∫ calculus

1. The problem is to find the absolute minimum of a function on the closed interval $[0,4]$. 2. To find the absolute minimum on a closed interval, we evaluate the function at criti

1. We are given the derivative of a function: $f'(x) = x^2 - 3 - 3 \sin(2x^2 - x)$ and the domain $x \in [0,4]$. 2. The problem is to understand or analyze this derivative function

1. The problem gives the derivative of a function as $f'(x) = (x^2 - 2) \sin(x)$ and an initial condition $f(-1) = 8$. We are asked to find the original function $f(x)$ on the inte

1. **Problem Statement:** Given the derivative of a function $f'(x) = (x^2 - 2) \sin(x)$ and the initial condition $f(-1) = 8$, find the original function $f(x)$. 2. **Formula and

1. **State the problem:** We are given the derivative of a function $$f'(x) = x^{2} \sin(x)$$ and the value $$f(-1) = -6$$. We need to find the absolute minimum value of $$f$$ on t

1. We are given the derivative of a function: $f'(x) = x^2 \sin(3x)$ and the initial condition $f(0) = 5$. We want to find the original function $f(x)$ on the interval $[-2, 2]$. 2

1. **State the problem:** We need to find the intervals where the function $f(x) = x^4 - 4x^3$ is decreasing. 2. **Recall the rule:** A function is decreasing where its derivative

1. **State the problem:** We are given the function $$f(x) = -x^4 - 12x^3 - 36x^2$$ and need to find the intervals where $$f$$ is increasing. 2. **Recall the rule:** A function is

1. **State the problem:** We need to find the intervals where the function $f(x) = x^3 - 21x$ is increasing. 2. **Recall the rule:** A function is increasing where its derivative $

1. **State the problem:** We need to find the intervals where the function $$f(x) = -x^3 + 12x^2 - 36x$$ is increasing. 2. **Recall the rule:** A function is increasing where its d

1. **State the problem:** We are given the derivative of a function $f'(x) = -x^3 - 6x^2 - 9x$ and need to find the intervals where the original function $f$ is decreasing. 2. **Re

1. مسئله: بررسی صحت معادله $$\sec x \frac{\partial w}{\partial x} + \sec y \frac{\partial w}{\partial y} = 1$$ برای تابع $$w = \sin y + f(\sin x - \cos y)$$ که در آن $$f$$ تابعی حق

1. The problem is to find the derivative of the function $f(x) = 4 - |x|$ at $x=0$. 2. The absolute value function $|x|$ is defined as:

1. The problem states: "The third derivative is under 16." We need to understand what this means and how to work with it. 2. The third derivative of a function $f(x)$ is denoted as

1. **Problem statement:** Find the Taylor polynomial of degree 3, $T_3(x)$, for $f(x) = \ln(1 + 2x)$ centered at $c=1$, and find an expression for the remainder $R_3(x)$. 2. **Form

1. **State the problem:** Find the area under the curve $y=\frac{11}{x}$ above the x-axis over the interval $[1,2]$. 2. **Set up the integral:** The area $A$ is given by the defini

1. **State the problem:** Find the derivative with respect to $x$ of the function $$f(x) = \frac{x}{2} + x.$$ 2. **Rewrite the function:** $$f(x) = \frac{x}{2} + x = \frac{x}{2} +

1. **State the problem:** Find the derivative of the function $f(x) = \frac{x}{2} + x$ with respect to $x$. 2. **Recall the derivative rules:**

1. **Problem Statement:** Find the left-hand limit, right-hand limit, and the limit at $a=2$ for the function $$f(x) = \frac{x^2 + 3x + 2}{x^2 - 4}$$ 2. **Recall the limit definiti

1. **Problem statement:** Find the derivative $f'(x)$ for $x < 1$ where the function is defined as $$f(x) = x^2 \text{ for } x \leq 1.$$ 2. **Formula used:** The derivative of a fu

1. **State the problem:** Find the right-hand derivative of the piecewise function $$f(x) = \begin{cases} x^2 - 1, & x < 0 \\ 2x - 1, & x \geq 0 \end{cases}$$ at $$x=0$$. 2. **Reca