∫ calculus

1. Problem: Find the derivatives and analyze curves based on given problems. 2. (3a) Let $y = \tan^{-1}\left(\frac{4\sqrt{x}}{1 - 4x}\right)$. Use the chain rule and derivative of

1. **State the problem:** We want to check if the function $$f(x) = \begin{cases} x^3, & x \leq 1 \\ 3x - 2, & x > 1 \end{cases}$$

1. **State the problem:** We are given a piecewise function: $$f(x) = \begin{cases} x^3, & x \leq 1 \\

1. لنبدأ بكتابة المسألة: نريد إيجاد مشتقة الدالة $$y = e^{x^3 - 3x + 1}$$. 2. نستخدم قاعدة السلسلة في التفاضل، حيث أن \(y = e^u\) و\(u = x^3 - 3x + 1\).

**Problem:** Find the expression for $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}$ given different functions for $u$. Specifically,

1. Problem: Given the function $$y = (-3 - 3x^7)^5$$, we need to find its derivative $$f'(x)$$. 2. Use the chain rule to find the derivative. The outer function is $$u^5$$ where $$

1. We are tasked with finding the limit $$\lim_{x\to +\infty} x \cos(x)$$

1. We are asked to find the indefinite integral of the polynomial function $$3x^2 + 7x - 2$$ with respect to $$x$$. 2. Recall the power rule for integration: $$\int x^n dx = \frac{

1. We are asked to evaluate the limit: $$\lim_{x \to +\infty} -x \sqrt{\frac{x^4 - 3}{9x^3 + 2x}}$$

1. **State the problem:** We need to find the shape of the largest rectangular beam that can be cut from a log of fixed size. The strength $S$ of the beam is proportional to the br

1. **State the problem:** Determine if the piecewise function $$f(x)=\begin{cases}4x-1 & \text{if } x<2 \\ 4 & \text{if } x=2 \\ 2x & \text{if } x>2\end{cases}$$ is continuous at $

1. **State the problem:** We need to find the derivative $\frac{dy}{dx}$ of the function $$y=\frac{3+x^{2}}{x+6}.$$\n\n2. **Apply the quotient rule:** For $y=\frac{u}{v}$, $$\frac{

1. The problem describes a function $b(x) = 2g(x) - 5$ and states a limit situation. 2. It says that as the input to $b(x)$ approaches 6 from either side (from values less than 6 a

1. Problem: Evaluate $$\int_0^{\frac{\pi}{2}} \sin^n x \, dx$$ and find specific values for $$n=6$$ and $$n=5$$. Step 1: Use the reduction formula for $$I_n = \int_0^{\frac{\pi}{2}

1. **State the problem:** We want to evaluate the definite integral $$\int_0^{\pi/2} \sin^4 \theta \cos^2 \theta \, d\theta.$$\n\n2. **Use trigonometric identities:** To integrate

1. The problem states that $L = \{a \in [-3, \infty) : \lim_{x \to a^-} f(x) \text{ exists}\}$ and $R = \{a \in [-3, \infty) : \lim_{x \to a^+} f(x) \text{ exists}\}$. We need to f

1. The problem asks to find the equation of the tangent line to the curve $$y = x^{3/2} + 3x + 1$$

1. **Problem Statement:** Understand the difference between partial derivatives and implicit differentiation. 2. **Partial Derivatives:** These are derivatives of multivariable fun

1. The problem is to find the limit of the floor function $\lfloor x \rfloor$ as $x$ approaches 1 from the left and from the right. 2. Recall the floor function $\lfloor x \rfloor$

1. Let's clarify the problem: We want to find the limit of the floor of the function $f(x)$ as $x$ approaches some value $a$. 2. The floor function, denoted $\lfloor y \rfloor$, gi

1. We are asked to find the limit of the function $f(x)=\lfloor x \rfloor$ (the floor function) as $x \to 1$. 2. Recall that the floor function $\lfloor x \rfloor$ gives the greate