∫ calculus

1. **Problem statement:** Given the function $f(x) = x^3 - 4x^2 - 4x - 1$, find the first derivative $f'(x)$. 2. **Formula and rules:** The derivative of a function $f(x)$ with res

1. **State the problem:** Find the area under the curve $$y = \frac{x^2 + 3}{4x - x^2}$$ from $$x=1$$ to $$x=3$$. 2. **Formula used:** The area under a curve from $$a$$ to $$b$$ is

1. **Problem statement:** Find the volume of the solid formed by rotating the region under the curve $$y=\frac{5}{x^2+5x+6}$$ from $$x=0$$ to $$x=1$$ about the x-axis. 2. **Formula

1. **State the problem:** Find the equation of the tangent line to the curve $y = f(x) = -2 \sin^2(x)$ at $x = \frac{11\pi}{6}$.\n\n2. **Formula and rules:** The tangent line at $x

1. **State the problem:** Find the equation of the tangent line to the function $f(x) = 3 \sec(x)$ at $x = 0$. 2. **Recall the formula for the tangent line:** The tangent line to $

1. **State the problem:** Evaluate the limit $$\lim_{x \to \infty} \frac{4x^2 + 20x}{5x^2 + 18}$$. 2. **Recall the rule for limits of rational functions:** When $x$ approaches infi

1. **State the problem:** We have a function $h(s)$ representing the height above sea level of a road at position $s$ miles, with domain $[0,10]$. The function has a single critica

1. **Stating the problem:** We are given limits of a function $f(x)$ approaching points $a=2$ and $b=-1$ from the left and right, as well as limits at infinity. We need to find: a)

1. **State the problem:** Differentiate $\sin^3(3x^2 - 5)$ with respect to $x$. 2. **Recall the formula:** If $y = [f(x)]^n$, then by the chain rule, $\frac{dy}{dx} = n[f(x)]^{n-1}

1. **State the problem:** Find the area of Region S bounded by the curves $f(x) = 2 - \frac{1}{4}x^2$ and $g(x) = -\cos\left(\frac{\pi}{3}x\right)$ from $x=0$ to $x=2$.

1. **Problem statement:** Find the area of region S bounded by $f(x) = -\cos\left(\frac{\pi}{3} x\right)$, $g(x) = 2 - \frac{1}{4} x^2$, the y-axis ($x=0$), and the vertical line $

1. نبدأ بكتابة المعادلة المعطاة: $$y = 2 + \sin x$$ 2. نحسب المشتقة الأولى بالنسبة لـ $x$:

1. نبدأ بكتابة الدالة المعطاة: $$h(x) = \cos(ax) - \sin(ax)$$ 2. نحسب المشتقة الأولى لـ $$h(x)$$ باستخدام قواعد الاشتقاق للدوال المثلثية:

1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$$. 2. **Recall the formula and rules:** When direct substitution leads to an indeterminate form like

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{\tan(6x)}{\sin(2x)}.$$\n\n2. **Recall important formulas and rules:**\n- As $x \to 0$, $\tan x \approx x$ and $\sin

1. **State the problem:** We want to approximate the area under the curve of the function $f(x) = \frac{3}{x+1}$ from $x=0$ to $x=3$ using a Riemann sum with 3 rectangles and right

1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$$. 2. **Recall the formula and rules:** The expression is a rational function. Direct substitution o

1. **State the problem:** Find the derivative $y'$ of the function $$y = \ln(\cos(x^2 + 1)).$$ 2. **Recall the formula:** The derivative of $\ln(u)$ with respect to $x$ is $$\frac{

1. **State the problem:** Find the average value of the function $g(x) = 1 - 6x^2$ on the interval $[-1, 3]$. 2. **Formula for average value:** The average value of a function $f(x

1. **Planteamiento del problema:** Queremos aproximar $\sin(0.12)$ usando un polinomio de Taylor de orden 2.

1. **State the problem:** We need to find values of $a$ and $b$ such that the function $$g(x) = \begin{cases} a^2 x + 2 & \text{if } x > 3 \\ 5 & \text{if } x = 3 \\ x^2 - b x + a