∫ calculus

1. The problem is to find the limit $$\lim_{x \to -\infty} \frac{5x + 1}{2x}$$. 2. The formula for limits involving rational functions as $x$ approaches infinity or negative infini

1. **State the problem:** We are given a function involving $\tan(y+ax)$ and $(y-ax)^{3/2}$ and asked to find the value of $$\frac{\partial^2 z}{\partial x^2} - a^2 \frac{\partial^

1. **Problem statement:** Given the curve $$y = \frac{(2x^2 + 10)^{3/2}}{x - 1}$$ for $$x > 1$$, (a) Show that $$\frac{dy}{dx}$$ can be written as $$\frac{(2x^2 + 10)^{1/2}}{(x - 1

1. **Stating the problem:** We are given two functions representing acceleration, $R''(t) = -16.8t + 23.2$ and $R''(t) = -330t + 266$, and asked to interpret their meaning. 2. **Un

1. Задача: Обчислити визначений інтеграл $$\int_{-\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{\sin^2 x}$$. 2. Формула: Використаємо, що $$\frac{1}{\sin^2 x} = \csc^2 x$$, а первісна ф

1. The problem gives two functions $R(t)$ and their first and second derivatives $R'(t)$ and $R''(t)$. 2. The function $R(t)$ represents a quantity depending on time $t$ (e.g., rev

1. **State the problem:** Water is flowing into a conical vessel with a depth of 15 cm and a top radius of 3.75 cm. The water level is rising at a rate of 12 cm/min. We need to fin

1. The problem is to find the limit $$\lim_{x \to 0^+} \left(x - \sqrt{x}\right).\n\n2. We start by understanding the behavior of each term as $x$ approaches 0 from the right (posi

1. The problem is to find the limit $$\lim_{x \to 0^+} (x - x)$$. 2. The expression inside the limit is $x - x$.

1. **State the problem:** Find the limit $$\lim_{x \to 2} \frac{x^2 + 4}{x - 2}$$. 2. **Understand the problem:** We want to find the value that the expression approaches as $x$ ge

1. **State the problem:** Find the limit $$\lim_{x \to 2} \frac{x-2}{x^2+4}$$. 2. **Recall the limit rule:** If direct substitution does not lead to an indeterminate form, substitu

1. Stating the problem: Calculate the left-hand limit as $x$ approaches 1 from the left for the function $$\frac{2x + 3}{x^2 - 1}$$. 2. Recall the formula and rules: The limit $$\l

1. **Problem Statement:** Find the local maximum and minimum values of the function $$y = x^5 - 5x^4 + 5x^3 - 10$$. 2. **Formula and Rules:** To find local maxima and minima, we us

1. **Problem:** Differentiate the function $y = (2x^2 + 3)^4 (x - 3)^3$. 2. **Formula and rules:** Use the product rule: $\frac{d}{dx}[uv] = u'v + uv'$ where $u = (2x^2 + 3)^4$ and

1. **Problem:** Differentiate the function $$y = (2x^2 + 3)^4 (x^4 - 3)^3$$. 2. **Formula and rules:** Use the product rule: $$\frac{d}{dx}[u v] = u' v + u v'$$ where $$u = (2x^2 +

1. **Stating the problem:** Calculate the limit $$\lim_{x \to 2} \frac{3x^2 - 5x + 1}{4x - 2}$$. 2. **Formula and rules:** To find limits of rational functions as $x$ approaches a

1. The problem asks to find the general antiderivative (indefinite integral) of the function $$f(x) = \sin^2 2x \cos^2 2x$$. 2. We use the formula for integration and trigonometric

1. The problem is to find the first and second derivatives of a function, but the function itself was not specified. 2. To find the first derivative $f'(x)$ of a function $f(x)$, w

1. **State the problem:** Evaluate the limit $$\lim_{x \to 0} \log \left(1 + x\right)^{\frac{1}{x}}$$. 2. **Rewrite the expression:** Using logarithm properties, $$\log \left(1 + x

1. **Problem statement:** Given the rate function of drug entering the bloodstream: $$R(t) = \frac{80t}{t^2 + 4}, \quad t \geq 0$$

1. **State the problem:** We need to evaluate the integral $$\int (\sqrt{x} + 1)(x - \sqrt{x} + 1) \, dx$$. 2. **Expand the integrand:** Multiply the two binomials: