∫ calculus

1. Problem: Find the derivative (уламжлал) of the function \(y = \frac{\sqrt{x+2}}{\arctan(x-2)}\). 2. Formula and rules: To differentiate a quotient \(\frac{u}{v}\), use the quoti

1. **State the problem:** We want to evaluate the limit $$\lim_{x \to 0} \left((1+x)^{\frac{1}{x}} - e^x\right).$$ 2. **Recall the formulas and rules:**

1. The problem is to evaluate the integral of the function $e^x$ from $x=0$ to $x=-\infty$. 2. The integral of $e^x$ with respect to $x$ is given by the formula:

1. **Problem Statement:** Given the limits \( \lim_{x \to 2} s(x) = 0 \) and \( \lim_{x \to 2} h(x) = -2 \), find the following limits: (i) \( \lim_{x \to 2} (s(x) + h(x)) \)

1. The problem asks to find intervals where both $h(x)<0$ and $h'(x)<0$. 2. $h(x)<0$ means the function is below the x-axis.

1. **Problem Statement:** Find all local maximum and minimum points of the function $$f(x,y) = -x^2 - 4y^2 - 2x + 8y - 1.$$\n\n2. **Formula and Rules:** To find local maxima and mi

1. **State the problem:** Show that $$\frac{1}{2}(e - \frac{1}{e}) = 1 + \frac{1}{3!} + \frac{1}{5!} + \ldots$$

1. **State the problem:** Given $\epsilon = 2023$, find $\delta > 0$ such that for all $(x,y)$, we have

1. The problem is to understand what differential equations are and how to solve a simple example. 2. A differential equation is an equation that relates a function with its deriva

1. The problem is to find the integral of a function, but the function is not specified in the question. 2. To solve an integral, we need the function to integrate, denoted as $f(x

1. Find an antiderivative of the function $2x$. 2. The formula for the antiderivative of $x^n$ where $n \neq -1$ is:

1. **State the problem:** Determine which of the given functions are well-behaved. A function is considered well-behaved if it is continuous, differentiable, and does not have abru

1. **Stating the problem:** Solve a linear equation using the DI (Derivative-Integral) method or tabular method. Since the user did not specify the exact equation, I will demonstra

1. **State the problem:** We need to find the integral $$\int xe^{2x} \, dx$$. 2. **Formula and method:** We will use integration by parts, which states:

1. **State the problem:** Find the limit $$\lim_{x \to +\infty} 3^{\left(\frac{1}{2}\right)^x} 4^x + 2$$.

1. The problem asks to find the derivative $\frac{dy}{dx}$ of the function $y = 2x^2$ using differentiation rules. 2. Recall the power rule for derivatives: if $y = x^n$, then $\fr

1. **State the problem:** Evaluate the limit expressions and simplify the given expressions involving $x$ approaching 0. 2. **Given expressions:**

1. **Problem Statement:** Find the critical numbers, intervals of increase/decrease, and relative extrema values for the function $$f(x) = x^3 - 3x^2 + 1$$ on the interval $$[-3,3]

1. **State the problem:** We need to solve the integral $$\int 4x \cos^2(x) \, dx$$. 2. **Use a trigonometric identity:** Recall that $$\cos^2(x) = \frac{1 + \cos(2x)}{2}$$.

1. **Problem statement:** Find the integral $$\int x \sin(x) \cos(x) \, dx$$. 2. **Formula and identities:** Use the double-angle identity for sine: $$\sin(2x) = 2 \sin(x) \cos(x)$

1. **State the problem:** Verify Rolle's theorem for the function $f(x) = \sin^2(x)$ on the interval $[0, \pi]$. 2. **Recall Rolle's theorem:** If a function $f$ is continuous on $