∫ calculus

1. The problem is to find the derivative of the function $$f(x) = \frac{y^2}{x}$$ with respect to $x$. 2. Assume $y$ is a function of $x$, so we use the quotient rule and chain rul

1. **State the problem:** We need to find the integral $$\int \frac{x^2 - 29x + 5}{(x - 4)^2 (x^2 + 3)} \, dx.$$\n\n2. **Set up partial fraction decomposition:** Since the denomina

1. **Problem Statement:** We are given the graph of the derivative function $f'$ of some function $f$. We need to find: - (a) The $x$-values where $f$ has points of inflection.

1. **State the problems:** We have two graphs with their skeletons of function $f(x)$, and we want to match the derivative $f'(x)$ for each based only on the sign of $f'(x)$ in the

1. The problem asks to find the $x$-value where the second derivative $f''(x)$ changes sign from negative to positive. 2. This point is called an inflection point, where the concav

1. **Problem Statement:** Evaluate the integrals: (a) $\int (y^2 + y^{-2})\, dy$

1. **Problem (a):** Calculate $$\int (1 - \frac{1}{w}) \cos(w - \ln(w)) \, dw$$ Step 1: Rewrite the integral

1. **State the problem:** Find critical points, concavity intervals, points of inflection, and classify critical points for $$f(x) = 3x + 3\sin(x)\text{ on }[0,2\pi].$$ 2. **Find t

1. **State the problem:** Compute the limit $$\lim_{x \to \infty} \sqrt[3]{x^3 - 7x^2 - x}$$ using L'Hospital's rule if appropriate. 2. **Analyze the expression:** As $$x \to \inft

1. Evaluate the integral $$\int_0^1 \int_y^1 dx \, dy$$. The inner integral with respect to $x$ is $$\int_y^1 dx = 1 - y$$.

1. Problem a) I: Find the convergence and interval of convergence for the power series: $$\sum_{n=1}^\infty \frac{(-1)^n 10^n}{n!} (x-10)^n$$ 2. Use the Ratio Test to determine co

1. Stating the problem: We need to evaluate the integral $$\int \frac{150^3 - 320^2 + 21\theta}{50 - 4} \, d\theta$$. 2. Simplify the constant denominator: $$50 - 4 = 46$$.

1. **Stating the problem:** We need to compute the integral $$\int \frac{16x^2}{(x-18)(x+6)^2} \, dx.$$

1. State the problem. From the graph we observe a vertical asymptote at $x=0$ with the left branch going to $+\infty$ as $x\to0^-$ and the right branch going to $-\infty$ as $x\to0

1. **Stating the problem:** We analyze the function $f$ based on the graph provided to find the infinite limits, limits at infinity, and equations of vertical and horizontal asympt

1. Given the function $f(x) = y \tan^{-1}(x) - 1$, we need to analyze its behavior as $x$ approaches $+\infty$ and $-\infty$. 2. Recall that the inverse tangent function $\tan^{-1}

1. **Problem a:** Evaluate $$\lim_{n \to \infty} \sqrt{4^n + 5^n}$$ - Step 1: Identify the dominant term inside the square root as $$n \to \infty$$.

1. نحدد المشكلة: نحسب النهاية $$\lim_{x \to \frac{\pi}{4}} \frac{\tan x - 1}{\sin x - \frac{\sqrt{2}}{2}}$$. 2. نعوض مباشرةً بـ $$x = \frac{\pi}{4}$$.

1. The problem is to find the derivative $y'(x)$ of the function $y(x) = x^x$. 2. To differentiate $y = x^x$, first rewrite it using logarithms:

1. **State the problem:** We are given the function $y = x^2 e^{2x}$ and need to find the first derivative $y'$, the second derivative $y''$, and then calculate the price elasticit

1. **State the problem:** Find the limit $$\lim_{x \to +\infty} \frac{3x^2 + 7}{5x - 1}$$.\n\n2. **Analyze the degrees:** The numerator is a polynomial of degree 2 ($3x^2 + 7$) and