5$ then $f^{-1}(5)=2$. | |
| inverse trigonometric | مثلثية عكسية | $\sin^{-1}, \cos^{-1}, \tan^{-1}, \ldots$ — they return the angle whose sine/cosine/tangent is given. | 9.2 |
| L'Hôpital's rule | قاعدة لوبيتال | For $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$ limits: $\displaystyle\lim\frac{f}{g}=\lim\frac{f'}{g'}$. | 5.5 8.1 8.5 10.2 |
| Leibniz notation | رمز لايبنتز | Writing derivatives as $\dfrac{dy}{dx}$. | |
| limit | نهاية | The value $f(x)$ gets closer and closer to as $x$ approaches a number (it need not equal $f(a)$). | 5.2 5.3 5.5 6.1 |
| limit laws | قوانين النهايات | Rules for limits of sums, products, quotients and powers — the limit of a sum is the sum of the limits, etc. | 8.1 8.2 |
| linear approximation | التقريب الخطي | Estimating $f(x)$ near $a$ by the tangent line: $f(x)\approx f(a)+f'(a)(x-a)$. | 9.3 |
| linear function | دالة خطية | A function whose graph is a straight line: $f(x)=mx+b$. | |
| local maximum | القيمة العظمى المحلية | A hill-top: $f(c)$ is bigger than all nearby values. | |
| local minimum | القيمة الصغرى المحلية | A valley-bottom: $f(c)$ is smaller than all nearby values. | |
| logarithm | لوغاريتم | The inverse of an exponential: $\log_b y = x$ means $b^x = y$. | 5.3 8.3 8.4 8.6 |
| logarithmic differentiation | الاشتقاق اللوغاريتمي | Take $\ln$ of both sides first, then differentiate — handy for $x^x$-type functions. | |
| Mean Value Theorem | مبرهنة القيمة المتوسطة | If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then somewhere in $(a,b)$ the tangent slope equals the average slope $\dfrac{f(b)-f(a)}{b-a}$. | 6.4 7.1 9.3 10.2 |
| natural logarithm | اللوغاريتم الطبيعي | The logarithm with base $e\approx 2.718$, written $\ln x$. | 5.3 |
| normal line | الخط العمودي | The line through a point of a curve perpendicular to the tangent line there; slope $-\dfrac{1}{f'(a)}$. | |
| one-sided limit | نهاية من جهة واحدة | A limit taken from only the left ($x\to a^-$) or only the right ($x\to a^+$). | |
| open interval | فترة مفتوحة | An interval that does NOT include its end points, written $(a,b)$. | 7.2 7.3 9.1 9.2 |
| optimization | الأمثلية (التحسين) | Finding the largest or smallest possible value of a quantity (area, cost, time …). | |
| parabola | قطع مكافئ | The U-shaped graph of a quadratic function. | 5.4 6.1 6.2 6.4 |
| parameter | وسيط (بارامتر) | An extra variable (often $t$) that both $x$ and $y$ depend on. | 7.2 10.1 10.2 10.3 |
| parametric equations | معادلات وسيطية (بارامترية) | $x$ and $y$ are both given as functions of a third variable $t$. | 10.1 10.2 10.3 10.4 |
| piecewise | دالة متعددة القواعد | A function given by different formulas on different parts of its domain. | |
| polynomial | كثيرة الحدود | A sum of terms like $3x^2-5x+1$ using only whole-number powers of $x$. | 5.3 5.4 7.4 9.1 |
| position | الموقع | Where an object is at time $t$, written $s(t)$. | 6.2 6.3 7.1 8.5 |
| power rule | قاعدة القوة | $\dfrac{d}{dx}\left(x^n\right)=n\,x^{n-1}$ — bring the power down, reduce it by one. | 5.1 |
| product rule | قاعدة الضرب | $(fg)'=f'g+fg'$. | 7.1 8.2 9.3 10.3 |
| quadratic | تربيعية | Of degree 2, e.g. $x^2-3x+2$; its graph is a parabola. | 5.2 5.3 5.4 6.4 |
| quotient rule | قاعدة القسمة | $\left(\dfrac{f}{g}\right)'=\dfrac{f'g-fg'}{g^2}$. | 8.3 |
| radian | راديان | Angle unit used in calculus: $2\pi$ radians $=360^\circ$. | 5.1 5.2 10.1 10.3 |
| range | مدى الدالة | All the $y$-values (outputs) a function actually produces. | 7.2 9.1 9.3 10.1 |
| rate of change | معدل التغيّر | How fast one quantity changes compared with another. | 7.1 7.3 10.2 |
| rational function | دالة نسبية | A fraction whose top and bottom are polynomials, e.g. $\dfrac{x+1}{x-5}$. | 5.3 8.4 8.6 |
| related rates | المعدلات المرتبطة | Problems where two changing quantities are linked, so their rates are linked too. | |
| Rolle's theorem | مبرهنة رول | If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$ and $f(a)=f(b)$, then somewhere between $a$ and $b$ the tangent is horizontal ($f'(c)=0$). | 9.3 |
| root | جذر / صفر | A value of $x$ that makes $f(x)=0$ — where the graph meets the $x$-axis. | 5.2 6.3 7.3 7.4 |
| secant line | القاطع | A line through two points of a curve. | 5.1 5.2 |
| second derivative | المشتقة الثانية | The derivative of the derivative, $f''(x)$ — tells how the slope itself changes. | 9.2 9.3 10.2 |
| second derivative test | اختبار المشتقة الثانية | At a critical point with $f'=0$: $f''>0$ gives a min, $f''<0$ gives a max. | |
| slope | الميل | How steep a line is: $\dfrac{\text{rise}}{\text{run}}$. | 5.2 5.4 6.4 7.1 |
| speed | السرعة (المقدار) | The size of velocity, $|v(t)|$ — never negative. | 5.4 5.5 6.5 7.1 |
| squeeze theorem | مبرهنة الشطيرة (الحصر) | If $g\le f\le h$ and $g$, $h$ have the same limit, then $f$ has that limit too. | 8.1 |
| substitution | التكامل بالتعويض | Replace an inner expression by $u$ (and its derivative by $du$) to simplify an integral. | 5.1 5.2 5.3 6.3 |
| tangent line | المماس (خط مماس) | The line that just touches the curve at a point and has the same direction there. | 5.1 5.2 5.3 5.4 |
| theorem | مبرهنة (نظرية) | A mathematical statement that has been proved to be always true. | 5.1 5.2 5.3 5.4 |
| trigonometric function | دالة مثلثية | $\sin, \cos, \tan, \cot, \sec, \csc$ — functions built from angles in a right triangle / unit circle. | 5.1 5.2 9.2 10.1 |
| unit circle | دائرة الوحدة | The circle of radius $1$ centred at the origin; the point at angle $\theta$ is $(\cos\theta, \sin\theta)$. | 7.2 10.1 10.2 |
| variable | متغيّر | A letter such as $x$ or $t$ that stands for a quantity that can change. | 5.2 5.3 6.2 6.3 |
| velocity | السرعة المتجهة | Rate of change of position, $s'(t)$; its sign shows the direction of motion. | 5.5 6.1 7.1 7.2 |
| vertical asymptote | خط تقارب رأسي | A vertical line $x=a$ where the graph shoots up or down to $\pm\infty$. | 5.3 5.5 |