∫ calculus

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 $x \geq 0$

1. \textbf{المسألة:} لدينا الدالة $$f(x) = \frac{x^2 + x + 1}{x + 1}$$ ونريد حساب الحدود التالية: $$\lim_{x \to -\infty} f(x), \quad \lim_{x \to 0^-} f(x), \quad \lim_{x \to 0^+} f

1. **State the problem:** Calculate the definite integral $$5 \int_1^5 \frac{2x^2}{3} \, dx$$. 2. **Formula and rules:** The integral of a function $f(x)$ over $[a,b]$ is given by

1. **State the problem:** Estimate the area under the curve of the function $f(x) = x^2$ from $x=0$ to $x=1$ using the midpoint rule with first 2 rectangles, then 4 rectangles. 2.

1. **State the problem:** Evaluate the integral $$\int 6 \sqrt[3]{x} \left( \frac{x^3}{3} + \frac{7}{2} x^{-\frac{1}{3}} \right) dx.$$\n\n2. **Rewrite the integral:** Recall that $

1. The problem is to evaluate the integral $$\int x^3 \sin x + y^n \, du$$. 2. Since the integral is with respect to $u$, and the integrand contains $x^3 \sin x + y^n$ which are co

1. **Problem statement:** Given the function $y = f(x) = ax^3 + bx^2 + cx + d$ with the derivative sign table and function behavior: - $f'(x)$ changes sign: positive on $(-\infty,

1. **Problem statement:** Differentiate the function $f(x) = x^{100}$ using the binomial formula. 2. **Recall the binomial formula:** The binomial theorem states that for any integ

1. **Problem Statement:** Given that $$\lim_{x \to 3} f(x) = 7$$, determine which of the following statements must be true: I. $$f$$ is continuous at $$x=3$$

1. The problem asks to find the acceleration of the particle at time $t=\pi$ given the position function $x(t) = \sin(2t) - \cos(3t)$ for $t \geq 0$. 2. Acceleration is the second

1. **Problem statement:** Find the area under the curve $y=4-x^2$ on the interval $[-2,2]$ using rectangles. 2. **Formula and explanation:** The area under a curve can be approxima

1. **State the problem:** Find the maximum and minimum values of the function $$f(x) = 3x^4 - 2x^3 - 6x^2 + 6x + 1$$. 2. **Formula and rules:** To find maxima and minima, we first

1. **Problem statement:** Evaluate the limits (i) $$\lim_{x \to 0} \left( \frac{1}{\sin x} - \frac{1}{x} \right)$$

1. **Problem 1: Define and find values of the function** Given the function:

1. Problem (a): A balloon is rising vertically at 3 m/s. A boy cycles beneath it at 5 m/s. When the balloon is 40 m high, find the rate of change of the distance between them. 2. L

1. **State the problem:** Find the area of the shaded region bounded by the curve, the line $y=2$, and the vertical line $x=\pi$. 2. **Identify the boundaries:** The region is boun

1. **Problem:** Find the limit $\lim_{x \to 0} x \cot x$. 2. **Formula and rules:** Recall that $\cot x = \frac{\cos x}{\sin x}$ and near zero, $\sin x \approx x$ and $\cos x \appr