∫ calculus

1. **Problem:** Find the derivative of the function $y = 2 \ln x$. 2. **Formula and rules:** The derivative of $\ln x$ with respect to $x$ is $\frac{1}{x}$ for $x > 0$.

1. **Problem:** Express the limit \(\lim_{\|P\|\to 0} \sum_{k=1}^n c_k^2 \Delta x_k\) as a definite integral, where \(P\) is a partition of \([0, 2]\). 2. **Formula and Explanation

1. State the problem: Use the Constant Multiple Rule in limits, which states that $$\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)$$ where $c$ is a constant. 2. Exampl

1. **Problem:** Find the value of $c$ that satisfies Rolle's Theorem for $$f(x) = \frac{x^2 + 4x - 12}{x^2 + 2x - 3}$$

1. **State the problem:** We are given the function $f(x) = x^2 + 1$ and asked to analyze it using calculus. 2. **Recall the formula for the derivative:** The derivative of a funct

1. **Problem:** Find the value of $c$ that satisfies Rolle's Theorem for $$f(x) = \frac{x^2 + 4x - 12}{x^2 + 2x - 3}$$ on the interval $[-6, 2]$.

1. The problem is to find the derivative of the function $f(x) = 3x^3 - 2$ using calculus. 2. The formula for the derivative of a power function $x^n$ is given by the power rule: $

1. The problem is to analyze the function $f(x) = x^2 - 4$ using calculus. 2. We start by finding the critical points where the derivative is zero or undefined. The derivative of $

1. The problem is to understand and apply the theorem on limits: $$\lim_{x \to a} c = c$$ where $c$ is a constant and $a$ is the point $x$ approaches. 2. This theorem states that t

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