∫ calculus

1. **Stating the problem:** We want to create an integration system in Desmos that fits a polygon under any function, such as $f(x)=mx+b$ or $f(x)=x^2$, to approximate the area und

1. **State the problem:** Evaluate the expression $$\int_0^3 1 \, dx + \lim_{n \to \infty} \left(3 - \frac{1}{n}\right) - 3 \over \cos(0)$$. 2. **Evaluate the integral:** The integ

1. Тодорхойл: Функцийн өсөх, буурах завсарыг олох нь функцийн графикийн ямар хэсэгт функц өсч, ямар хэсэгт буурч байгааг тодорхойлох явдал юм. 2. Функцийн өсөх, буурах завсарыг оло

1. The problem is to simplify the integrand $$\frac{\csc^2 x \sin x - \sin x}{\cos x}$$ and understand why it becomes $$\frac{1 - \sin^2 x}{\sin x \cos x}$$. 2. Recall that $$\csc

1. Problem: Find the derivative of $f(x) = \frac{3}{4}$ using the constant rule. The constant rule states that the derivative of a constant is zero.

1. نبدأ بكتابة المسألة: حساب التكامل من الفترة $-1$ إلى $3$ للدالة $2-5x$ باستخدام التعريف العام للتكامل. 2. التعريف العام للتكامل هو حساب نهاية مجموع ريمان:

1. **سؤال 40: ميل المماس لمنحنى الدالة ص = ٢^س عند النقطة (١، ٠٣)** نريد حساب ميل المماس لمنحنى الدالة عند النقطة المعطاة.

1. **State the problem:** We are given the differential equation $$\frac{dy}{dx} = \frac{1}{2y + 1}$$ with the initial condition $$y(0) = 0$$. We need to find the particular soluti

1. **Problem statement:** We have the differential equation $\frac{dy}{dt} = ky$, where $k$ is a constant and $t$ is time in years. The population doubles every 10 years. We need t

1. **State the problem:** A puppy's weight $W(t)$ grows proportionally to its current weight over time $t$ (in months). Given $W(0)=2.0$ pounds and $W(2)=3.5$ pounds, find $W(3)$.

1. **State the problem:** Find the radius and interval of convergence for the series $$\sum_{n=1}^\infty \frac{2n! (4x - 5)^n}{17 n^3}$$ with center at $$x = \frac{5}{4}$$.

1. **Énoncé du problème :** Calculer le volume du solide dont la base est la région elliptique définie par $$9x^2 + 4y^2 = 36$$ et dont les sections transversales perpendiculaires

1. **Énoncé du problème** : Trouver le volume du solide obtenu en faisant tourner la région délimitée par $y=\sqrt{x}$, $y=0$, et $x=4$ autour de la droite $x=6$ en utilisant la mé

1. Énoncé du problème : Trouver une expression intégrale pour le volume du solide obtenu en faisant tourner la région délimitée par $y=\sqrt{x}$, $y=0$, et $x=4$ autour de la droit

1. Énoncé du problème : Trouver le rayon d'un disque pour calculer le volume d'un solide de révolution en utilisant la méthode des disques en Calcul 2. 2. Formule utilisée : Le vol

1. **State the problem:** Find the volume of the solid formed by rotating the region bounded by the curves $x = -8 + y^2$ and $x = -2y$ about the line $x = -9$. 2. **Identify the m

1. **State the problem:** Find the equations of the tangent lines to the curve given by $$x^4y^2 + x^2y^4 = 26x^6 - 6$$ at $$x=1$$. 2. **Implicit differentiation:** Differentiate b

1. **State the problem:** Find the limit $$\lim_{h \to 0} \frac{1}{h} \left( \frac{1}{\sqrt{1+h}} - 1 \right)$$ and use it to find $$f'(1)$$ for $$f(x) = \frac{1}{\sqrt{x}}$$. Also

1. Problem: Find the derivative $f'(x)$ for each given function without simplifying. 2. (a) Given $f(x) = 5x^{12} - \frac{3}{7}x^7 + x^6 - 10e^x + 5\sqrt[4]{x}$.

1. **Problem statement:** Find the derivative $f'(x)$ using the limit definition of the derivative: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

1. **Problem statement:** (a) Find the slope of the secant line through points (1,2) and (2,17) for the function $f(x) = 4x^2 + 3x - 5$.