∫ calculus

1. Problem: Find the derivative of the function \(x^3 + \sqrt{2x^2y} - x = 2x^2\).\n\n2. We will differentiate both sides with respect to \(x\). Note that \(y\) may be a function o

1. We are asked to find the derivatives of the given functions. 2. Recall the derivative rules we will use:

1. **Problem statement:** Given the vector function $$\mathbf{r}(t) = (t\sin t + \cos t)\mathbf{i} + (\sin t - t\cos t)\mathbf{j}$$ for $$t > 0$$, find the acceleration vector $$\m

1. **State the problem:** We are given the vector function $$\mathbf{r}(t) = (t\sin t + \cos t) \mathbf{i} + (\sin t - t\cos t) \mathbf{j}$$ for $$t > 0$$. We want to analyze or fi

1. **State the problem:** Find the limit $$\lim_{x \to 2} (8 - 3x + 12x^2)$$. 2. **Recall the limit rule for polynomials:** The limit of a polynomial function as $x$ approaches a v

1. **State the problem:** Find the limit $$\lim_{x \to 2} (8 - 3x + 12x^2)$$. 2. **Recall the limit rule:** For polynomial functions, the limit as $x$ approaches a value is simply

1. **Problem:** Evaluate $$\lim_{x \to -1} 2e^{3x+4}$$ 2. **Formula and rules:** The exponential function $$e^x$$ is continuous everywhere, so $$\lim_{x \to a} e^{f(x)} = e^{\lim_{

1. Problema: Se consideră funcția $f : \mathbb{R} \to \mathbb{R}, f(x) = x^5 + x^3 + 2x$. Arătați că $$\int_{-1}^1 (f(x) - x^3 - 2x) \, dx = 0.$$ 2. Formula folosită: Integrală def

1. **Énoncé du problème :** Nous avons la fonction $f(x) = \frac{1}{\arcsin(x)}$ définie sur l'intervalle $(-1,1)$ sauf en $x=0$ où $\arcsin(x)=0$.

1. The problem is to explore the expression $0^0$ using limits and understand if it can be represented as 0. 2. The expression $0^0$ is an indeterminate form in mathematics, meanin

1. **State the problem:** We want to understand the expression $\frac{0}{0}$ and see if it can be represented as 0 using limits. 2. **Important note:** The expression $\frac{0}{0}$

1. **Problem:** Find the derivative of $y = \sin^2 3x - 5$. 2. **Formula:** Use the chain rule and power rule. If $y = (\sin u)^2$, then $\frac{dy}{dx} = 2 \sin u \cdot \cos u \cdo

1. **Problem:** Find the limit $$\lim_{n \to \infty} \frac{(n+7)^2 + (3n+1)^2}{(n+4)^3 - (1+n)^3}$$. 2. **Formula and rules:** For limits at infinity involving polynomials, divide

1. The problem asks to differentiate the function $$y=\frac{x^2}{x^2+1}$$ with respect to $$x$$. 2. We use the quotient rule for differentiation: if $$y=\frac{u}{v}$$, then $$\frac

1. **Problem Statement:** Calculate the definite integral of the function $f(x) = x^2 + 3x + 2$ from 0 to 1. 2. **Formula Used:** The definite integral of a polynomial function $f(

1. **Problem statement:** Find the derivative of the function $$f(x) = \frac{\ln x}{x^2}$$. 2. **Formula used:** We will use the quotient rule for derivatives, which states:

1. The problem is to evaluate the integral $$\int \frac{1}{\sin x + \cos x} \, dx.$$\n\n2. First, use the identity $$\sin x + \cos x = \sqrt{2} \sin \left(x + \frac{\pi}{4}\right)$

1. **Problem:** Evaluate the integral $$\int \left(\frac{\sec x}{\csc x} + \frac{\csc x}{\sec x}\right) dx$$. 2. **Formula and rules:** Recall that $$\sec x = \frac{1}{\cos x}$$ an

1. **State the problem:** We need to find the limit as $x$ approaches $+\infty$ of the expression $\left(\sqrt{x^2 + 2} - x\right)^5$. 2. **Recall the formula and approach:** When

1. **State the problem:** Find the local and absolute maxima and minima of the function $$f(x) = 3x - 6 \cos(x)$$ on the interval $$[-\pi, \pi]$$. 2. **Find the derivative:** To fi

1. The problem is to list the basic derivative formulas used in calculus. 2. The derivative of a function measures how the function's output changes as its input changes.