∫ calculus

1. **State the problem:** Find the first derivative of the function $z = x^3 \log x$ with respect to $x$. 2. **Recall the formula:** To differentiate a product of two functions, us

1. The problem is to find the indefinite integral of the function $x^2 - 6x + 2$ with respect to $x$. 2. The formula for integrating a polynomial term $x^n$ is $$\int x^n dx = \fra

1. **Problem statement:** Find the second Taylor polynomial $P_2(x)$ for $f(x) = e^x \cos x$ about $x_0 = 0$. Use $P_2(0.5)$ to approximate $f(0.5)$, find an upper bound for the er

1. **State the problem:** Determine if the function $$f(x) = \frac{x^2 - x - 2}{x - 2}$$ is continuous at $$x=2$$. 2. **Recall the definition of continuity at a point:** A function

1. **Problem statement:** Given the function $f(x) = 2x \cos(2x) - (x - 2)^2$ and $x_0 = 0$, find the third Taylor polynomial $P_3(x)$ around $x_0=0$ and use it to approximate $f(0

1. Problem: Calculate the derivative of the function \(f(x) = -5x^6 + 4x^3 - \frac{1}{2}x + 4\). 2. Formula: The derivative of a power function \(x^n\) is given by \(\frac{d}{dx}x^

1. **Problem Statement:** We need to model the elevation $f(x)$ of a pedestrian bridge spanning 60 meters horizontally, peaking at 8 meters at the midpoint $x=30$, with slope const

1. **State the problem:** We are given the function $f(x) = (x-1)^3 (x+1)$ and asked to use the second derivative test to find where $f(x)$ is concave up, concave down, and to loca

1. **Problem Statement:** Evaluate the integral $$\int_{2}^{4} \int_{x}^{4} 6x^{2} \, dy \, dx$$ and reverse the order of integration.

1. The problem is to understand the limit notation $\lim_{x \to a}$ and what it represents. 2. The limit $\lim_{x \to a} f(x)$ describes the value that the function $f(x)$ approach

1. The problem is to understand the definite integral \(\int_a^b f(x)\,dx\), which represents the area under the curve of the function \(f(x)\) from \(x=a\) to \(x=b\). 2. The form

1. **Problem Statement:** Evaluate the line integral $$\oint_C (2x^2 - y^2)\,dx + (x^2 + y^2)\,dy$$ where $C$ is the boundary of the region enclosed by the $x$-axis and the semicir

1. **Problem Statement:** Find the area of the surface generated by revolving the curve $y=\sqrt{9 - x^2}$ for $-1 \leq x \leq 1$ about the y-axis. 2. **Formula:** The surface area

1. The problem: Learn the product rule for differentiation. 2. The product rule states that if you have two functions $u(x)$ and $v(x)$, the derivative of their product is given by

1. مسئله: مقدار مشتق تابع $$f(x) = \frac{(x+2)^3 (x^2+1)^4}{(x^3+1)^2}$$ را در نقطه $$x=1$$ بیابید. 2. برای مشتق‌گیری از تابعی که به صورت کسر است، از قاعده مشتق کسر استفاده می‌کنیم

1. The problem is to find the limit of a function as the variable approaches a certain value. 2. A common formula used is $$\lim_{x \to a} f(x) = L$$ where $L$ is the value the fun

1. **State the problem:** Find the limit $$\lim_{x \to -2} \frac{\sqrt[3]{x-6} + 2}{x^3 + 8}.$$\n\n2. **Recall the formula and important rules:** The limit involves a cube root and

1. **State the problem:** Find the limit as $x$ approaches 3 of the expression $$\frac{\sqrt{x+13} - 2\sqrt{x+1}}{x^2 - 9}.$$\n\n2. **Recall the formula and rules:** The denominato

1. **Problem Statement:** Explain why the expression $1^\infty$ is an indeterminate form in calculus, using the example $$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x$$ and p

1. **State the problem:** We want to find the limit $$\lim_{x \to -\infty} \left( \sqrt{x^2 - 5x + 1} - 2x \right).$$

1. The problem is to find the limit $$\lim_{x \to -\infty} \frac{5x + 1}{2x}$$. 2. The formula for limits involving rational functions as $x$ approaches infinity or negative infini