∫ calculus

1. Το πρόβλημα είναι να υπολογίσουμε τη δεύτερη παράγωγο μιας συνάρτησης. 2. Η δεύτερη παράγωγος είναι η παράγωγος της πρώτης παραγώγου, δηλαδή αν $f(x)$ είναι η αρχική συνάρτηση,

1. Το πρόβλημα ζητά να βρούμε τη σειρά Taylor της συνάρτησης $f(x)=\frac{x}{x^2+4x+5}$ μέχρι και τον τρίτο όρο (τρίτο παράγοντα).\n2. Η σειρά Taylor μιας συνάρτησης $f(x)$ γύρω από

1. The problem asks to identify which function corresponds to the given slope field shown in the top-right graph. 2. A slope field represents the direction of the tangent (slope) t

1. **State the problem:** Find the limit of a function as the variable approaches a certain value. Since the user did not specify the exact function or limit, let's consider a gene

1. **Problem Statement:** We are given the function $$h(x) = x^4 - 3x^3 - 10x + 2$$ and need to find its first and second derivatives, critical points, and plot the function with t

1. Muammo: $y = x^2$, $x = 2$, va $y = 0$ chiziqlari bilan chegaralangan shaklni $Oy$ o‘qi atrofida aylantirishdan hosil bo‘lgan jism hajmini topish. 2. Formulalar va qoidalar: $Oy

1. Muammo: $y = x^2$, $x = 2$, va $y = 0$ chiziqlari bilan chegaralangan shaklni $Oy$ o'qi atrofida aylantirishdan hosil bo'lgan jism hajmini topish. 2. Formulalar va qoidalar: $Oy

1. **Problem Statement:** Find the area of the region bounded by the parabola $y = x^{2}$, the vertical line $x = 2$, and the horizontal line $y = 0$. 2. **Formula and Explanation:

1. **Problem statement:** A window is shaped as a rectangle surmounted by an equilateral triangle. The total perimeter is 12 m. Find the rectangle dimensions that maximize the wind

1. Muammo: Berilgan parametrik tenglamalar $x = 6 \cos^3 t$, $y = 6 \sin^3 t$ va $0 \leq t \leq \frac{\pi}{3}$ oraliqdagi egri chiziqning yoyi uzunligini topish. 2. Egri chiziqning

1. Muammo: Parametrik tenglamalar $x=3\cos t$, $y=8\sin t$ bilan berilgan egri va $y=4\sqrt{3}$ chiziq bilan chegaralangan va $y \geq 4\sqrt{3}$ sharti ostidagi shakl yuzasini topi

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ for the equation $$\sqrt{x} + \sqrt{y} = 4.$$\n\n2. **Recall the formula and rules:** We will use implicit differentia

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ for the equation $$\sqrt{x} + \sqrt{y} = 4.$$\n\n2. **Rewrite the equation:** Recall that $\sqrt{x} = x^{\frac{1}{2}}$

1. **State the problem:** We need to find the slope of the tangent line to the curve defined by the implicit equation $$x^2 + 4xy - 3y^2 = 7$$ at the point $(2,1)$. 2. **Formula an

1. **Problem:** Find the integral $\int e^{2x} \, dx$. 2. **Formula and rule:** The integral of $e^{ax}$ with respect to $x$ is $\frac{1}{a} e^{ax} + C$, where $a$ is a constant an

1. We are asked to determine if the function $$g(x) = \frac{7x^2}{10 + 2x}$$ is continuous on various intervals. 2. Recall that a rational function is continuous everywhere its den

1. **Problem statement:** Find the local maxima and minima (Hoch- und Tiefpunkte) of the function $f(x)$ using the sign change criterion for the derivative. 2. **Given functions:**

1. The problem asks to find the limit of the function $f(x)$ at various points and determine continuity at given points or intervals based on the graph description. 2. **Limits**:

1. **Problem statement:** We want to find the limit $$\lim_{(x,y) \to (0,0)} \frac{xy}{\sqrt{x^2 + y^2}}$$ and show that it equals zero without using polar coordinates. 2. **Recall

1. **State the problem:** Differentiate the function $$y = (x^2 + 4x + 6)^5$$ using the chain rule. 2. **Recall the chain rule formula:** If $$y = [u(x)]^n$$, then $$\frac{dy}{dx}

1. The problem asks to find the instantaneous velocity $v(t_0)$ for the function $s(t)$ at the given time $t_0$. 2. The instantaneous velocity is the derivative of the position fun