∫ calculus

1. Сформулюємо задачу: потрібно побудувати рівняння дотичної до графіка функції $f(x) = \ln(x^2 + 1)$ у точці $T = [1, f(1)]$ та знайти диференціал функції в точці $x_0 = 1$, тобто

1. **State the problem:** Find the derivative of the function $$y = x^{2} (\sqrt{x} + 1)$$. 2. **Rewrite the function:** Recall that $$\sqrt{x} = x^{\frac{1}{2}}$$, so the function

1. **Problem statement:** Given a function $f(x) = g(x) - r(x)$ with conditions at $x=2$: $g'(2) = r'(2)$ and $g''(2) > r''(2)$, determine the nature of $f$ at $x=2$. 2. **Recall t

1. **Problem Statement:** (i) Evaluate the definite integral $$\int_{-1}^1 \sqrt{3x^2 - 2x + 3(3x - 1)} \, dx$$

1. The problem asks for a function that has a maximum at $x=12$ and includes an exponential component. 2. A common way to create a function with a maximum at a specific point is to

1. **Problem:** Find the derivative of $f(x) = (\ln x)^2$. 2. **Formula:** Use the chain rule: If $f(x) = [g(x)]^2$, then $f'(x) = 2g(x)g'(x)$.

1. **Problem 1: Find the turning (stationary) points of** $y = x^3 - 3x + 5$ **and distinguish between them.** 2. To find turning points, we first find the derivative $y' = \frac{d

1. **State the problem:** We need to find the integral of the function $$\frac{6x}{3x^2 + k}$$ with respect to $x$. 2. **Recall the formula and rules:** The integral of a function

1. **State the problem:** We have two functions representing water flow rates into and out of a tank over time $t$ hours: $$H(t) = 12(0.93)^t$$

1. **State the problem:** Determine if the function $f(x) = x^4 - x^2$ has a horizontal asymptote. 2. **Recall the definition of horizontal asymptotes:** A horizontal asymptote is

1. **State the problem:** We want to show that the equation $x^5 - x^3 + 3x - 5 = 0$ has at least one solution in the interval $(1, 2)$ using the Intermediate Value Theorem (IVT).

1. **State the problem:** Given the implicit function defined by the equation $$z(z^2+3x)+3y=0,$$ we need to prove that $$\frac{d^2z}{dx^2}+\frac{d^2z}{dy^2}=\frac{2z(x-1)}{(z^2+x)

1. The problem asks to evaluate the definite integral $$\int_{-2011}^{2011} (x^3 + x) \, dx$$. 2. Use the property that the integral of an odd function over symmetric limits is zer

1. Masala: $\int_{-25}^{5} (x^3 + x + 1) \, dx + \int_{5}^{25} (x^3 + x + 1) \, dx$ ni hisoblang. Integralni hisoblash uchun avvalo umumiy integralni topamiz:

1. Masala: $$\int_0^{\frac{\pi}{2}} \cos x \, dx$$ ni hisoblang. 2. Formulalar: $$\int \cos x \, dx = \sin x + C$$ va belgilangan integralni hisoblash uchun $$\int_a^b f(x) \, dx =

1. **Problem Statement:** Consider the infinite series $$\sum_{n=1}^{\infty} 5^{1-n} (-1)^n$$.

1. The problem is to understand and analyze the inequality $x' - 2 \leq x \leq 5$. 2. Here, $x'$ typically denotes the derivative of $x$ with respect to some variable, often time $

1. **Problem Statement:** Evaluate the integral $$\int x \sqrt{x+1} \, dx$$. 2. **Formula and Approach:** To solve integrals involving products of polynomials and roots, substituti

1. **Problem Statement:** Given $g(x) = \int_0^x f(t) \, dt$ where $f$ is a piecewise linear function with points $(0,0), (1,1), (3,4), (6,1), (9,-1), (10,1)$.

1. **Problem Statement:** Find the derivative of the function $$f(x) = \frac{1 - x}{2x}$$ using the limit definition of the derivative. 2. **Limit Definition of the Derivative:** T

1. **Problem statement:** Find the volume of the solid bounded by the surface $f(x,y)$ over the given region $R$. We will solve problems 3, 4, and 12 from Exercise 2.2.