∫ calculus

1. The problem is to find the partial derivative with respect to $x$ of the function $fff$. 2. To differentiate a function with respect to $x$, we apply the partial derivative oper

1. **Problem Statement:** Find the x- and y-intercepts, all asymptotes, first and second derivatives, extreme points, intervals of increase/decrease, inflection points, and concavi

1. **Problem a:** Evaluate $$\lim_{q \to 1} \frac{q^3 - 1}{4(q^2 - 1)}$$. 2. **Formula and rules:** When direct substitution leads to an indeterminate form like $$\frac{0}{0}$$, fa

1. **Problem Statement:** Find the $x$- and $y$-intercepts, all asymptotes, first and second derivatives, extreme points, intervals of increase/decrease, inflection points, and con

1. Problem 1: Match the functions in Column A with their types in Column B and their graphs in Column C. 1.1. Given functions: $f(x)=3\ln(x)-2$, $f(x)=10-4x$, $f(x)=2^x+7$, $f(x)=x

1. **Problem Statement:** We have a piecewise function $$g(t) = \begin{cases} \frac{1}{2}t + 1 & \text{if } t < 2 \\ \sqrt{t} - 2 & \text{if } t \geq 2 \end{cases}$$ We need to:

1. **Problem Statement:** Find the derivative $f'(x)$ for the function $f(x) = x + \sqrt{x}$. 2. **Recall the derivative rules:** The derivative of $x$ is 1. For $\sqrt{x}$, rewrit

1. **State the problem:** We need to find $\frac{d^5y}{dx^5}$ for the function $y=3x^4+2x^3-4x^2+x+5$. 2. **Recall the formula:** The $n$th derivative of a polynomial term $ax^m$ i

1. **State the problem:** Find $\frac{d^5y}{dx^5}$ for the function $y=3x^4+2x^3-4x^2+x+5$.\n\n2. **Recall the formula:** The $n$th derivative of a polynomial term $ax^m$ is given

1. **Problem 15:** Solve the differential equation $$\frac{dy}{dx} = \frac{1}{x - x^3}$$ for the general solution. 2. **Step 1:** Factor the denominator:

1. **Problem statement:** Calculate the first three terms of each sequence $(u_n)_n$ defined by the given general term, then find expressions for $u_{2n}$ and $u_{n+1}$. ---

1. **Problem Statement:** Find the second derivative $f''(x)$, the ratio $\frac{f'(x)}{f(x)}$, and the expression $f''(x) - 2$ for the function $f(x)$ defined by the triangle verti

1. Problem 17: Given $F(x) = \int_0^{x^2} \sqrt{2t} + 2 \, dt$, find $F'(1)$. 2. To find $F'(x)$ when $F(x)$ is defined as an integral with a variable upper limit, use the Leibniz

1. **State the problem:** We want to find the limit as $x$ approaches infinity of the function $$\lim_{x \to \infty} \frac{-5x + 7}{2x^4 + x^2 + 8x}.$$\n\n2. **Recall the rule for

1. **State the problem:** Evaluate the limit \( \lim_{z \to 1} \frac{6 - 3z + 10z^2}{-2z^4 + 7z} \). 2. **Recall the limit evaluation rule:** If direct substitution results in a de

1. The problem asks to define the term limit $L$ of a continuous function $f(x)$ about a point $x=p$. 2. The limit of a function $f(x)$ as $x$ approaches $p$ is the value that $f(x

1. **Planteamiento del problema:** Calcular el área bajo la curva de la función $f(x) = 3x^2 - 4x + 1$ en el intervalo $[0,3]$ usando la integral definida $$\int_0^3 (3x^2 - 4x + 1

1. **State the problem:** We want to find the maximum and minimum profit values for the profit function $$f(x) = 4x^5 - 20x^4 + 20x^3 + 20$$ where $x$ is the number of units sold (

1. **Problem Statement:** (a) Explain the difference between 'Derivative' and 'Differentiation'.

1. The problem is to evaluate the definite integral $$\int_{0}^{\frac{\pi}{4}} (2 - \tan x)^2 \, dx.$$\n\n2. We start by expanding the integrand using the formula $$(a - b)^2 = a^2

1. سوال شما درباره دیفرانسیل است، که شاخه‌ای از ریاضیات است که به مطالعه نرخ تغییرات و مشتق‌ها می‌پردازد. 2. در دیفرانسیل، ما معمولاً به دنبال یافتن مشتق تابعی هستیم که نشان‌دهنده