∫ calculus

1. **Problem:** Determine if the function $f(x) = x^4 + 3x^2 - 6x + 2$ is continuous at $x=3$. 2. **Formula and rules:** A polynomial function is continuous everywhere. To check co

1. **State the problem:** Differentiate the function $$y = (\sin x)^{x^3}$$ with respect to $$x$$. 2. **Formula and approach:** When differentiating a function of the form $$y = f(

1. **State the problem:** Differentiate the function $$y = (\sin x)^{x^3}$$ with respect to $$x$$. 2. **Formula and approach:** When differentiating a function of the form $$y = u(

1. **Problem statement:** Find the limits of the following functions as $x \to 0$: (i) $\lim_{x \to 0} \frac{(1+x)^n - 1}{x}$

1. **State the problem:** We want to analyze the function $$y(t) = \int_0^t 62.5 \sin(2\pi \cdot 60 \tau) e^{-(t-\tau)} d\tau$$ which represents a convolution integral involving a

1. **State the problem:** Evaluate the definite integral $$\int_0^1 3x \sqrt{1 - x} \, dx.$$\n\n2. **Rewrite the integral:** Note that $$\sqrt{1 - x} = (1 - x)^{1/2}.$$ So the inte

1. **Problem statement:** We want to find the maximum area of a symmetric trapezoid inscribed under the parabola given by the function $$f(x) = -\frac{1}{2}x^2 + 2$$.

1. **Problem Statement:** We have a software development cost function given by $$C(h) = 5000 + 150h - 2h^2 + 0.01h^3$$ where $h$ is the number of developer hours.

1. **Problem Statement:** We have a server response time modeled by the function $$T(n) = \alpha n^2 + \beta n + \gamma$$ where $n$ is the number of concurrent users. We need to fi

1. **State the problem:** Find the derivative of the function $$y = (x^2 + 3)^4 (2x^3 - 5)^3$$. 2. **Formula used:** We will use the product rule and the chain rule.

1. **State the problem:** Find the derivative of the function $$f(\theta) = 80\sin(\theta) + 20$$ with respect to $$\theta$$. 2. **Recall the derivative rules:**

1. The problem is to find the derivative of a function, but the function is not specified. 2. The derivative of a function $f(x)$, denoted $f'(x)$ or $\frac{d}{dx}f(x)$, measures t

1. **Problem 5.1:** Given $f(2)=5$ and $f(0)=1$, evaluate $$\int_0^2 f'(x) U(f(x)) \, dx$$ where $U$ is the unit step function. 2. **Step 1:** Recall the Fundamental Theorem of Cal

1. The problem asks how to get the expression $dl = \frac{d\ell(w)}{dw}$. 2. This expression represents the derivative of a function $\ell(w)$ with respect to the variable $w$.

1. **State the problem:** We need to evaluate the integral $$\int \tan^2(4x) \sec^2(2x) \, dx$$. 2. **Recall formulas and identities:**

المطلوب: دراسة الدالة $f(x)=\frac{2x-3}{x^2+|x|-3}$ وإعطاء تمثيل بياني. 1. المجال.

1. **Problem statement:** Given the function $y = (\sqrt{4x+1})^3$, find the derivative $\frac{dy}{dx}$.

1. **Problem:** What number exceeds its square by the maximum amount? 2. **Step 1: Define the function.** Let the number be $x$. The amount by which the number exceeds its square i

1. **Problem Statement:** Evaluate the double integral $$\int_0^1 \int_0^x y \, dy \, dx$$. 2. **Formula and Explanation:** The integral is over the region where $y$ goes from 0 to

1. **State the problem:** We are given the differential equation $$y'' = 32^3$$ with initial conditions $$y(4) = 1$$ and $$y'(4) = 4$$. We need to find the function $$y(x)$$. 2. **

1. Find the derivatives of the following functions: 1.a. Given $F(x) = 2x^2(3x^4 - 2)$, use the product rule: $\frac{d}{dx}[u v] = u' v + u v'$. Here, $u = 2x^2$, $v = 3x^4 - 2$.