∫ calculus

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

1. The problem asks for the value of $k$ such that the function $$f(x) = \begin{cases} kx + 1, & x \leq \pi \\ \cos x, & x > \pi \end{cases}$$ is continuous at $x = \pi$. 2. For co

1. **State the problem:** Find the limit \(\lim_{x \to 0} \frac{(x+1)^5 - 1}{x}\).

1. Differentiate each function as requested. **(a) $y = e^{\sin^2 5x}$**

1. Differentiate (a) $y = e^{\sin^2 5x}$: Use the chain rule: $\frac{dy}{dx} = e^{\sin^2 5x} \cdot \frac{d}{dx}(\sin^2 5x)$.

1. Stating the problem: We need to find various limits involving functions $f(x)$ and $g(x)$ using their graphs. 2. Analyze each limit step-by-step:

1. The problem asks us to find \(\frac{dy}{dx}\) given that \(\tan(x y^{2}) = (2x + y)^{3}\). We must differentiate both sides implicitly with respect to \(x\). 2. To differentiate

1. The problem states that for a differentiable function $y(x)$, the equation $x + y^4 = 10$ holds with $y \neq 0$. We want to find $\frac{dy}{dx}$. 2. Differentiate both sides imp

1. The problem involves the expression $x^2 \int_a^b f(x)\,dx$ where $x$ is a variable and $\int_a^b f(x)\,dx$ represents the definite integral of the function $f(x)$ from $a$ to $

1. The problem is to find the $x$-coordinates where the gradient (derivative) of the curve $y=4x^3 - 8x + 5$ equals $\frac{1}{3}$. 2. First, find the derivative of $y$ with respect

1. The problem asks to evaluate the indefinite integral $$\int xe^{3x} \, dx$$. 2. To solve this, we use integration by parts. Recall the formula: $$\int u \, dv = uv - \int v \, d

1. Problem 1: Differentiate the function $y = \frac{3}{\sqrt{5x - 2}}$ using the Chain Rule. 2. Rewrite the function in exponent form: $$y = 3(5x - 2)^{-\frac{1}{2}}$$.