∫ calculus

1. **Problem statement:** Evaluate the following limits and analyze the given functions and expressions step-by-step. ---

1. **State the problem:** Find the equation of a curve passing through the point (1, 1) such that the perpendicular distance of the origin from the normal at any point $P(x,y)$ on

1. **State the problem:** We are given the differential equation $$\frac{dy}{dx} = y e^x$$ with the initial condition $$x=0, y=e$$. We need to find the value of $$y$$ when $$x=1$$.

1. Problem: Find $$\lim_{x \to 2} \frac{x - 2}{4 + 2x}$$. Step 1: Substitute $x = 2$ directly.

1. **State the problem:** We want to find the value of $$\ln y = \lim_{x \to 0} \frac{\tan x - \sin x}{3x^2}$$ and then find $$y$$ itself. 2. **Recall series expansions:** For smal

1. **State the problem:** We have the function $$f(x) = \frac{1}{1-x}$$ for $$0 \leq x < 1$$ and $$f(1) = 0$$. We want to explain why $$f$$ has a minimum value but no maximum value

1. **State the problem:** Evaluate the definite integral $$\int_{-2}^3 (3x^2 - 2x - 12) \, dx$$. 2. **Find the antiderivative:** Integrate each term separately:

1. **State the problem:** Find $\frac{dy}{dx}$ for the implicit function defined by $$e^x - e^y = x - y.$$\n\n2. **Differentiate both sides with respect to $x$: **\nUsing implicit

1. **State the problem:** We need to evaluate the integral $$\int \cot^7(3\theta) \, d\theta$$. 2. **Rewrite the integral:** Recall that $$\cot x = \frac{\cos x}{\sin x}$$, so $$\c

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.