∫ calculus

1. We are asked to find the limits as $n \to \infty$ for each sequence $x_n$ in problems 3.1 to 3.20. 2. For each, we simplify the expression and analyze the dominant terms for lar

1. **State the problem:** Differentiate the implicit equation $$7y^2 + \sin(3x) = 12 - \frac{y}{4}$$ with respect to $$x$$. 2. **Rewrite the equation:** $$7y^2 + \sin(3x) = 12 - \f

1. لنبدأ بتحديد الدالة التي تريد حساب متسلسلة تايلور لها. 2. لنفترض أن الدالة هي $f(x)$ ونريد توسيعها حول نقطة معينة $a$.

1. We are asked to evaluate the definite integral $$\int_0^{\pi/3} 2 \sec^{2} x \, dx$$. 2. Recall that the integral of $\sec^{2} x$ with respect to $x$ is $\tan x$: $$\int \sec^{2

1. **State the problem:** We have a function $$f : \mathbb{R} \to \mathbb{R}$$ defined by $$f(x) = x - \frac{x^3}{3}$$.

1. The problem asks us to evaluate the expression $$\int_1^{\pi} 4x \, dx - \int_1^{\pi} \tan x \, dx.$$ 2. First, calculate $$\int_1^{\pi} 4x \, dx.$$

1. Let's analyze the function $y = 4x - x^2$. 2. This is a quadratic function, which is a parabola opening downwards since the coefficient of $x^2$ is negative (-1).

1. The problem asks: Determine whether the sequence \(\left\{ \frac{\sqrt{n + 47} - \sqrt{n}}{1} \right\}_{n=0}^\infty\) converges or diverges and compute its limit if it converges

1. Let's start by stating the problem: Find the partial derivative of a given function with respect to a specified variable. 2. Suppose the function is $f(x,y) = x^2 y + 3xy^2$ and

1. Stating the problem: We need to evaluate the definite integral $$\int_{-1}^2 (x^2 - 5x - 4)\,dx$$. 2. Find the antiderivative: The integral of $$x^2$$ is $$\frac{x^3}{3}$$, the

1. **State the problem:** We have the piecewise function $$f(x)=\begin{cases}\sqrt{-x} & \text{if } x<0\\3 - x & \text{if } 0 \leq x < 3\\(x - 3)^2 & \text{if } x \geq 3\end{cases}

1. Let's start by stating the problem clearly: We need to evaluate the integral $$\int_{\pi}^{2\pi} \cos(2x)\, dx.$$\n\n2. Recall the integral formula for cosine: $$\int \cos(ax)\,

1. **Stating the problem:** We have the function $$f_k(x)=x(x-k)(x-2)$$ with $$0<k<2$$ and want to express the area $$A(k)$$ of the shaded region between $$x=0$$ and $$x=2$$ bounde

1. **State the problem**: Find the integral of the polynomial function $$x^5 + 5x^2 - 6x + 7$$. 2. **Recall the integration rule**: The integral of $$x^n$$ with respect to $$x$$ is

1. The problem is to find the indefinite integral of $x^2$ with respect to $x$. 2. Recall that the integral of $x^n$ with respect to $x$ is given by the formula:

1. Problem statement: Evaluate the limit $\lim_{x\to+\infty} (x^2+3x+5)$. 2. Idea: Recognize that the polynomial is a parabola that opens upward and the highest power $x^2$ dominat

1. **State the problem:** Find the one-sided limits of the function $$f(x) = \frac{\sqrt{2x}(x-1)}{|x-1|}$$ as $$x \to 1^+$$ and $$x \to 1^-$$ respectively. 2. **Rewrite the functi

1. Find the fourth derivative of the function $f(x) = \ln \sqrt[3]{x^2} - \frac{42}{\sqrt[4]{x}} + \sinh(\sqrt[5]{x})$. Step 1: Simplify each term.

1. The problem asks for the value of $1 \div \infty$ where $\infty$ represents infinity. 2. Infinity is not a regular number; it is a concept representing an unbounded value that g

1. Let's start with the basic idea: A limit in calculus describes the value that a function approaches as the input approaches some value. 2. For example, consider the function $f(

1. Given the equation: $$xy + \frac{x}{y} = 0$$ 2. Differentiate both sides with respect to $x$, using implicit differentiation.