∫ calculus

1. **State the problem:** Find the limit $$\lim_{x\to 1} \frac{\sqrt{x+3} - \sqrt{2x-1}}{x - 1}$$

1. **Stating the problem:** We want to find the limit $\lim_{z \to n\pi} \frac{z - n\pi}{\sin z}$. 2. **Substitution check:** Direct substitution yields $\frac{n\pi - n\pi}{\sin(n\

1. Najprv si uvedomme vzorec pre Taylorovu radu funkcie $\sin x$ okolo bodu $z$: $$\sin x = \sin z + \cos z (x-z) - \frac{\sin z}{2!} (x-z)^2 - \frac{\cos z}{3!} (x-z)^3 + \cdots$$

1. Stating the problem: We analyze multiple limit problems based on given piecewise and plotted graphs to find values of $a$ or certain limits and function values. 2. For problem 1

1. Evaluate $$ \int \frac{x^3}{\sqrt{x^2 + 1}} \, dx $$. 2. Evaluate $$ \int \sin^2{(x)} \cos^2{(x)} \, dx $$.

1. **Problem (a):** Evaluate $$\int \frac{x^3}{\sqrt{x^2 + 1}} \, dx$$ Step 1: Use substitution. Let $$u = x^2 + 1$$, so $$du = 2x \, dx$$ which gives $$x \, dx = \frac{du}{2}$$.

1. **Problem (a):** Evaluate the integral $$\int_0^1 \frac{\ln(x)}{\sqrt{x}} \, dx$$. 2. Let us use the substitution $$x = t^2$$, then $$dx = 2t \, dt$$ and $$\sqrt{x} = t$$. Also,

1. **State the problem:** We need to find the derivative of the function $$y = \sin^{-1}\left(\cos(2x^3 - 3x - 5)\right).$$ 2. **Recall the derivative formula:** If $$y = \sin^{-1}

1. Problem statement: Find the derivative of $y = \sin^{-1}(\cos(2x^3 - 3x - 5))$. 2. Strategy: We treat this as a composition $y = f(g(h(x)))$ where $h(x)=2x^3-3x-5$, $g(t)=\cos t

1. The problem is to find the points on the curve $y = x^3 + 3x^2 - 9x + 4$ where the tangent line is horizontal. 2. A tangent line is horizontal where the derivative $y'$ is zero.

1. **State the problem:** Find the derivative of the function $$r = \sin(0^2) \cos(20)$$ with respect to the variable (assumed as $x$). 2. **Simplify the function:** Since $0^2 = 0

1. The problem asks to produce the power series for $\cos^2(2x)$ up to the term in $x^6$. 2. Recall the double-angle identity:

1. Differentiate $f(x) = x^2 + \frac{1}{x^2}$. - Rewrite $\frac{1}{x^2}$ as $x^{-2}$.

1. We are given the function $$g(t) = \frac{(1 + \sin 3t)^{-1}}{3 - 2t} = \frac{1}{(1 + \sin 3t)(3 - 2t)}$$ and asked to find its derivative with respect to $t$. 2. Rewrite functio

1. Use the definition of the derivative to find the derivative of the following functions: 1.1. Given $V(t) = \sqrt{14 + 3t}$.

1. **Problem 1:** Find the range of values for $k$ such that $f(x) = 1 - 2kx + kx^2 - x^3$ is decreasing over all real numbers. 2. **Step:** To determine when $f$ is decreasing eve

1. The problem states that the function \( f(X) = aX^3 - 2X^2 + 4 \) has an inflection point at \( x = \frac{1}{3} \). We need to find the value of \( \frac{f(1)}{f'(1)} \). 2. Sta

1. **Problem:** Use the definition of the derivative to find the derivatives of the functions: - $$V(t) = \sqrt{14 + 3t}$$

1. The problem is to find the derivative of the function $$g(t) = \frac{(1 + \sin 3t)^{-1}}{3 - 2t}$$. 2. Rewrite $g(t)$ as $$g(t) = \frac{1}{(1 + \sin 3t)(3 - 2t)}$$ for clarity.

1. المشكلة: لدينا الدالة $$f(x) = \frac{1}{\sqrt[3]{4x^2 - 5}}$$ ونريد إيجاد مشتقتها $$f'(x)$$. 2. كتابة الدالة بصيغة أسية لتسهيل الاشتقاق:

1. **Statement of the problem:** Find the derivative of the function: $$f(x) = \left(\sqrt[3]{4x^{2} - 5}\right)^{-1} = \frac{1}{(4x^{2} - 5)^{1/3}}$$