∫ calculus

1. Problem: Calculate the limits of the sequences given by their general terms. 2. (a) $a_n = \frac{n^3 + 2n - 1}{4n^4 + n^2 - n}$

1. The problem is to evaluate the integral $\int 9 \, dx$. 2. The integral of a constant $c$ with respect to $x$ is given by the formula:

1. **State the problem:** Differentiate the function $f(t) = 3t^2 t = 3t^3$ with respect to $t$. 2. **Recall the power rule:** The derivative of $t^n$ with respect to $t$ is $\frac

1. The problem is to find the derivative of a function, which measures how the function changes as its input changes. 2. The derivative of a function $f(x)$ is denoted as $f'(x)$ o

1. **State the problem:** Find the derivative of the given function (please provide the function if not specified). 2. **Recall the derivative rules:**

1. **Problem Statement:** We are given two functions to analyze and graph:

1. **State the problem:** Differentiate the implicit equation $$e^{xy} = xy$$ with respect to $$x$$. 2. **Recall the rules:** We will use implicit differentiation and the product r

1. **State the problem:** Calculate the limit $$\lim_{x \to \infty} \frac{8x + 1}{\sqrt{7 + x^2}}.$$\n\n2. **Recall the formula and rules:** When dealing with limits involving infi

1. **Problem:** Find the integral $$\int \frac{x^2 - 9x - 35}{(x+1)(x-2)(x+3)} \, dx$$ **Step 1:** Use partial fraction decomposition. Assume:

1. **Тодорхойлолт:** f(x) функцийн өсөх, буурах завсар, локал минимум, максимум, хотгор, гүдгэр завсар, нугарлын цэгийг олох. 2. **Формул ба дүрэм:**

1. Тодорхойлолт: f функц нь [1,5] битүү завсарт тасралтгүй бөгөөд дараах экстремум цэгүүдтэй байна. 2. Абсолют максимум гэдэг нь тухайн завсарт хамгийн их утгатай цэг, абсолют мини

1. The problem is to find the derivative of the function $f(x) = \cos^2(x)$. 2. We recognize that $\cos^2(x)$ means $(\cos(x))^2$, so we will use the chain rule for differentiation

1. Tehtävä: Derivoi funktio $f(x) = \sin(x^3)$.\n\n2. Käytämme ketjusääntöä, koska funktio on yhdistelmä kahdesta funktiosta: $\sin(u)$, missä $u = x^3$.\n\n3. Ketjusäännön mukaan

1. The problem is to find the derivative of the function $f(x) = \sin(x^3)$.\n\n2. We use the chain rule for differentiation, which states that if $f(x) = \sin(g(x))$, then $f'(x)

1. Problem: Calculate the integrals given in parts a, b, c, and d. 2. Part a: \(\int \sqrt{16x} \sin(1 + x^{3/2}) \, dx\)

1. We are asked to compute limits using the (\epsilon - \delta) definition of limits for the following functions: (a) $\lim_{x \to 1} (x^2 + 1)$

1. The problem is to find the derivative of the function $f(x) = \tan x$. 2. Recall the formula for the derivative of the tangent function: if $f(x) = \tan x$, then

1. Problem: Find the implicit derivatives for the given equations. (c) $$\sqrt{x} + y = x^4 + y^4$$

1. **State the problem:** We need to find the constant $\lambda$ such that the piecewise function $$

1. Асуудлыг тодорхойлох: $f(x) = |x^2 - 9|$ функцийн $x$-ийн ямар утганд дифференциалчлагдахыг олж, $f'(x)$-ийн томъёог бичнэ.

1. **Problem statement:** (a) Find the domain where the function $f(x) = |x^2 - 9|$ is differentiable and find the formula for $f'(x)$.