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 used 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 most.
Basics
| English | العربية | Meaning | Used in |
|---|---|---|---|
| absolute value | القيمة المطلقة | The distance of a number from $0$; it is never negative. $|-3|=3$. | 0.1 3.4 |
| closed interval | فترة مغلقة | An interval that INCLUDES its end points, written $[a,b]$. | 1.3 3.5 3.6 |
| coefficient | معامل | The number multiplying a variable, e.g. $7$ in $7x^2$. | 1.4 2.2 |
| composite function | دالة مركّبة | A function inside another, $f(g(x))$: first apply $g$, then apply $f$. | 2.3 0.1 |
| conjugate | المرافق | For $\sqrt{a}-b$, the conjugate is $\sqrt{a}+b$; multiplying them removes the square root. | 1.2 2.1 0.1 |
| constant | ثابت | A fixed number that does not change, e.g. $5$, $\pi$ or $C$. | 2.2 1.2 3.3 |
| degree | درجة | For a polynomial: the highest power of $x$. For angles: a unit of measure ($360^\circ$ in a circle). | 1.4 0.1 2.1 |
| domain | مجال الدالة | All the $x$-values you are allowed to put into a function. | 0.1 1.3 3.4 |
| equation | معادلة | A statement that two expressions are equal, e.g. $2x+1=7$. | 2.4 2.2 2.5 |
| exponential function | دالة أُسّية | A function with the variable in the power, e.g. $e^x$ or $2^x$. | 2.2 4.3 |
| factor | التحليل إلى عوامل | Writing an expression as a product, e.g. $x^2-4=(x-2)(x+2)$. | 0.1 1.2 1.3 |
| 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. | 2.2 1.3 2.1 |
| graph | التمثيل البياني | The picture of all points $(x, f(x))$ of a function drawn on axes. | 1.4 1.1 2.2 |
| identity | متطابقة | An equation true for every allowed value, e.g. $\sin^2 x+\cos^2 x=1$. | 0.1 |
| interval | فترة | All the numbers between two end points, e.g. $[1,3]$ or $(0,\infty)$. | 1.3 3.4 3.5 |
| inverse function | دالة عكسية | A function that undoes another: if $f(2)=5$ then $f^{-1}(5)=2$. | 0.1 |
| inverse trigonometric | مثلثية عكسية | $\sin^{-1}, \cos^{-1}, \tan^{-1}, \ldots$ — they return the angle whose sine/cosine/tangent is given. | 2.2 0.1 |
| linear function | دالة خطية | A function whose graph is a straight line: $f(x)=mx+b$. | |
| logarithm | لوغاريتم | The inverse of an exponential: $\log_b y = x$ means $b^x = y$. | 0.1 2.4 2.3 |
| natural logarithm | اللوغاريتم الطبيعي | The logarithm with base $e\approx 2.718$, written $\ln x$. | 2.4 3.7 |
| open interval | فترة مفتوحة | An interval that does NOT include its end points, written $(a,b)$. | 1.1 2.1 3.4 |
| parabola | قطع مكافئ | The U-shaped graph of a quadratic function. | 1.1 2.1 0.1 |
| parameter | وسيط (بارامتر) | An extra variable (often $t$) that both $x$ and $y$ depend on. | 2.5 |
| piecewise | دالة متعددة القواعد | A function given by different formulas on different parts of its domain. | 1.2 1.3 |
| polynomial | كثيرة الحدود | A sum of terms like $3x^2-5x+1$ using only whole-number powers of $x$. | 1.2 1.3 3.5 |
| quadratic | تربيعية | Of degree 2, e.g. $x^2-3x+2$; its graph is a parabola. | 0.1 2.1 3.5 |
| radian | راديان | Angle unit used in calculus: $2\pi$ radians $=360^\circ$. | 0.1 1.1 |
| range | مدى الدالة | All the $y$-values (outputs) a function actually produces. | 0.1 3.6 4.3 |
| rational function | دالة نسبية | A fraction whose top and bottom are polynomials, e.g. $\dfrac{x+1}{x-5}$. | 1.3 1.2 1.4 |
| root | جذر / صفر | A value of $x$ that makes $f(x)=0$ — where the graph meets the $x$-axis. | 1.4 2.1 |
| theorem | مبرهنة (نظرية) | A mathematical statement that has been proved to be always true. | 3.5 2.2 3.4 |
| trigonometric function | دالة مثلثية | $\sin, \cos, \tan, \cot, \sec, \csc$ — functions built from angles in a right triangle / unit circle. | 2.2 1.3 2.3 |
| unit circle | دائرة الوحدة | The circle of radius $1$ centred at the origin; the point at angle $\theta$ is $(\cos\theta, \sin\theta)$. | 0.1 |
| variable | متغيّر | A letter such as $x$ or $t$ that stands for a quantity that can change. | 3.6 3.3 4.3 |
Limits
| English | العربية | Meaning | Used in |
|---|---|---|---|
| asymptote | خط تقارب | A line the graph gets closer and closer to but (usually) never reaches. | 1.4 1.1 1.2 |
| continuity | الاتصال | The property of being continuous — no holes, jumps or breaks. | 1.3 2.1 1.2 |
| continuous | متصلة | A function you can draw without lifting your pencil: $f(a)$ exists, the limit exists, and they are equal. | 1.3 3.5 2.1 |
| discontinuity | عدم الاتصال | A point where the graph has a hole, jump or break. | 1.3 3.4 |
| horizontal asymptote | خط تقارب أفقي | A horizontal line $y=L$ that the graph approaches as $x\to\pm\infty$. | 1.4 |
| indeterminate form | صيغة غير معيّنة | An expression like $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$ that tells you nothing yet — more work is needed. | 1.2 3.7 |
| infinity | ما لا نهاية | Not a number but the idea of growing without bound ($\infty$). | 1.4 1.3 0.1 |
| limit | نهاية | The value $f(x)$ gets closer and closer to as $x$ approaches a number (it need not equal $f(a)$). | 1.2 1.1 1.3 |
| 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. | 1.2 1.3 1.4 |
| one-sided limit | نهاية من جهة واحدة | A limit taken from only the left ($x\to a^-$) or only the right ($x\to a^+$). | 1.1 1.2 1.3 |
| squeeze theorem | مبرهنة الشطيرة (الحصر) | If $g\le f\le h$ and $g$, $h$ have the same limit, then $f$ has that limit too. | 1.4 |
| vertical asymptote | خط تقارب رأسي | A vertical line $x=a$ where the graph shoots up or down to $\pm\infty$. | 1.4 1.1 1.2 |
Derivatives
| English | العربية | Meaning | Used in |
|---|---|---|---|
| acceleration | التسارع | Rate of change of velocity, $a(t)=v'(t)=s''(t)$. | 3.2 4.1 2.2 |
| average rate of change | معدل التغيّر المتوسط | Total change $\div$ time (or $x$) elapsed — the slope of a secant line. | 2.1 3.2 3.5 |
| chain rule | قاعدة السلسلة | Derivative of $f(g(x))$ is $f'(g(x))\cdot g'(x)$: outside derivative $\times$ inside derivative. | 2.3 2.4 4.3 |
| derivative | مشتقة | The instantaneous rate of change of a function — the slope of its tangent line at a point. | 2.2 2.1 2.3 |
| difference quotient | خارج قسمة الفروق | $\dfrac{f(a+h)-f(a)}{h}$ — the slope of a secant line. | 2.1 1.2 2.2 |
| differentiable | قابلة للاشتقاق | Has a derivative there — the graph is smooth, with no corner, cusp or break. | 2.1 3.5 2.2 |
| differentiation | الاشتقاق (التفاضل) | The process of finding a derivative. | 2.4 2.2 3.3 |
| implicit differentiation | الاشتقاق الضمني | Differentiating an equation in $x$ and $y$ without solving for $y$ first; every $y$-term gets a $\dfrac{dy}{dx}$. | 2.4 2.3 2.5 |
| instantaneous | لحظي | At one exact moment, not averaged over an interval. | 2.1 3.2 3.5 |
| Leibniz notation | رمز لايبنتز | Writing derivatives as $\dfrac{dy}{dx}$. | 2.1 2.3 |
| logarithmic differentiation | الاشتقاق اللوغاريتمي | Take $\ln$ of both sides first, then differentiate — handy for $x^x$-type functions. | 2.4 |
| parametric equations | معادلات وسيطية (بارامترية) | $x$ and $y$ are both given as functions of a third variable $t$. | 2.5 2.4 3.3 |
| position | الموقع | Where an object is at time $t$, written $s(t)$. | 3.2 4.1 2.1 |
| power rule | قاعدة القوة | $\dfrac{d}{dx}\left(x^n\right)=n\,x^{n-1}$ — bring the power down, reduce it by one. | 2.2 2.3 4.1 |
| product rule | قاعدة الضرب | $(fg)'=f'g+fg'$. | 2.2 2.4 4.4 |
| quotient rule | قاعدة القسمة | $\left(\dfrac{f}{g}\right)'=\dfrac{f'g-fg'}{g^2}$. | 2.2 2.4 2.3 |
| rate of change | معدل التغيّر | How fast one quantity changes compared with another. | 2.1 3.3 3.2 |
| secant line | القاطع | A line through two points of a curve. | 2.1 3.5 2.2 |
| second derivative | المشتقة الثانية | The derivative of the derivative, $f''(x)$ — tells how the slope itself changes. | 2.2 3.4 2.5 |
| slope | الميل | How steep a line is: $\dfrac{\text{rise}}{\text{run}}$. | 2.1 2.2 2.4 |
| speed | السرعة (المقدار) | The size of velocity, $|v(t)|$ — never negative. | 3.2 2.5 3.3 |
| tangent line | المماس (خط مماس) | The line that just touches the curve at a point and has the same direction there. | 3.1 2.1 |
| velocity | السرعة المتجهة | Rate of change of position, $s'(t)$; its sign shows the direction of motion. | 3.2 2.1 2.2 |
Applications
| English | العربية | Meaning | Used in |
|---|---|---|---|
| absolute maximum | القيمة العظمى المطلقة | The largest value of $f$ on the whole interval. | 3.4 |
| absolute minimum | القيمة الصغرى المطلقة | The smallest value of $f$ on the whole interval. | 3.4 |
| concave down | مقعّرة للأسفل | Curving downward like a cap $\cap$; $f''(x)<0$. | 3.4 2.1 |
| concave up | مقعّرة للأعلى | Curving upward like a cup $\cup$; $f''(x)>0$. | 3.4 |
| concavity | التقعّر | Which way the graph bends — up ($\cup$) or down ($\cap$). | 3.4 |
| critical point | نقطة حرجة | An interior $x$ where $f'(x)=0$ or $f'(x)$ does not exist — a candidate for a max or min. | 3.4 3.6 |
| decreasing | تناقصية | Going down from left to right; $f'(x)<0$. | 3.4 2.1 3.2 |
| Extreme Value Theorem | مبرهنة القيمة القصوى | A continuous function on a closed interval $[a,b]$ always has an absolute max and min. | 3.4 3.6 |
| extremum | قيمة قصوى | A maximum or a minimum. | 3.4 3.6 |
| first derivative test | اختبار المشتقة الأولى | At a critical point: $f'$ changes $+\to-$ gives a max; $-\to+$ gives a min. | 3.4 |
| increasing | تزايدية | Going up from left to right; $f'(x)>0$. | 3.4 3.3 3.5 |
| inflection point | نقطة انعطاف | A point where the graph changes from concave up to concave down, or vice versa. | 3.4 |
| L'Hôpital's rule | قاعدة لوبيتال | For $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$ limits: $\displaystyle\lim\frac{f}{g}=\lim\frac{f'}{g'}$. | 3.7 3.6 4.1 |
| linear approximation | التقريب الخطي | Using the tangent line $L(x)=f(a)+f'(a)(x-a)$ to estimate $f(x)$ for $x$ near $a$. | 3.1 |
| local maximum | القيمة العظمى المحلية | A hill-top: $f(c)$ is bigger than all nearby values. | 3.4 |
| local minimum | القيمة الصغرى المحلية | A valley-bottom: $f(c)$ is smaller than all nearby values. | 3.4 |
| 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 $\dfrac{f(b)-f(a)}{b-a}$. | 3.5 3.4 3.6 |
| normal line | الخط العمودي (العمودي على المماس) | The line through a point of a curve perpendicular to the tangent line there; its slope is $-1/f'(a)$. | 3.1 |
| optimization | الأمثلية (التحسين) | Finding the largest or smallest possible value of a quantity (area, cost, time …). | 3.6 3.4 3.5 |
| related rates | المعدلات المرتبطة | Problems where two changing quantities are linked, so their rates are linked too. | 3.3 2.5 3.4 |
| 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 ($f'(c)=0$). | 3.5 |
| second derivative test | اختبار المشتقة الثانية | At a critical point with $f'=0$: $f''>0$ gives a min, $f''<0$ gives a max. | 3.4 |
Integration
| English | العربية | Meaning | Used in |
|---|---|---|---|
| antiderivative | الدالة الأصلية (عكس المشتقة) | A function $F$ whose derivative is $f$: $F'(x)=f(x)$. | 4.1 4.2 4.3 |
| area | المساحة | The amount of flat space inside a region, in square units. | 3.6 4.2 4.3 |
| average value of a function | القيمة المتوسطة للدالة | $f_{\text{ave}}=\dfrac{1}{b-a}\displaystyle\int_a^b f(x)\,dx$ — the height of the rectangle on $[a,b]$ with the same area as the region under $f$. | 4.2 |
| constant of integration | ثابت التكامل | The $+\,C$ added to an antiderivative, because constants differentiate to $0$. | |
| definite integral | التكامل المحدد | $\displaystyle\int_a^b f(x)\,dx$ — a number: the signed area under $f$ from $a$ to $b$. | 4.2 4.3 4.1 |
| differential equation | معادلة تفاضلية | An equation involving an unknown function and its derivatives, e.g. $\dfrac{dy}{dx}=6x^2$. | 4.1 |
| Fundamental Theorem of Calculus | المبرهنة الأساسية في التفاضل والتكامل | Differentiation and integration undo each other; $\displaystyle\int_a^b f\,dx=F(b)-F(a)$. | 4.2 |
| indefinite integral | التكامل غير المحدد | $\displaystyle\int f(x)\,dx=F(x)+C$ — the whole family of antiderivatives. | 4.1 4.2 4.3 |
| initial condition | الشرط الابتدائي | A known value such as $y(x_0)=y_0$ used to find $C$. | 4.1 |
| initial-value problem | مسألة القيمة الابتدائية | A differential equation plus a starting condition like $y(0)=5$ that fixes $C$. | 4.1 |
| integral | تكامل | Either a family of antiderivatives (indefinite) or a signed area (definite). | 4.1 4.2 4.4 |
| integrand | الدالة المكامَلة | The function being integrated — the $f(x)$ inside $\displaystyle\int \ldots\,dx$. | 4.3 4.4 4.1 |
| integration | التكامل | The process of finding an integral — the reverse of differentiation. | 4.4 4.2 4.3 |
| integration by parts | التكامل بالتجزئة | $\displaystyle\int u\,dv=uv-\int v\,du$ — the product rule run backwards. | 4.4 4.3 |
| substitution | التكامل بالتعويض | Replace an inner expression by $u$ (and its derivative by $du$) to simplify an integral. | 4.3 1.2 4.4 |