∫ calculus

1. **State the problem:** Find the derivative of the function $$y = \frac{(x+2)^2}{(x+1)^3 (x+3)^4}$$. 2. **Formula and rules:** We will use the quotient rule for derivatives: $$\l

1. We are asked to find the derivative of the function \( y = \frac{\sqrt{9+3x-x^{2}}}{e^{x}} \). 2. Recall the quotient rule for derivatives: if \( y = \frac{u}{v} \), then \( y'

1. **State the problem:** We are given a cube whose side length is increasing at a rate of 12 m/s. We need to find the rate at which the volume of the cube is increasing when the s

1. The problem is to understand and interpret the expression $$\frac{\partial y^n}{\partial y} = y \delta u$$. 2. Here, $$\frac{\partial y^n}{\partial y}$$ denotes the partial deri

1. **Problem:** Find the limit $$\lim_{n \to +\infty} \frac{4n^2}{1 - 3n^2}$$. 2. **Formula and rules:** When evaluating limits of rational functions as $$n$$ approaches infinity,

1. مسئله: بررسی مشتق‌پذیری یک تابع در یک نقطه خاص. 2. تعریف مشتق‌پذیری: تابع $f$ در نقطه $x=a$ مشتق‌پذیر است اگر حد زیر وجود داشته باشد:

1. مسئله: مشتق تابع $p$ را پیدا کنید. 2. مشتق تابع به معنی نرخ تغییرات تابع نسبت به متغیر مستقل است.

1. **Problem:** Find the volume of the solid generated by revolving the region bounded by $y = x^2$, $x=1$ to $x=2$, and $y=0$ about the y-axis using the method of cylindrical shel

1. **Problem statement:** Find the volume of the solid generated by revolving the region bounded by the parabola $x = y^2 + 1$ (left boundary) and the line $x = 5$ (right boundary)

1. **Problem (九): Solve the differential equation** $s'(t) = 0.07s(t) + 40000$ with initial condition $s(0) = 1000000$. 2. **Step 1: Identify the type of differential equation.**

1. مسئله: حد $$\lim_{x \to -1} \frac{\sqrt{7x+3} - \sqrt{7x+4}}{1 + \sqrt{x}}$$ را بیابید. 2. ابتدا بررسی می‌کنیم که آیا جایگذاری مستقیم ممکن است:

1. 問題陳述：分析並繪製函數 $f(x) = x^3 - 4x^{2/3}$ 的圖形，定義域為 $x \in \mathbb{R}$。 2. 公式與規則：

1. Masalah: Hitung integral \( \int \frac{dx}{\sqrt{16 - x^2}} \) menggunakan substitusi. 2. Rumus dan aturan penting: Integral bentuk \( \int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin

1. **State the problem:** Find the Maclaurin series (Taylor series at $x=0$) for the function $f(x) = \sin 3x$. 2. **Recall the Maclaurin series formula:**

1. Tentukan integral tak tentu dari $\int \frac{1}{x^2} \, dx$. 2. Gunakan aturan integral pangkat: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ untuk $n \neq -1$.

1. **State the problem:** We need to find the limit $$\lim_{x \to 0} \frac{\tan(2x) - \sin(2x)}{x^3}.$$\n\n2. **Recall the formulas and expansions:** For small $x$, use the Taylor

1. **State the problem:** Find the slope of the tangent line and the equation of the tangent line for the function $f(x) = 3x + 4$ at the point $(1,7)$. 2. **Recall the formula for

1. **State the problem:** Find the coordinates of the point of inflection of the function $f(x) = x^3 - 3x + 2$. 2. **Recall the formula and rules:** The point of inflection occurs

1. **Problem Statement:** Find the area between the graph of the function $f(x) = 1 - x^2$ and the x-axis from $x = -1$ to $x = 1$. 2. **Formula:** The area between the curve and t

1. مسئله: مقدار حد $$\lim_{x \to 0^-} \frac{\sqrt{2+3x} - \sqrt{2-x}}{\sqrt{1 - \cos x}}$$ را بیابید. 2. فرمول‌ها و نکات مهم:

1. **State the problem:** We want to find the integral $$\int \log(x) \, dx$$. 2. **Use integration by parts formula:** $$\int u \, dv = uv - \int v \, du$$.