∫ calculus

1. **Problem:** Find the limit $$\lim_{x \to 1} \frac{x^3 - 3x}{x - 1}$$ using the values $$x = \{0.9, 0.99, 0.999, 1.001, 1.01, 1.1\}$$. 2. **Formula and rules:** The limit of a f

1. **Problem Statement:** Evaluate the definite integrals of the function $f(x)$ given its graph: a) $\int_{-2}^{3} f(x) \, dx$

1. **State the problem:** We need to solve for $h$ given the expression for the derivative: $$\frac{1}{3\pi} \times \frac{1}{2} h^2 \times h.$$ 2. **Simplify the expression:** Mult

1. **State the problem:** We need to sketch the curves $y = x^2 + 3$ and $y = 7 - 3x$ and find the area enclosed between them using integration. 2. **Find the points of intersectio

1. **State the problem:** We need to sketch the curves $y = x^2 + 3$ and $y = 7 - 3x$ and find the area enclosed between them using integration. 2. **Find the points of intersectio

1. We are asked to find the first and second derivatives of the function $$f(x) = \frac{1}{2 + \sin x}$$. 2. The first derivative of a function of the form $$\frac{1}{g(x)}$$ is gi

1. We are asked to find the derivative of the function $$f(x) = \sin^2 x + \frac{4}{\sin^2 x}$$ with respect to $$x$$. 2. Recall the derivative rules:

1. **Problem statement:** Show that $$\lim_{n \to \infty} \frac{1}{2n - 1} = 0$$ using the definition of the limit. 2. **Definition of limit for sequences:** For a sequence $(a_n)$

1. **State the problem:** Given the implicit equation $$x^{2} + y^{2} = A e^{B x}$$ where $A$ and $B$ are constants, find the resulting differential equation involving $y', y''$. 2

1. **State the problem:** We are given the implicit equation $$x^2 + y^2 = A e^{B x}$$ and asked to find the resulting differential equation by differentiating with respect to $x$.

1. **State the problem:** We are given two functions $y_1 = \sin(x^2)$ and $y_2 = x \cos(x^2)$ and asked to evaluate their Wronskian $W(y_1,y_2)$ at $x = \sqrt{\pi}$. 2. **Recall t

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