∫ calculus

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.

1. **State the problem:** We need to find the 2025th derivative of the function $g(x) = \cos(x)$. 2. **Recall the derivatives of cosine:**

1. **Stating the problem:** We are given that $y=f(x^2)$ and the derivative $\frac{dy}{dx} = 2x^3 - 4x$. We need to find $f'(4)$. 2. Since $y = f(x^2)$, by the chain rule,

1. **State the problem:** Find the derivative of the function $$y = (x^3 + 5)^x$$. 2. **Understand the function:** This is a function where the base and the exponent both depend on

1. **State the problem:** Find the derivative of the function $$y = \frac{x+1}{x-1}$$ with respect to the variable $$t = \sqrt{2x+1}$$ at $$x = 4$$. 2. **Explain the relation:** We

1. The problem is to find the derivative of the inverse function $g(x)$ of $f(x) = 5x + 2$ at $x=12$. 2. Recall that if $g$ is the inverse of $f$, then $g(f(x))=x$ and also $g'(y)

1. We are given two functions: $f(x) = \tan(x)$ and $g(x) = \frac{\pi}{x}$. We need to find the derivative of $f$ evaluated at $g(4)$, which is $f'(g(4))$. 2. First, evaluate $g(4)

1. **State the problem:** We need to find the derivative of the composition function $(f \circ g)(x) = f(g(x))$ evaluated at $x = \sqrt{\pi}$ given $f(x) = \sin(x)$ and $g(x) = x^{