∫ calculus

1. The problem is to find the derivative of the function $$f(x) = \left(2e^{-x} + e^{3x}\right)^3$$. 2. We use the chain rule for differentiation: if $$f(x) = [g(x)]^n$$, then $$f'

1. The problem is to find the second-order Taylor expansion of the function $$f(x,y) = e^x \cos y$$ about the point $$(0,0)$$. 2. The Taylor expansion formula for a function of two

1. **Problem Statement:** Calculate the integral $$\int e^{-t} \cos(\omega t) \, dt$$ where $\omega$ is a constant.

1. **State the problem:** We want to evaluate the definite integral $$\int_0^2 5x \, dx$$ using the definition of the integral as a limit of Riemann sums. 2. **Find the width of ea

1. **State the problem:** Express the integral $$\int_4^6 \sqrt{6 + x^2} \, dx$$ as a limit of Riemann sums using right endpoints, without evaluating the limit. 2. **Find the width

1. The problem is to evaluate the definite integral $$\int_4^6 \sqrt{6 + x^2} \, dx$$. 2. We use the formula for the integral of $$\sqrt{a^2 + x^2}$$:

1. **Problem:** Find the second derivative of the function $$f(x) = \sqrt{2x^2 + 3x^{-3} + 7x^{-1}}$$. 2. **Step 1: Rewrite the function**

1. **State the problem:** Find the second derivative $y''$ of the function $y = x^2 \ln(2x)$.\n\n2. **Recall the formula and rules:** We will use the product rule for derivatives s

1. **State the problem:** Find the partial derivatives of the function $$u = e^{\frac{x}{y}} + e^{\frac{y}{x}}$$ with respect to $x$ and $y$. 2. **Recall the formula:** For a funct

1. **State the problem:** We need to evaluate the definite integral $$\int_3^5 \frac{x^3}{x^2 - 3x + 2} \, dx.$$ 2. **Factor the denominator:** The quadratic in the denominator fac

1. **State the problem:** Evaluate the definite integral $$\int_0^1 \frac{1}{\sqrt{4 - x^2}} \, dx$$. 2. **Recall the formula:** The integral $$\int \frac{1}{\sqrt{a^2 - x^2}} \, d

1. **Problem a:** Evaluate the limit as $x$ approaches $-3$ of $\frac{x+3}{x^2-9}$. 2. **Recall the formula:** The limit of a function $f(x)$ as $x$ approaches a value $a$ is $\lim

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

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