∫ calculus

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $$y = 5x + \sqrt{3 - x}$$ using the definition of the derivative. 2. **Recall the definition of the de

1. **State the problem:** We need to determine which of the given limit expressions agree with the behavior of the graph near the vertical asymptotes at $x = 1$, $x = 3$, and $x =

1. **State the problem:** We are asked to determine which limit expressions agree with the graph of the function $f$ that has vertical asymptotes near $x = -5, 1, 3,$ and $5$. Spec

1. The problem asks which limit expressions agree with the graph of function $f$ that has vertical asymptotes at $x=-5$, $x=-1$, $x=3$, and $x=7$. 2. Recall that a vertical asympto

1. **Problem statement:** We want to find the one-sided limits of the function $$g(x) = \frac{1}{\tan^2(x)}$$

1. **Problem statement:** Calculate the derivative of the function $$f(x) = \frac{2x + 4}{x - 2}$$ defined on the interval $$l = ]2 ; +\infty[,$$ and verify the derivative using an

1. Problème 4 : Coût de fabrication C(x,y) = 2x^2 + 3y^2 + xy 1. Calcul des dérivées partielles premières :

1. **State the problem:** We are asked to find the limits of the function $f(x)$ at various points based on the graph description. 2. **Recall the definition of limits:** The limit

1. The problem is to solve an integral using the integral table method. 2. Integral tables provide formulas for common integrals, which can be used to find antiderivatives quickly.

1. The problem is to find the derivative of the function $f(\sin 5x)$.\n\n2. We use the chain rule for derivatives, which states that if $f(g(x))$ is a composite function, then its

1. Задача: Найти максимум функции $f(x) = x^3 - 6x^2$ на некотором интервале. 2. Формула для нахождения экстремумов: необходимо найти производную функции и приравнять её к нулю, то

1. **State the problem:** We are given the implicit equation $$ue^y = 2x + 8y$$ and asked to find $$\frac{dy}{dx}$$.

1. **Stating the problem:** We are given the implicit equation $$y^2 + 3xy = 2x + 3$$ and asked to find the derivative $$\frac{dy}{dx}$$ and the equation of the tangent line at a s

1. **Stating the problem:** We are given the implicit equation $$y + 1 = \sqrt{y^2 - x^2}$$ where $y$ is a function of $x$. We want to find the derivative $y'$ and determine which

1. Il problema chiede di trovare la derivata della funzione $$f(x) = \frac{x\sqrt{x} + \sqrt[3]{x}}{x^2}$$ e verificare la correttezza della derivata data. 2. La funzione può esser

1. **State the problem:** We need to evaluate the double integral

1. **State the problem:** We are given the differential equation $$\frac{dy}{dx} = \frac{1}{x + e}$$ for $$x > -e$$, and the curve passes through the point $$(e, 2 + \ln 2)$$. We n

1. **State the problem:** Find the derivative of the function $f(x) = x^{2x}$. 2. **Recall the formula:** When differentiating functions of the form $y = u(x)^{v(x)}$, use logarith

1. **State the problem:** We need to evaluate the integral

1. **State the problem:** Find the derivative of the function $f(x) = x(a x^2 + b x + 3)$. 2. **Rewrite the function:** Distribute $x$ to get $f(x) = a x^3 + b x^2 + 3 x$.

1. The problem asks to estimate the limit $$\lim_{x \to -10} h(x)$$ using the given table of values for $$h(x)$$ near $$x = -10$$. 2. The limit $$\lim_{x \to a} f(x)$$ is the value