∫ calculus

1. The problem is to understand the concept of limits and continuity in calculus. 2. The limit of a function $f(x)$ as $x$ approaches a value $a$ is the value that $f(x)$ gets clos

1. Diberikan fungsi $f(x) = 2(2x - \frac{5}{7})^4$. 2. Gunakan aturan rantai untuk turunan fungsi bentuk $g(x)^n$, yaitu $\frac{d}{dx}[g(x)^n] = n g(x)^{n-1} g'(x)$.

1. Diberikan fungsi $f(x) = 2(2x - \frac{5}{7})^4$. Kita diminta mencari turunan fungsi ini. 2. Gunakan aturan rantai untuk turunan fungsi berbentuk $f(x) = a(g(x))^n$, yaitu:

1. **Problem statement:** Find the limit $$\lim_{x \to -\frac{1}{2}^-} \frac{x^7 + ax + \frac{1}{x^7}}{x - \frac{1}{x}} = b$$ where $a$ and $b$ are constants. 2. **Rewrite the expr

1. **Problem Statement:** Calculate the following limits for the piecewise function $f(x)$ based on the given graph: a) $\lim_{x \to 3^+} f(x)$

1. The problem asks which limit expression equals the derivative of the function $f(x) = \sqrt{x}$ at $x=4$. 2. Recall the definition of the derivative at a point $a$:

1. **State the problem:** We want to find the derivative of the function $$f(x) = x^2 + 1$$ at the point $$x=2$$. 2. **Recall the definition of the derivative:** The derivative at

1. **Постановка задачі:** Дослідити неперервність функції $$y=\begin{cases}-0.5x, & x<0 \\\ x^2+1, & 0\leq x<1 \\\ 2, & x\geq 1 \end{cases}$$

1. Soal: Turunkan fungsi $f(x) = x^3 + 5x^2 - 2x + 7$.\n\n2. Kaidah yang digunakan: kaidah turunan fungsi polinomial, yaitu $\frac{d}{dx} x^n = nx^{n-1}$.\n\n3. Turunan fungsi: $$f

1. **Problem:** Evaluate the integral $$\int \sin^4 x \cos^3 x \, dx$$ 2. **Formula and rules:** When integrating powers of sine and cosine, if one power is odd, save one sine or c

1. **Problem:** Find the value of $a$ such that the function $$f(t) = \begin{cases} \sqrt{4 + t^2} - 2 \, / \, a t^2, & t \neq 0 \\ 0, & t = 0 \end{cases}$$

1. Let's first understand what a turning point is in the context of a function. 2. A turning point occurs where the function changes direction from increasing to decreasing or vice

1. **Problem:** Given a function $f$ with $f(3) = -5$ and slope at any point $(x,y)$ on the graph given by $\frac{2x^2}{y}$, find the equation of the tangent line at $x=3$ and use

1. **Problem:** Given a function $f$ with $f(3) = -5$ and the slope of the tangent line at any point $(x,y)$ on the graph is given by $\frac{2x^2}{y}$, find the equation of the tan

1. **State the problem:** We need to find the limit $$\lim_{x \to 0} \frac{2f(3x) - x^2}{2x^2 + x}$$ where $f$ is a piecewise linear function given by points and lines. 2. **Analyz

1. The problem asks to find the derivative of the function given in the image (assuming the function is $f(x) = x^3 - 5x^2 + 6x - 2$ as a typical calculus-level question). 2. The f

1. مسئله: باید حد $$\lim_{x \to 0^+} \sqrt{x} \left[1 + \sin^2 \left(\frac{2\pi}{x}\right)\right]$$ را محاسبه کنیم. 2. فرمول و نکات مهم: تابع $$\sin^2(\theta)$$ همیشه بین 0 و 1 قرا

1. مسئله: ثابت کنید که $$\lim_{x \to 0} f(x) = 0$$ برای تابع $$f(x) = \begin{cases} x^2 & \text{اگر } x \text{ گویا باشد} \\ 0 & \text{اگر } x \text{ گنگ باشد} \end{cases}$$

1. The problem asks: When does a tangent line approximation underestimate the value of a function? 2. The tangent line approximation at a point $x=a$ uses the linearization formula

1. The problem asks: At which point does a tangent line approximation underestimate the function value? 2. Tangent line approximation uses the tangent line at a point to estimate n

1. **State the problem:** Find the area in the first quadrant bounded by the curve $f(x) = 4x - x^2$ and the x-axis. 2. **Identify the region:** The first quadrant means $x \geq 0$