∫ calculus

1. The problem is to find the derivative of the function $f(x) = x^2 + 3x$. 2. The derivative of a function $f(x)$, denoted $f'(x)$, represents the rate of change or slope of the f

1. مسئله را بیان می‌کنیم: می‌خواهیم حد عبارت $$\lim_{n \to 3} \frac{\sqrt{5n + 1}}{3n - n^2}$$ را محاسبه کنیم. 2. ابتدا مقدار تابع را در نقطه $n=3$ جایگذاری می‌کنیم تا ببینیم آیا م

1. **Define Homogeneous Function:** A function $f(x,y)$ is called homogeneous of degree $n$ if for all $t > 0$, it satisfies $$f(tx, ty) = t^n f(x,y).$$

1. Problem: Find $\frac{dy}{dx}$ if $f(x,y) = c$ (constant). Formula: For implicit functions, $\frac{dy}{dx} = -\frac{f_x}{f_y}$ where $f_x = \frac{\partial f}{\partial x}$ and $f_

1. **State the problem:** Calculate the limit $$\lim_{n \to \infty} \frac{3 + (-3)^n}{4^n}$$ by definition. 2. **Recall the limit definition and properties:** For sequences, if the

1. **Problem Statement:** Find the derivatives of all trigonometric functions including simple, inverse, hyperbolic, and inverse hyperbolic functions. 2. **Formulas and Rules:** Th

1. **State the problem:** We need to find the limit $$\lim_{x \to 1^+} (x^2 - 1) \tan \frac{\pi x}{2}$$. 2. **Recall the formula and behavior:** As $x \to 1^+$, note that $x^2 - 1

1. **Problem:** Find the limit $$\lim_{x \to 0} \frac{x - \sin x}{x^3}$$ without using L'Hospital's rule. 2. **Recall the Taylor series expansion:** For small $x$, $$\sin x = x - \

1. **Problem statement:** Evaluate the line integral $$\int_C yz\,dx - xz\,dy + xy\,dz$$ where the curve $C$ is given by the parametric equations $x=\sin t$, $y=\cos t$, $z=t^2$ fo

1. **Stating the problem:** We want to evaluate the expression $S = Syzdx - xzdy + xydz$ where $x = \sin t$, $y = \cos t$, and $z = t^2$ for $0 \leq t \leq \frac{\pi}{2}$. The prob

1. The problem asks: What does $\lim_{x \to a} f(x) = L$ mean? 2. The limit definition states: As $x$ approaches $a$, the function values $f(x)$ approach $L$.

1. The problem asks: What does $\lim_{x \to a} f(x) = L$ mean? 2. The limit notation $\lim_{x \to a} f(x) = L$ means that as $x$ gets closer and closer to the number $a$, the value

1. **Problem Statement:** Evaluate the line integral $$\int_C (x+2y)\,dx + (x - y)\,dy$$ where the curve $$C$$ is given by the parametric equations $$x=2\cos t, y=2\sin t, 0 \leq t

1. **Problem Statement:** Evaluate the line integral $$\int_C (x + 2y)\,dx + (x - y)\,dy$$ where the curve $C$ is given by the parametric equations $$x = 2\cos t, \quad y = 4\sin t

1. **State the problem:** Evaluate the integral $$\int_0^\infty e^{-n} n^5 \, dn$$. 2. **Recall the formula:** This integral is a form of the Gamma function $$\Gamma(k) = \int_0^\i

1. **Problem Statement:** Evaluate the line integral $$\int_C (x + 2y)\,dx + (x - y)\,dy$$ where the curve $C$ is given by the parametric equations $$x = 2\cos t, \quad y = 4\sin t

1. **Problem Statement:** Evaluate the line integral $$\int_C (x + 2y)\,dx + (x - y)\,dy$$ along the curve $C$. To make the problem smoother and easier, let's change the curve $C$

1. **Problem Statement:** Evaluate the line integral along the curve $C$ for the function $f(x + 2y)dx + (x - y)dy$. To make the problem smoother and easier, let's change the funct

1. **Problem Statement:** Evaluate the triple integral $$\int_0^2 \int_0^{\sqrt{4-x^2}} \int_{-5+x^2+y^2}^{3-x^2-y^2} x \, dz \, dy \, dx$$. 2. **Change to simplify:** Change the o

1. The problem is to find the third derivative $f^{(3)}(x)$ of the function $f(x) = 3e^{-2x} + 5x^{4}$. 2. Recall the rules for derivatives:

1. **State the problem:** We need to find the derivative of the function $f(x) = \ln\left( (e^x)^x \right)$. 2. **Simplify the function:** Recall the power rule for exponents: $(a^