∫ calculus

1. **Énoncé du problème :** Calculer le volume du solide obtenu par la révolution de la région délimitée par $y = x^2$ et $y = x + 6$ autour de l'axe $y=0$. 2. **Détermination des

1. **State the problem:** Find the limit $$\lim_{x \to \pi} \frac{x + \pi \sec x}{x^2 - \pi^2}.$$\n\n2. **Recall the formula and rules:** The limit is of the form $$\frac{f(x)}{g(x

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{6e^x - 2x^2 - 4}{\sin(2x^2)}.$$\n\n2. **Recall important formulas and rules:**\n- The Taylor expansion of $e^x$ near

1. **State the problem:** We have the function $g(x) = \sqrt{1 - \sin^2 x}$ on the interval $0 \leq x \leq \pi$. We want to determine which of the given statements (A, B, C, D) cou

1. **State the problem:** We are given values of a function $f$ at points $x=0,1,2,3$ and asked which statements about the Intermediate Value Theorem (IVT), Mean Value Theorem (MVT

1. **State the problem:** Find the points of inflection of the function $$f(x) = - \frac{x^3 - 2x^2 + x - 1}{x - 2}, \quad x \neq 2.$$ 2. **Recall the definition:** Points of infle

1. **Énoncé du problème** : Calculer le volume du solide obtenu par la révolution de la région délimitée par $y=\sqrt{x}$, $y=0$ et $x=9$ autour de l'axe de rotation $y=-1$. 2. **F

1. **Problem:** Find the derivative functions $f'$ for the given functions. 2. **Recall the derivative rules:**

1. **State the problem:** We want to write the Taylor series of the function $f(x) = e^{2x}$ about the point $x_0 = \frac{1}{2}$ using sigma notation. 2. **Recall the Taylor series

1. **State the problem:** We are given parametric equations $x = 5t$ and $y = \sqrt{t}$, and we want to find the second derivative $\frac{d^2y}{dx^2}$ at $t = \frac{1}{9}$. 2. **Re

1. **Problem Statement:** Find the derivative $\frac{dy}{dx}$ of the function $y = x^{n-1}$, where $n$ is a natural number. 2. **Formula Used:** The power rule for differentiation

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ for the function $y = x - 4 - \frac{1}{x}$. 2. **Recall the derivative rules:**

1. The problem is to determine if the expression is an indeterminate form. 2. Indeterminate forms occur in limits when substitution leads to expressions like $\frac{0}{0}$, $\frac{

1. Problemet handler om at differentiere hvert led i et polynomium. 2. Reglen for differentiation af et led $ax^n$ er: $$\frac{d}{dx}(ax^n) = a \cdot n \cdot x^{n-1}$$ hvor $a$ er

1. The problem involves integrating with respect to $z$ from $0$ to $\frac{L}{2}$, not $t$ or $1$. 2. We need to set up the integral correctly with the limits $0$ to $\frac{L}{2}$.

1. **Problem statement:** Find the first three derivatives of the function $$f(x) = \frac{1}{5} (e^x - e^{-x})$$. 2. **Recall the derivative rules:**

1. Задача: Найти интеграл \( \int \frac{\sin 2x}{\cos x} \, dx \). 2. Используем формулу двойного угла для синуса: \( \sin 2x = 2 \sin x \cos x \).

1. المشكلة: إيجاد مشتقة الدالة $$f(x) = 1 - \frac{x}{2}$$. 2. القاعدة المستخدمة: مشتقة الفرق هي فرق المشتقات، ومشتقة الثابت صفر، ومشتقة الدالة $$\frac{x}{2}$$ هي مشتقة $$x$$ مقسومة

1. المشكلة: تصحيح المشتقة الثانية. 2. نبدأ بتحديد الدالة الأصلية التي نريد إيجاد مشتقتها الثانية.

1. **State the problem:** We need to find the differential $df$ of the function $f(x,y) = \sqrt{x^3 + y^2}$ at the point $(2,3)$.

1. نبدأ بحل السؤال (2) حول نقاط الانعطاف للدالة $f(x) = x + \frac{1}{x}$. 2. نقاط الانعطاف تحدث عندما يتغير تقعر الدالة، أي عندما تكون مشتقة التانية تساوي صفر أو غير معرفة مع تغير