∫ calculus

1. **Stating the problem:** We want to evaluate the limit $$\lim_{x\to \frac{\pi}{2}} x \sec^2 x$$. 2. **Recall the definition:** $$\sec x = \frac{1}{\cos x}$$, so $$\sec^2 x = \fr

1. We are asked to evaluate the limit $$\lim_{x \to 3} \frac{x - 1}{x^2 - x - 6}$$. 2. First, let's factor the denominator:

1. **State the problem:** We need to find the limits of the function $$f(x) = \frac{1 - 2x}{x^2 - 1}$$ as $$x$$ approaches $$-1$$ from the left ($$x \to -1^-$$), from the right ($$

1. Evaluate the integral $$\int \frac{3y}{y^2 + 4} \; dy$$ Step 1: Recognize the form suitable for substitution. Let $$u = y^2 + 4$$.

1. Problem: Find $$\lim_{x \to 3} (2x + 1)$$ Solution: Substitute $x=3$ directly since the function is polynomial and continuous.

1. **Problem:** Given $u = f(2x - 3y, 3y - 4z, 4z - 2x)$, prove that $$\frac{1}{2} \frac{\partial u}{\partial x} + \frac{1}{3} \frac{\partial u}{\partial y} + \frac{1}{4} \frac{\pa

1. Use the definition of the derivative to find the derivatives. 1.a. Given $Q(t)=10+5t-t^2$, the definition of derivative is $$Q'(t)=\lim_{h\to0}\frac{Q(t+h)-Q(t)}{h}.$$ Calculate

1. The problem is to find the slope of the tangent line to the curve \(y=x^3+4x-7\) at the point where \(x=2\). 2. The slope of the tangent line at any point is given by the deriva

1. The user has input the expression $\frac{d}{dx}$ which signifies the derivative operator with respect to $x$. 2. This expression alone is not a complete problem but indicates di

1. **State the problem:** We want to find the derivative $\frac{dR}{dq}$ given the function $$R = q \sqrt{1000 - q^2}.$$\n\n2. **Rewrite the expression:** Express the square root a

1. Evaluate the integral $$\int \frac{3y}{5y^2 + 4} dy$$. Step 1: Substitute to simplify the integral. Let $$u = 5y^2 + 4$$, then $$du = 10y dy$$, so $$y dy = \frac{du}{10}$$.

1. **Evaluate** $$\int \frac{3y}{5y^2 + 1} dy$$ Step 1. Let us substitute $$u = 5y^2 + 1$$. Then, $$du = 10y dy$$ or $$dy = \frac{du}{10y}$$.

1. **Stating the problem:** We want to show that for a sequence $a_n > 0$ for all $n$, $\lim_{n\to\infty} a_n = \infty$ if and only if $\lim_{n\to\infty} \frac{1}{a_n} = 0$. 2. **R

1. Let's analyze the function A: $F(x) = x \sin x$. - Domain: $x$ can be any real number since $x$ and $\sin x$ are defined everywhere.

1. نبدأ بمسألة حساب نهاية الدالة عند $x \to 0$ للدالة $f(x) = \sin\left(\frac{3x^2}{5x}\right)$. \n2. يمكن تبسيط داخل دالة الجيب: $$\frac{3x^2}{5x} = \frac{3x}{5}.$$ \n3. إذن تصبح

1. **Problem statement:** Prove Lagrange's theorem (Mean Value Theorem), which states: If a function $f(x)$ is continuous on the closed interval $[a,b]$ and differentiable on the o

1. The problem: find the derivative of $\tan x$ with respect to $x$.\n\n2. Recall the definition: $\tan x = \frac{\sin x}{\cos x}$.\n\n3. Use the quotient rule: if $f(x) = \frac{g(

1. The problem asks us to differentiate the function $y=\tan x$. 2. Recall that the derivative of $\tan x$ with respect to $x$ is a standard derivative result from calculus.

1. The problem is to find the integral of the function $\sec^2(x)$. 2. Recall that the derivative of $\tan(x)$ is $\sec^2(x)$, so the antiderivative (integral) of $\sec^2(x)$ is $\

1. The problem is to find the integral of $\tan^2 x$ with respect to $x$. 2. Recall that $\tan^2 x = \sec^2 x - 1$ from the Pythagorean identity.

1. We are asked to find the integral of the function $\tan x$. 2. Recall that $\tan x = \frac{\sin x}{\cos x}$.