∫ calculus

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

1. **State the problem:** Find the limit as $x$ approaches 3 of the function $$\frac{x^2 - 9}{x - 3}$$. 2. **Recall the formula and rules:** The limit of a rational function where

1. **State the problem:** Find the limit as $x$ approaches 2 of the function $$\frac{x^2 - 4}{x - 2}$$. 2. **Recall the formula and rules:** The direct substitution of $x=2$ gives

1. **Problem 7:** Find the minimum value of the function $g(x) = x^5 - 5x^3 - 20x$ on the interval $[0,3]$. 2. To find the minimum, we first find the critical points by computing t

1. **State the problem:** Find the limit as $x$ approaches 0 of the expression $$\frac{x^2 - x}{\sqrt{3} - \sqrt{3} - x}.$$ 2. **Simplify the denominator:** Notice that $\sqrt{3} -

1. **State the problem:** Differentiate the function $$f(x) = \frac{\ln x}{x^3}$$. 2. **Rewrite the function:** Using the property $$\frac{1}{x^3} = x^{-3}$$, rewrite the function

1. **Problem:** Find the area between $y = e^x$ and $y = 1$ over the interval $[0,2]$. 2. **Formula:** The area between two curves $y = f(x)$ and $y = g(x)$ over $[a,b]$ is given b

1. **State the problem:** We need to find the derivative of the function $f(x) = 3x^2 - 7x + 1$ using the limit definition of the derivative: $$f'(x) = \lim_{h \to 0} \frac{f(x+h)

1. **Problem:** Evaluate the integral $$\int \frac{x^2 + 3x + 1}{(x+2)(x-3)^2 (x^2 + 4)^2} \, dx$$ 2. **Formula and rules:** For rational functions with polynomial denominators, us

1. **Problem Statement:** We have a piecewise function \( j(x) \) defined as:

1. **State the problem:** Find the area enclosed by the curves given by the equations: $$2y=5\sqrt{x}, \quad y=5, \quad 2y+3x=8$$

1. **Problem:** Find the derivative of $f(x) = (6x^2 + 7x)^4$ using the chain rule. 2. **Formula:** The chain rule states that if $f(x) = (g(x))^n$, then