∫ calculus

1. **State the problem:** Find the limit as $n$ approaches positive infinity of the expression $$\frac{3n^2 + 7n + 1}{3n^2 - n + 2}$$. 2. **Recall the rule for limits of rational f

1. **State the problem:** We are given the function $$f(x) = -2x^4 + 36x^2$$ and need to find all intervals where $$f$$ is both increasing and concave down. 2. **Recall definitions

1. **Problem:** Find the $n$th derivative of the function $f(x) = \cos(ax + b)$.\n\n2. **Formula and rules:** The derivative of $\cos(u)$ with respect to $x$ is $-\sin(u) \cdot \fr

1. **State the problem:** We want to evaluate the integral $$\int \sec^3\left(\frac{1}{2}x + 3\right) \, dx.$$\n\n2. **Recall the formula and approach:** The integral of $$\sec^3 u

1. **Problem statement:** Find the coordinates of all absolute maximums and minimums of the function $$f(x) = (x^2 - 3)^{\frac{2}{3}}$$ on the interval $$[-8, 2]$$. 2. **Formula an

1. نبدأ بحل السؤال: حساب \( \lim_{x \to 1} (x - 1) \sqrt{x - 3} \). 2. نلاحظ أن الدالة \( (x - 1) \sqrt{x - 3} \) تحتوي على جذر تربيعي \( \sqrt{x - 3} \) والذي يتطلب أن يكون \( x -

1. **Problem Statement:** Find the first derivative $f'(x)$ and second derivative $f''(x)$ at $x=1$ using numerical differentiation from the given table: $$\begin{array}{c|cccccc}

1. The problem is to find the derivative of the function $u v$ where $u$ and $v$ are functions of $x$. 2. We use the product rule for differentiation, which states:

1. **State the problem:** We need to find the local maxima and minima of the function $$f(x)=x^3-6x^2+9x+1$$ using the second derivative test. 2. **Find the first derivative:** The

1. **State the problem:** Find the absolute minimum and absolute maximum of the function $$f(x) = x^2 + 4x + 5$$ on the interval $$[-3, 1]$$. 2. **Formula and rules:** To find abso

1. We are asked to find the absolute minimum and maximum of the function $f(x) = x^2 + 4x + 5$ on the interval $-3 \leq x \leq 1$. 2. The formula to find absolute extrema on a clos

1. **State the problem:** We want to find the integral $$\int \log(x) \, dx$$. 2. **Recall the integration by parts formula:**

1. **Problem statement:** Find the equation(s) of the tangent(s) to the curve $$x^2 y^2 + y^2 + x^3 = 5$$ at points where $$x=1$$ using implicit differentiation. 2. **Formula and r

1. **Problem statement:** Evaluate the limits (i) $$\lim_{x \to 0} \left( \frac{1}{\sin x} - \frac{1}{x} \right)$$

1. **State the problem:** A conical vase has a height of 20 cm and a radius of 3 cm at the top. It is being filled with liquid at a rate of 10 cm³/s. We need to find how fast the h

1. The problem asks for the meaning of the notation $$\frac{\partial^2 f}{\partial y \partial x}$$ in terms of the order of partial differentiation. 2. This notation represents a s

1. **State the problem:** Find the partial derivative $\frac{\partial z}{\partial x}$ for the function $$z = x^4 \sin(x y^3).$$ 2. **Recall the formula:** To find $\frac{\partial z

1. The problem asks what the partial derivative $\frac{\partial I}{\partial T}$ represents when $I$ is a function of temperature $T$ and humidity $H$, written as $I = f(T, H)$. 2.

1. **State the problem:** Find the derivative of the function $$y = \frac{(x+2)^2}{(x+1)^3 (x+3)^4}$$. 2. **Formula and rules:** We will use the quotient rule for derivatives: $$\l

1. We are asked to find the derivative of the function \( y = \frac{\sqrt{9+3x-x^{2}}}{e^{x}} \). 2. Recall the quotient rule for derivatives: if \( y = \frac{u}{v} \), then \( y'

1. **State the problem:** We are given a cube whose side length is increasing at a rate of 12 m/s. We need to find the rate at which the volume of the cube is increasing when the s