∫ calculus

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 $f(x) = 4x^2 + 3x - 5$.

1. **State the problem:** Find the values of $a$ and $b$ that make the piecewise function $$

1. **Problem statement:** Find the function $f(x)$ given that the slope of the tangent line to $f$ at $x=c$ is given by the limit $$\lim_{h \to 0} \frac{\tan\left(\frac{\pi}{4} + h

1. **Problem statement:** Find the derivative of the velocity function $$v(t) = (t - 5)(t - 2)^2$$ for time $$t \geq 0$$. 2. **Formula used:** To differentiate a product of two fun

1. **State the problem:** We have the position function of an object given by $$s(t) = 6 + 8t - t^2$$ for $$0 \leq t \leq 5$$. We need to determine when the object is moving forwar

1. **State the problem:** Find the second derivative $y''$ when $y = e^{\sqrt{x}}$. 2. **Recall the formula:** To find $y''$, we first find $y'$ using the chain rule, then differen

1. The problem asks to analyze the functions given in problems 12 and 13, focusing on finding inflection points (Wendepunkte) and other characteristics. 2. For problem 12, the func