∫ calculus

1. The problem asks us to find $$\lim_{x \to \infty} \cos\left(x^{2} + e^{\frac{x!}{2}}\right)$$. 2. Note that $x!$ (factorial of $x$) grows extremely fast as $x$ increases. Theref

1. The problem is to analyze or work with the implicit equation given by $$x e^y - 3 y \sin x = 1$$. 2. This is an implicit relation between $x$ and $y$, which does not easily solv

1. The critical value approach typically involves finding where the derivative of a function equals zero or does not exist, indicating potential maximum, minimum, or inflection poi

1. نُعطى الدالة $$f(x) = x^2 + 5x$$. 2. المطلوب هو إيجاد معادلة المماس للمنحنى حيث يكون المماس عموديًا على مستقيم يميل بزاوية $$\frac{\pi}{4}$$ مع محور $$x$$ السالب.

1. Pernyataan masalah: Cari nilai limit $$\lim_{x\to 2} \frac{x-2}{x^2 + 3 - 10}$$ ketika $$x$$ mendekati 2. 2. Sederhanakan penyebut: $$x^2 + 3 - 10 = x^2 - 7$$.

1. نبدأ بتحديد المعادلة المعطاة للمنحنى: $$f(x) = x^2 + 5x$$. 2. نُريد إيجاد معادلة المماس للمنحنى بحيث يكون المماس عمودياً ويُميل بزاوية $\frac{\pi}{4}$ على محور $x$ السالب.

1. **State the problem:** We want to find the second Taylor polynomial of the function $f(x) = \sqrt[4]{x} = x^{\frac{1}{4}}$ centered at $x = 7$. 2. **Find the derivatives:**

1. The problem is to analyze and understand the function $y = xe^{-2x}$. 2. This function is a product of $x$ and an exponential decay $e^{-2x}$, which means it will grow initially

1. The problem is to understand if I know how to differentiate functions in calculus. 2. Differentiation refers to finding the derivative of a function, which gives the rate at whi

1. **Stating the problem:** Find the derivative of the two given functions using the Chain Rule. ### Part (a)

1. Let's state the problem: We need to show that the function $g(x) = x^3 - 6x^2 + 18x - 2$ is always increasing. 2. To determine whether $g(x)$ is always increasing, we look at it

1. Evaluate each limit: (a) \( \lim_{x \to 2} \frac{\sqrt{x} - \sqrt{2}}{x^2 - 2x} \)

1. Stated problem: Find the equation of the tangent line to the curve $y = \log_e \sqrt{1} = \sin 2x$ at the point where $x = \frac{\pi}{2}$. 2. Simplify the function: $\log_e \sqr

1. **Stating the problem:** We are given the function $f(x) = x^2 - 1$ and a point $x_0 = -1$.

1. We are asked to evaluate the integral $$\int_0^2 \ln(x^2 + 1) \, dx$$. 2. For the first integral, we use integration by parts. Let $$u = \ln(x^2 + 1)$$ and $$dv = dx$$.

1. State the problem: We have the function $f(x) = x^2 - 1$ and a point $x_0 = -1$.

1. The problem is to derive the derivative of a function $f(x)$ using first principles, which means using the definition of the derivative as a limit. 2. The definition of the deri

1. **Evaluate** $\int_{-1}^1 \frac{|x|}{x} \, dx$, $x \neq 0$. Since $\frac{|x|}{x} = -1$ for $x < 0$ and $1$ for $x > 0$, split integral:

**Problem:** Find the derivative of the following functions. **2.3** $$f(x) = \left(\sqrt[3]{x} - \frac{2}{5\sqrt{x^6}}\right)^2$$

1. The problem asks to find the value of $k$ for which the function $$f(x) = \begin{cases} \frac{1-\cos 4x}{x^2}, & x < 0 \\ k, & x = 0 \end{cases}$$

1. Problem: Find the first order derivative of $f(x) = x^5$ using the first principle (definition of derivative). Step 1: The first principle of differentiation defines the derivat