∫ calculus

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

1. **Problem Statement:** Find the derivative $\frac{dy}{dx}$ for the functions using the definition of derivative and verify using derivative formulas. 2. **Definition of Derivati

1. **Problem:** Differentiate the function $y = 2x^2 + 3x - 20$. 2. **Formula:** The derivative of $x^n$ is given by $\frac{d}{dx} x^n = n x^{n-1}$.

1. **State the problem:** Find the volume of the solid generated by revolving the region bounded by the curves $y=2x-1$, $y=\sqrt{x}$, and $x=0$ about the y-axis using the shell me

1. **Problem statement:** Find the closed form of the power series \(\sum_{n=2}^\infty (n-1) x^{n+6}\) using the power series representation of \(\frac{1}{1-x}\). 2. **Recall the g

1. **State the problem:** Calculate the definite integral $$\int_{-1}^1 (x^5 + 7x^4) \, dx$$. 2. **Recall the integral rules:** The integral of a sum is the sum of the integrals, a

1. **State the problem:** We are given three differentiable functions $f(x)$, $g(x)$, and $h(x)$ where $f(x) = -10x^2 - 6$, the tangent line to $g(x)$ at $x = -2$ is $y = -10x - 8$

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, determine the critical points by solving $$h'

1. The problem is to find the derivative of the function $g(x) = -6x - x^3$ and understand why the derivative is $g'(x) = -6 - 3x^2$. 2. The formula for the derivative of a functio

1. **Problem:** Calculate the integral $$\int \frac{x^3 - 2x + 4}{x^2 - 1} \, dx.$$\n\n2. **Formula and rules:** When integrating rational functions where the degree of the numerat

1. **Problem Statement:** We analyze the function $$f(x) = 3 \ln(x + 2) - x$$ to find where it is concave up and concave down, and explain why concave-up regions help gradient desc