∫ calculus

1. **Problem Statement:** Evaluate the Laplace transforms of the given functions. 2. **Recall the Laplace transform definition:**

1. The question C asks you to show the analytic procedure for the solutions of parts a and b. 2. This means you need to manually find the derivatives of the given functions using d

1. **Problem Statement:** Find the first and second derivatives of the following functions: i. $y(x) = 5x^2$

1. **مقدمة للمشكلة:** لدينا الدالة $$f(x) = \frac{2a - x^2}{x}$$ حيث $$a > 0$$ و $$x \neq 0$$.

1. The problem asks if the Intermediate Value Theorem (IVT) can be used to conclude that the equation $g(x) = -\frac{3}{4}$ has a solution for $x$ in the interval $[100, 101]$. 2.

1. **Problem Statement:** Find the first, second, and third derivatives of the functions: i. $y(x) = e^{6x^2 + 3x - 2}$

1. We need to find the Laplace transform of each given function. The Laplace transform of a function $f(t)$ is defined as $$\mathcal{L}\{f(t)\} = \int_0^\infty e^{-st} f(t) \, dt.$

1. **Problem Statement:** Find the first, second, and third derivatives of the function $$y(x) = e^{6x^2 + 3x - 2}$$ with respect to $$x$$. 2. **Formula and Rules:** The derivative

1. **State the problem:** Find the limit $$\lim_{x \to -2} \frac{x^2 - 4}{x + 2}$$. 2. **Recall the formula and rules:** When direct substitution leads to an indeterminate form lik

1. **State the problem:** Find the limit $$\lim_{h \to 0} \frac{f(z_0+h) - f(z_0)}{h}$$ where $$f(z) = \frac{2z - 1}{3z + 2}$$. 2. **Recognize the limit:** This limit is the defini

1. **Problem Statement:** We want to find the nth order derivative of the exponential function $e^{ax}$, where $a$ and $x$ are constants and $n$ is a positive integer. 2. **Formula

1. **State the problem:** We want to find the limit $$\lim_{x \to 0} \frac{\sinh x - \sin x}{x^3}$$ using L'Hopital's rule. 2. **Recall definitions and formulas:**

1. مسئله: تابع $f(x) = \sqrt{x^{4} - x^{7}}$ را در نظر بگیرید و بررسی کنید که آیا در نقاط $x=0$ و $x=\pm 1$ حد دارد یا خیر. 2. ابتدا دامنه تابع را بررسی می‌کنیم. برای اینکه زیر راد

1. مسئله: بررسی حد تابع $$f(x) = \sqrt[3]{x - x^2}$$ در نقطه $$x=1$$ است. 2. فرمول و نکات مهم: برای بررسی حد در یک نقطه، باید حد چپ و حد راست را جداگانه محاسبه کنیم. اگر هر دو حد ب

1. **שאלה 1:** נתונים שלושה טורים A, B, C וטענות I, II, III:

1. **Problem:** Given the number of cars sold after $t$ days as $$N(t) = 4000 + 45t^2 - t^3$$ for $1 \leq t \leq 48$, find the day when the maximum rate of growth of sales occurs.

1. The problem is to understand what a "calculus cat" might mean in a mathematical or educational context. 2. Since "calculus cat" is not a standard mathematical term, it might be

1. We are asked to find the primitive (antiderivative) of the function $$f(x) = \frac{x^3 + 2x}{\sqrt{x^4 + 4x^2 + 2}}$$ over the real numbers. 2. To find the primitive, we look fo

1. **Problem statement:** Find the derivative $\frac{dy}{dx}$ of the function $y = \sqrt{x^2 + 2}$. 2. **Formula and rules:** Use the chain rule for derivatives. The derivative of

1. **Problem:** Find the indefinite integral (antiderivative) of the function $4x^3 - 6x + 9$. 2. **Formula:** The integral of a sum is the sum of the integrals, and the power rule

1. The problem is to understand how the gradients \(\nabla f = (1,1)\) and \(\nabla g = (2x, 2y)\) were calculated. 2. The gradient of a function \(f(x,y)\) is a vector of its part