∫ calculus

1. مسئله: حد یک تابع را پیدا کنیم. 2. تعریف حد: حد تابع $f(x)$ در نقطه $a$ مقداری است که $f(x)$ به آن نزدیک می‌شود وقتی $x$ به $a$ نزدیک می‌شود، یعنی:

1. **State the problem:** Find the limit $$\lim_{x \to -r} \frac{\frac{x}{r} - 1}{x + r}$$. 2. **Rewrite the expression:** The numerator is $$\frac{x}{r} - 1 = \frac{x - r}{r}$$.

1. The problem asks for the limiting value of the function $$f(x) = \sin x$$ as $$x$$ approaches $$\pi$$. 2. The limit of a function $$f(x)$$ as $$x$$ approaches a value $$a$$ is t

1. The problem asks for a reasonable estimate of the right-hand limit of the function $g$ as $x$ approaches 0, i.e., $\lim_{x \to 0^+} g(x)$. 2. The right-hand limit means we consi

1. **State the problem:** Find the derivative $f^\prime(x)$ of the function $f(x) = x^4 + 2x$.

1. **State the problem:** We need to evaluate the integral $$\int e^{-4x} \cos 3x \, dx$$. 2. **Formula and method:** For integrals of the form $$\int e^{ax} \cos(bx) \, dx$$, we u

1. **State the problem:** Evaluate the double integral $$\iint_R (x + y) \, dA$$ where the region $$R$$ is the rectangle defined by $$0 \leq x \leq 2$$ and $$1 \leq y \leq 3$$. 2.

1. **State the problem:** Evaluate the double integral $$\iint_R (x + y) \, dA$$ where the region $$R$$ is the rectangle defined by $$0 \leq x \leq 2$$ and $$1 \leq y \leq 3$$. 2.

1. **State the problem:** Evaluate the double integral $$\iint_R (x + y) \, dA$$ where the region $$R$$ is the rectangle defined by $$0 \leq x \leq 2$$ and $$1 \leq y \leq 3$$. 2.

1. **State the problem:** We need to find the radius of convergence of the power series $$\sum_{n=1}^\infty n x^n$$. 2. **Recall the formula:** The radius of convergence $R$ of a p

1. Let's start with a simple calculus problem: Find the derivative of the function $f(x) = x^2 + 3x + 5$. 2. The derivative of a function gives the rate at which the function's val

1. **Problem statement:** Find the Taylor series expansion of the function $f(x) = e^{2x}$ around $x=0$ up to the $x^3$ term. 2. **Formula:** The Taylor series of a function $f(x)$

1. **State the problem:** Determine whether the series $$\sum_{n=1}^\infty \frac{1}{n^2}$$ converges or diverges. 2. **Recall the p-series test:** A p-series $$\sum_{n=1}^\infty \f

1. **Problem Statement:** We have a battery level function $$B(t) = 100\left(1 - \frac{t}{10} e^{-0.2t}\right)$$ for $$t \in [0,10]$$ hours.

1. **Problem Statement:** We have a battery level function $$B(t) = 100\left(1 - \frac{t}{10}e^{-0.2t}\right)$$ for $$t \in [0,10]$$ hours. We need to find: a. The maximum and mini

1. مسئله: تعیین نقاطی که تابع در آن‌ها محدودیت‌پذیر است و نوشتن حد تابع در نقاط ۲، ۳ و ۵. 2. تعریف محدودیت‌پذیری: تابع در نقطه‌ای محدودیت‌پذیر است اگر حد چپ و حد راست آن نقطه وجود

1. مسئله: تعیین نقاطی که حد تابع در آنها وجود دارد یا قابل تعیین است و بررسی دامنه تابع و حدهای داده شده. 2. تعریف تابع و دامنه: طبق نمودار، تابع در نقاط 0 و 5 تعریف نشده (دایره‌ها

1. We are asked to determine if the improper integral $$\int_1^\infty \frac{1}{x-6} \, dx$$ converges or diverges, and if it converges, to find its value. 2. The integral is improp

1. Problem: Find the derivative of the function $y = x^2 - 16x - 2^4$. 2. Formula: The derivative of $x^n$ is $nx^{n-1}$, and the derivative of a constant is 0.

1. The problem is to find the integral of $\ln x$ with respect to $x$, i.e., $\int \ln x \, dx$. 2. We use integration by parts formula: $$\int u \, dv = uv - \int v \, du$$

1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{6 + x - x^2}{12 - x - x^2}$$. 2. **Recall the limit rule:** If direct substitution leads to an indeterminate form li