∫ calculus

1. **State the problem:** We need to solve the differential equation $$\frac{dy}{dx} = \frac{y - x + 2}{y - x - 4}.$$\n\n2. **Rewrite the equation:** Let us introduce a substitutio

1. **State the problem:** Evaluate the limit as $n$ approaches infinity of the expression $$6 \frac{1^2 + 2^2 + 3^2 + \cdots + n^2}{n^3}.$$\n\n2. **Recall the formula for the sum o

1. **State the problem:** We want to find the derivative of the function $f(x) = \sqrt{x+2}$ from first principles. 2. **Recall the definition of derivative from first principles:*

1. **Problem statement:** Determine if the function $g(f(x))$ where $f(x)=x^3+2$ and $g(x)=\cos x$ is even, odd, or neither.

1. مسئله: محاسبه حد تابع با استفاده از قاعده هوپیتال. 2. قاعده هوپیتال می‌گوید اگر حد $$\lim_{x \to a} \frac{f(x)}{g(x)}$$ به صورت $$\frac{0}{0}$$ یا $$\frac{\infty}{\infty}$$ باشد

1. مسئله: قانون از هوپیتال برای محاسبه حدهایی که به صورت \( \frac{0}{0} \) یا \( \frac{\infty}{\infty} \) هستند استفاده می‌شود. 2. قانون از هوپیتال می‌گوید اگر حد \( \lim_{x \to a}

1. مسئله: حد $$\lim_{x \to 0} \left( \frac{1}{x^2} - \frac{1}{\sin^2 x} \right)$$ را محاسبه کنید. 2. فرمول و قواعد مهم: برای محاسبه حدهایی که شامل توابع مثلثاتی و توان‌های کوچک هست

1. **Problem Statement:** We want to understand the concept of continuity and find the slope of the tangent line to a function at a point, using an intuitive example. 2. **Continui

1. **State the problem:** We are given the function $f(x) = 5x + 4$ and two points $x_0 = 3$ and $x_1 = 7$. We want to find the average rate of change of the function between these

1. **Problem statement:** Calculate the integral $$\int_0^{2\pi} f(x) \, dx$$ where $$f(x) = \begin{cases} \sin x & \text{for } x \leq \pi \\ -2 \sin x & \text{for } x > \pi \end{c

1. **State the problem:** We want to evaluate the double integral $$\iint_D F(x,y)\,dA$$ where $$F(x,y) = x^2 + 2xy + y^2$$ and the region $$D$$ is bounded by the curves $$y = x$$

1. **Problem statement:** Given the curve defined by the equation $$xy + y^2 e^{-x} = 4,$$ we need to (a) show that $$\frac{dy}{dx} = \frac{y^2 - y e^x}{x e^x + 2y}$$ and (b) find

1. **Problem statement:** Given the curve equation $$2x^2 y - xy^2 = a^3$$ where $a$ is a positive constant, we need to show there is only one point on the curve where the tangent

1. **Problem statement:** Given the curve defined by the equation $$xy + y^2 e^{-x} = 4,$$ we need to show that $$\frac{dy}{dx} = \frac{y^2 - y e^x}{x e^x + 2y}$$ and then find the

1. The problem is to evaluate the integral $$\int \frac{0 + 1 + xy + x^2 y^2}{x^2} \, dx$$. 2. First, simplify the integrand by dividing each term by $x^2$:

1. **State the problem:** We need to find the derivative $\frac{dz}{dt}$ where $z = f(x,y) = \sqrt{x^2 - y^2}$, with $x = e^{2t}$ and $y = e^{-t}$. 2. **Recall the chain rule for m

1. **State the problem:** We need to find the derivative $\frac{dz}{dt}$ where $z = f(x,y) = \sqrt{x^2 - y^2}$, with $x = e^{2t}$ and $y = e^{-t}$.\n\n2. **Recall the chain rule fo

1. **State the problem:** We are given the function $$L(x) = \frac{f(x) - \tan(x)}{x^3}$$ and we want to analyze or simplify it depending on the context (e.g., limit, derivative, e

1. The problem asks to show that for the curve $y = \sqrt{x} \sin 2x$ on $0 \leq x \leq \frac{1}{2} \pi$, the maximum point $M$ at $x = a$ satisfies $\tan 2a = -4a$. 2. To find the

1. **State the problem:** We are given the function $F(x) = \frac{1}{x}$ defined on the interval $0 < x < 4$. We need to determine whether this function is bounded on this interval

1. The problem is to find the antiderivatives (indefinite integrals) of certain trigonometric-related functions. 2. The formulas for these antiderivatives are: