∫ calculus

1. Let's state the problem: We want to understand which Riemann sum, left or right, generally overestimates the integral of a function. 2. The formula for the left Riemann sum over

1. **Problem Statement:** Calculate the LEFT(2), RIGHT(2), and MID(2) Riemann sums for the integral $$\int_1^2 \frac{1}{x} \, dx$$ using 2 subdivisions, and compare these approxima

1. **Problem statement:** We have the function $f(x) = \frac{1}{x} + \frac{\ln x}{x}$ defined for $x > 0$. We want to prove that the vertical line $x=0$ is an asymptote of the curv

1. **Problem statement:** (i) Find the derivative $\frac{dy}{dx}$ if $y = \ln(2x^2 + 5)$.

1. **Problem 10.1:** Use the Mean Value Theorem (MVT) to show that $$\cos b - \cos a \leq b - a$$ for all real numbers $$a < b$$. 2. **Mean Value Theorem statement:** If a function

1. **State the problem:** We are given the differential equation $y' = \frac{x^3}{x^2 + 1}$ and need to find the function $y(x)$. 2. **Recall the formula:** Since $y' = \frac{dy}{d

1. **Problem Statement:** Find the derivative of a given function (please specify the function if needed). 2. **Formula Used:** The derivative of a function $f(x)$ is given by $f'(

1. **Stating the problem:** We need to simplify and analyze the derivative expression given by $$y' = \sqrt{x}(x - x^2)(x - x^2)(x - x^2).$$ 2. **Formula and rules:** Recall that \

1. **State the problem:** Evaluate the integral $$\int_{\frac{\pi}{4}}^{\frac{\pi}{12}} \cos^2 x \, dx$$. 2. **Recall the formula:** To integrate $$\cos^2 x$$, use the power-reduct

1. **State the problem:** Find the intervals where the function (2) is increasing, decreasing, and constant. Then find its domain and range.

1. **State the problem:** We need to find the derivative $\frac{dy}{du}$ when $y = \frac{u^2 + 1}{u^2 - 1}$. 2. **Formula used:** For a function $y = \frac{f(u)}{g(u)}$, the deriva

1. **State the problem:** Find the asymptotes of the function $y=\frac{\sin x}{x}$.\n\n2. **Recall the definition of asymptotes:** Asymptotes are lines that the graph of a function

1. **State the problem:** We want to find the limit $$\lim_{x \to \infty} \frac{\sin x}{x}$$. 2. **Recall the properties:** The sine function, $\sin x$, oscillates between $-1$ and

1. **State the problem:** We want to find the limit $$\lim_{x \to \infty} \frac{\sin x}{x^2}$$. 2. **Recall the properties:** The sine function oscillates between -1 and 1 for all

1. The problem asks to find $\frac{dx}{d\theta}$ when $x = \cos \theta \sin \theta$. 2. We use the product rule for differentiation: if $x = u(\theta) v(\theta)$, then $\frac{dx}{d

1. **Problem Statement:** Calculate the integral $$\int e^x \cos x \, dx$$ using integration by parts. 2. **Formula Used:** Integration by parts formula:

1. **State the problem:** Find the derivative of the function $f(x) = \tan x$. 2. **Recall the formula:** The derivative of $\tan x$ with respect to $x$ is given by the formula:

1. **State the problem:** We want to find a value of $k$ and a substitution $w$ such that $$\int \frac{12x - 2}{(3x + 2)(x - 1)} \, dx = k \int \frac{dw}{w}.$$

1. Let's start by understanding the problem: We want to know if a function that has minima must also have maxima. 2. A minimum (plural: minima) of a function is a point where the f

1. We are given the function $$f(x,y) = x^2 - 3xy^2$$ and asked to find the partial derivative of $$f$$ with respect to $$y$$, denoted as $$\frac{\partial f}{\partial y}$$. 2. The

1. The problem asks for the 3rd derivative of the function $$y = e^{6x}$$. 2. Recall the rule for differentiating exponential functions: $$\frac{d}{dx} e^{ax} = a e^{ax}$$ where $a