∫ calculus

1. We are asked to find the limit: $$\lim_{x \to \left(\frac{1}{2}\right)^-} \frac{\ln(1 - 2x)}{\tan(\pi x)}.$$\n\n2. First, note the behavior of the function inside the logarithm

1. **Problem Statement:** Find the value of $x$ where the slope of the tangent to the curve $y = x^2 + 3x + 2$ is equal to 7. 2. **Formula and Rules:** The slope of the tangent to

1. The problem is to find the value of $x$ where the slope of the tangent to a curve is given or needs to be determined. 2. The slope of the tangent line to a curve at a point is g

1. The problem is to find the critical points of a function. 2. Critical points occur where the derivative of the function is zero or undefined.

1. **State the problem:** We want to analyze the function $y = -x^3 + 6x^2$ and understand its graph using the slope of the tangent line. 2. **Formula for slope of tangent line:**

1. The problem is to find the sixth order derivative of the function $y = \arctan(x)$. 2. Recall that the first derivative of $\arctan(x)$ is given by the formula:

1. Let's first state the problem: Determine if $x=0$ is a critical point of a given function. 2. Recall the definition: A critical point of a function $f(x)$ occurs where the deriv

1. **State the problem:** We want to find the intervals on which the function $$f(x) = \cos^2(4x) + 3 \cos(4x)$$ is increasing or decreasing for $$0 < x < \frac{\pi}{2}$$. 2. **Fin

1. **State the problem:** We need to find the derivative of the composite function $ (f \circ g)(x) = f(g(x)) $ at $ x=2 $. Given functions are $ f(x) = 3x^{2} - 2x + 1 $ and $ g(x

1. **State the problem:** Evaluate the limit $$\lim_{x \to 0} \frac{3x}{4 - 4\sec^2(5x)}.$$\n\n2. **Recall the formula and important rules:** The secant function is $$\sec(\theta)

1. **Problem statement:** We need to find the instantaneous velocity of an object moving in a straight line at time $t=3$ seconds, given its position function $$s(t) = 8\sqrt{t+1}

1. **State the problem:** Evaluate the integral $$\int \cos 4\theta \sqrt{2} - \sin 4\theta \, d\theta$$. 2. **Rewrite the integral:** Separate the integral into two parts:

1. The problem is to find the derivative of the inverse function of a given function. 2. The formula for the derivative of the inverse function $f^{-1}$ at a point $y$ is:

1. **بيان المسألة:** لدينا الدالة $f$ معرفة كما يلي:

1. **State the problem:** We need to find the limit of the function $f(x)$ as $x$ approaches $-3$, i.e., $\lim_{x \to -3} f(x)$. 2. **Recall the definition of a limit:** The limit

1. **Stating the problem:** We need to find the integral $$\int (x^2 + 4x + 7) \sin^{-1}(2x - 1) \, dx$$. 2. **Formula and approach:** This is an integral involving a product of a

1. **Statement of the problem:** We have a piecewise function defined as: $$f(x) = \begin{cases} e^x - 1 & x > 0 \\ \ln(x + 1) & x = 0 \\ x^3 e^{2x} & x < 0 \end{cases}$$

1. **State the problem:** We have the curve $$y = x^3 - 4x^{5/2} - kx^{1/2} + 28x - 44$$ for $$x \geq 0$$, where $$k$$ is a positive constant.

1. **Statement of the problem:** Calculate the derivatives of the following functions:

1. **State the problem:** Find the limit of the piecewise function $$f(x) = \begin{cases} x^2 & \text{if } x \leq 2 \\ 3x - 3 & \text{if } x > 2 \end{cases}$$

1. **State the problem:** Find the derivative of the function $$f(x) = \frac{e^{2x}}{x^{1/2}(x^2+5)^{1/4}}$$. 2. **Rewrite the function for clarity:**