∫ calculus

1. **Problem Statement:** We will explore key concepts in calculus including continuity, differentiability, chain rule, derivatives of inverse trigonometric functions, implicit dif

1. Find the derivative of each function: 1) Given $y = \ln(\sin^{-1}(2x))$.

1. The problem is to evaluate the definite integral $$\int_a^b f(x)\,dx$$ where $f(x)$ is a function and $a$, $b$ are the limits of integration. 2. The definite integral represents

1. **Problem:** Determine if the function $$y=\frac{7}{6-x}-9$$ is decreasing over its entire domain. 2. **Step 1:** Find the derivative $$y'$$ to analyze increasing/decreasing beh

1. **State the problem:** We have differentiable functions $A(t), B(t), C(t), D(t)$ related by the equation $$AB = \log(C^2 + D^2 + 1).$$

1. **State the problem:** Find the limit $$\lim_{x \to -\infty} \frac{\sqrt{5x - 2} - 2}{x + 3}$$. 2. **Analyze the expression:** As $x \to -\infty$, the term $5x - 2$ inside the s

1. The problem asks to state one property of a discontinuous function. 2. A discontinuous function is a function that is not continuous at one or more points in its domain.

1. **State the problem:** We have a piecewise function $$g(t) = \begin{cases} t - 2, & t < 0 \\ t^2, & 0 \leq t \leq 2 \\ 2t, & t > 2 \end{cases}$$

1. Problem 21: Find $$\lim_{y \to 6^+} \frac{y+6}{y^2-36}$$. 2. Factor the denominator: $$y^2-36 = (y-6)(y+6)$$.

1. نبدأ بحساب د(س) = طا (ص - س) - (π - ص) ونريد د(π / ٦). 2. نلاحظ أن د(س) دالة تعتمد على س و ص، ولكن بدون قيمة ص لا يمكننا حساب د(π / ٦) بدقة.

1. Problem 17: Find $$\lim_{x \to 3} \frac{x}{x-3}$$. Step 1: Substitute $x=3$ directly into the expression:

1. Problem 17: Find $$\lim_{x \to 3} \frac{1}{x-3}$$. As $$x$$ approaches 3, the denominator $$x-3$$ approaches 0.

1. The problem asks to find all intervals where the function $f(x)$ is concave down on the open interval $(-9,9)$. 2. Recall that concavity is determined by the second derivative $

1. Problem 13: Find $$\lim_{t \to 2} \frac{t^3 + 3t^2 - 12t + 4}{t^3 - 4t}$$. 2. Substitute $t=2$ directly:

1. The problem is to find the derivative of the function $f(t) = te^{-2t}$. 2. We will use the product rule for differentiation since the function is a product of two functions: $t

1. **State the problem:** We want to evaluate the integral $$\int y e^y \, dy$$. 2. **Choose a method:** Use integration by parts, where we let:

1. **Problem statement:** We are given several limit problems and a function $f(x) = ax + b - \sqrt{x^2 + 1}$ to analyze. 2. **Inequality bounds:** For $m \leq \frac{1}{2 - \sin x}

1. **Evaluate the limits:** a) \(\lim_{x \to 1} \frac{x^3 - 1}{x^2 - 1}\)

1. The problem is to find the derivative of the function $f(x) = e^{-x^2}$.\n\n2. Recall the chain rule for derivatives: if $f(x) = e^{g(x)}$, then $f'(x) = e^{g(x)} \cdot g'(x)$.\

1. **Find** $\frac{dy}{dx}$ for $$ y = \frac{x}{(x^6 - 2)^3} \cdot \left( \frac{x^2 - 3}{\sqrt{x+1}} \right)^{\frac{4}{3}} \left(2 + \frac{3}{x}\right)^7 \sqrt{x^2 + 2 + x^2} $$

1. The problem is to find the derivative of the function $$f(s) = \sqrt{s^3 + 1}$$. 2. Rewrite the square root as a power: $$f(s) = (s^3 + 1)^{\frac{1}{2}}$$.