∫ calculus

1. The problem is to find the derivative of the function $f(x) = \sin(x^3)$.\n\n2. We use the chain rule for differentiation, which states that if $f(x) = \sin(g(x))$, then $f'(x)

1. Problem: Calculate the integrals given in parts a, b, c, and d. 2. Part a: \(\int \sqrt{16x} \sin(1 + x^{3/2}) \, dx\)

1. We are asked to compute limits using the (\epsilon - \delta) definition of limits for the following functions: (a) $\lim_{x \to 1} (x^2 + 1)$

1. The problem is to find the derivative of the function $f(x) = \tan x$. 2. Recall the formula for the derivative of the tangent function: if $f(x) = \tan x$, then

1. Problem: Find the implicit derivatives for the given equations. (c) $$\sqrt{x} + y = x^4 + y^4$$

1. **State the problem:** We need to find the constant $\lambda$ such that the piecewise function $$

1. Асуудлыг тодорхойлох: $f(x) = |x^2 - 9|$ функцийн $x$-ийн ямар утганд дифференциалчлагдахыг олж, $f'(x)$-ийн томъёог бичнэ.

1. **Problem statement:** (a) Find the domain where the function $f(x) = |x^2 - 9|$ is differentiable and find the formula for $f'(x)$.

1. **State the problem:** Find the critical numbers of the function $$p(x) = \frac{x^2 + 2}{2x - 1}$$. Critical numbers occur where the derivative is zero or undefined, but the fun

1. Асуудлыг тодорхойлъя: (с) $y = x^4 + 1$ муруйн 32x - y = 15 шулуунтай параллел шүргэгч шулууны тэгшитгэлийг олох.

1. **State the problem:** Find the critical numbers of the function $$f(x) = 3x^4 + 4x^3 - 12x^2$$. 2. **Recall the formula and rules:** Critical numbers occur where the derivative

1. **State the problem:** We need to determine if the piecewise function $$f(x) = \begin{cases} 0 & \text{if } x < 0 \\ x & \text{if } x \geq 0 \end{cases}$$ is continuous. 2. **Re

1. **Тодорхойлолт:** Функцийн уламжлал гэдэг нь функцийн өөрчлөлтийн хурдыг заадаг бөгөөд тодорхойлолтоор $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ гэж тодорхойлогдоно. 2.

1. Statement of the problem. Decide whether the function $f(x)=0$ for $x<0$ and $f(x)=x$ for $x\ge 0$ is continuous on the real line.

Problem: Determine whether the function $f(x)=\begin{cases}0 & x<0\\ x & x\ge 0\end{cases}$ is continuous. 1. Recall the definition: a function is continuous at a point $a$ if thre

1. Бодлогыг тодорхойлъя: $F(x) = \frac{5x}{1+x^2}$ функцийн $x=2$ дээрх дериватив $F'(2)$-ийг олж, үүнийгээ ашиглан $y=\frac{5x}{1+x^2}$ муруйг $(2,2)$ цэг дээр шүргэгчийн тэгшитгэ

1. **State the problem:** Differentiate the function $y = x^2$ with respect to $x$. 2. **Recall the differentiation rule:** The power rule states that if $y = x^n$, then $\frac{dy}

1. The problem asks to explain why each piecewise function is discontinuous at the given point $a$ and to sketch the graph. 2. For each function, we check continuity at $x=a$ by ve

1. **Differentiate** $y = x^2$. Using the power rule $\frac{d}{dx} (x^n) = nx^{n-1}$, we get

1. **Problem statement:** We want to find a number $\delta$ such that if $|x - 4| < \delta$, then $|\sqrt{x} - 2| < 0.4$. This means we want to control how close $\sqrt{x}$ is to 2

1. **Problem Statement:** We are given a function $G(x)$ graphed as a sinusoidal wave oscillating between $-2$ and $2$ on the interval approximately $[0,5]$. We need to find: (a) $