∫ calculus

1. **State the problem:** We want to find the limit as $x$ approaches positive infinity of the expression $$\left( \frac{\sqrt[x]{2} + \sqrt[x]{4} + \sqrt[x]{8}}{3} \right)^x.$$\n\

1. **State the problem:** Find the value of $a$ such that the limit $$\lim_{x \to 0} \frac{a \sin x - \sin 2x}{\tan^3 x}$$ is finite. 2. **Recall important formulas and approximati

1. **State the problem:** Find the limit as $x \to \infty$ of $$\left(\frac{\sqrt[3]{2} + \sqrt[3]{4} + \sqrt[3]{8}}{3}\right)^2$$. 2. **Understand the expression:** The expression

1. Differentiate $y = x \sin x$ with respect to $x$. Use the product rule: $\frac{d}{dx}[uv] = u'v + uv'$ where $u = x$ and $v = \sin x$.

1. **State the problem:** Find the derivative $y'$ if $y = -e^{\csc(x^2)}$. 2. **Recall the chain rule:** If $y = f(g(x))$, then $y' = f'(g(x)) \cdot g'(x)$.

1. **State the problem:** Evaluate the integral $$\int e^{\tan\theta} (\sec\theta - \sin\theta) \, d\theta.$$\n\n2. **Recall relevant formulas and rules:** We will use substitution

1. The problem involves the expression $$\oint 2xy \frac{dy}{dx} + 2y^2 - 3x - d$$ which appears to be a line integral or a differential expression involving $x$, $y$, and their de

1. **Problem Statement:** Given the graph of the derivative function $f'(x)$ for $-1 \leq x \leq 3$, determine the number of inflection points of the original function $f(x)$. 2. *

1. The problem asks to find the interval where the derivative $f'$ of the function $f$ is negative. 2. The derivative $f'$ is negative where the function $f$ is decreasing.

1. The problem asks us to identify which graph could represent the second derivative $f''(x)$ of a function $f(x)$ given its shape. 2. The original function $f(x)$ starts below the

1. The problem involves identifying the derivative function $f'(x)$ of a cubic function $f(x)$ shown in the top-right graph. 2. The top-right graph shows $f(x)$, a cubic curve incr

1. **Problem Statement:** We are given a curve $\hat{f}(x)$ and asked to identify which of the statements (a) to (d) about the function $f$ are correct, except one. 2. **Understand

1. **Problem statement:** We have a function $f$ defined on the interval $[a,b]$ with values in $\mathbb{R}^-$ (meaning $f(x) < 0$ for all $x \in [a,b]$). We want to determine for

1. **Problem Statement:** We are given the graph of the first derivative $f'(x)$ of a continuous function $f$ on $\mathbb{R}$. We need to identify which statement among (a), (b), (

1. The problem asks us to determine the correct relationship between the function $f(x)$ and the tangent line $g(x)$ to the curve $y=f(x)$ at any point $(x,y)$. 2. Recall that the

1. **Problem Statement:** Given that $f'(x) < 0$ and $f''(x) > 0$ for each $x \in [a,b]$, determine which graph represents the function $f$ on the interval $[a,b]$. 2. **Understand

1. **Problem Statement:** We analyze the function $f$ defined on the interval $[1,5]$ and determine which of the given statements (a) to (d) about its critical points, inflection p

1. **Problem Statement:** Given the graph of the second derivative $f''(x)$, determine which statement about the function $f$ is true. 2. **Understanding the problem:** The graph s

1. **Problem Statement:** We are given the graph of the derivative function $f'(x)$ of a continuous function $f$ on $\mathbb{R}$. The graph is a parabola opening upward with vertex

1. **Problem:** Evaluate the limit $$\lim_{x \to 2} \frac{\sqrt{6 - x^2}}{3 - x - 1}$$ 2. **Formula and rules:** To evaluate limits involving square roots and rational expressions,

1. **مسئله:** محاسبه مشتق تابع داده شده است. 2. برای حل مشتق، ابتدا باید تابع مورد نظر را مشخص کنید. لطفاً تابعی که می‌خواهید مشتق بگیرید را ارسال کنید.