∫ calculus

1. **State the problem:** We are given a function $f(p)$ representing the number of shoes sold at price $p$. We know $f(120) = 9000$ and $f'(120) = -60$. The revenue function is $R

1. Stated problem: Calculate the integral $$\int (1 - x) \sin x \, dx$$. 2. Formula and rules: Use integration by parts, where $$\int u \, dv = uv - \int v \, du$$.

1. **State the problem:** We are given functions $j(x)$ and $k(x)$ with their values and derivatives at $x=-1$, and a function $$f(x) = \frac{1 - k(x)}{j(x)^2 - 4}.$$ We need to fi

1. **State the problem:** We want to find the volume $V$ of the solid generated by rotating the region bounded by the curve $y = \frac{x^4}{2}$ and the line $y = 2$ between $x=0$ a

1. **State the problem:** We need to evaluate the integral $$I = \int 2 \cos^2 x \, dx$$ and determine which of the given options is correct. 2. **Recall the formula:** Use the dou

1. **State the problem:** We are given the function $f(x) = 6\sqrt[3]{x}$ and asked to find the linearization $L(x)$ at $x=1$ and then use it to estimate $f(1.1)$. 2. **Recall the

1. **State the problem:** We have a position function for a drone's height given by $$s(t) = -t^3 + 6t^2 + 15t + 10$$ for $$t \geq 0$$. We want to find the velocity $$v(t)$$ and ac

1. **State the problem:** Find the limit $$\lim_{x \to -1} \frac{2x^2 - x - 3}{x + 1}$$. 2. **Recall the limit rule:** If direct substitution leads to an indeterminate form like $$

1. **State the problem:** We need to find the integral $$\int \frac{2x^3 + 5x^2 + 3x - 5}{2x - 1} \, dx$$. 2. **Use polynomial division:** Since the degree of the numerator (3) is

1. **State the problem:** We have a function $\int(x)$ such that its derivative satisfies $$\int'(x) = 6\sqrt{2} \sin(x) (f'(x))^2$$ for all real $x$, and the initial condition $$\

1. **بيان المسألة:** نريد إيجاد نهاية الدالة

1. **Stating the problem:** We analyze the function $$y = xe^{-x^2}$$ to find its domain, intercepts with axes, positivity intervals, limits, and study the first and second derivat

1. **Stating the problem:** We analyze the function $$Y = e^x(e^x - 1)$$ by finding its domain, intercepts, sign study, limits, first and second derivatives.

1. Знайдемо критичні точки функції $$y = -\frac{1}{3}x^3 + 9x$$. Критичні точки знаходяться там, де похідна функції дорівнює нулю або не існує.

1. **State the problem:** Find the limit $$\lim_{x \to 5} \frac{x^2 - 6x + 5}{x - 5}$$. 2. **Recall the formula and rules:** When direct substitution results in an indeterminate fo

1. **State the problem:** Evaluate the limit $$\lim_{x \to 0} \frac{e^x - x - 1}{5x^2}.$$\n\n2. **Recall the formula and rules:** When direct substitution results in an indetermina

1. **State the problem:** Evaluate the limit $$\lim_{x \to 1} \frac{\sqrt{x} - \sqrt[3]{x}}{x - 1}$$. 2. **Recognize the indeterminate form:** Substituting $x=1$ gives $$\frac{\sqr

1. The problem is to find the derivative of a geometric function, typically involving shapes or geometric formulas. 2. The derivative measures how a function changes as its input c

1. **State the problem.** Find $\lim_{x\to 0}\dfrac{\sin(3x)-3x}{x^3}.$

1. Vamos resolver o primeiro integral: $$\int x e^{-x^2} \, dx$$ 2. Para resolver, usamos substituição. Seja $$u = -x^2$$, então $$du = -2x \, dx$$ ou $$-\frac{1}{2} du = x \, dx$$

1. **State the problem:** Find the limit $$\lim_{x \to 1} \frac{\frac{2}{x} + 3 + x - 1}{x - 5} \div (x - 1).$$ 2. **Rewrite the expression:** The expression can be written as