∫ calculus

1. **State the problem:** We are given the function $f(x) = 5x + 4$ and two points $x_0 = 3$ and $x_1 = 7$. We want to find the average rate of change of the function between these

1. **Problem statement:** Calculate the integral $$\int_0^{2\pi} f(x) \, dx$$ where $$f(x) = \begin{cases} \sin x & \text{for } x \leq \pi \\ -2 \sin x & \text{for } x > \pi \end{c

1. **State the problem:** We want to evaluate the double integral $$\iint_D F(x,y)\,dA$$ where $$F(x,y) = x^2 + 2xy + y^2$$ and the region $$D$$ is bounded by the curves $$y = x$$

1. **Problem statement:** Given the curve defined by the equation $$xy + y^2 e^{-x} = 4,$$ we need to (a) show that $$\frac{dy}{dx} = \frac{y^2 - y e^x}{x e^x + 2y}$$ and (b) find

1. **Problem statement:** Given the curve equation $$2x^2 y - xy^2 = a^3$$ where $a$ is a positive constant, we need to show there is only one point on the curve where the tangent

1. **Problem statement:** Given the curve defined by the equation $$xy + y^2 e^{-x} = 4,$$ we need to show that $$\frac{dy}{dx} = \frac{y^2 - y e^x}{x e^x + 2y}$$ and then find the

1. The problem is to evaluate the integral $$\int \frac{0 + 1 + xy + x^2 y^2}{x^2} \, dx$$. 2. First, simplify the integrand by dividing each term by $x^2$:

1. **State the problem:** We need to find the derivative $\frac{dz}{dt}$ where $z = f(x,y) = \sqrt{x^2 - y^2}$, with $x = e^{2t}$ and $y = e^{-t}$. 2. **Recall the chain rule for m

1. **State the problem:** We need to find the derivative $\frac{dz}{dt}$ where $z = f(x,y) = \sqrt{x^2 - y^2}$, with $x = e^{2t}$ and $y = e^{-t}$.\n\n2. **Recall the chain rule fo

1. **State the problem:** We are given the function $$L(x) = \frac{f(x) - \tan(x)}{x^3}$$ and we want to analyze or simplify it depending on the context (e.g., limit, derivative, e

1. The problem asks to show that for the curve $y = \sqrt{x} \sin 2x$ on $0 \leq x \leq \frac{1}{2} \pi$, the maximum point $M$ at $x = a$ satisfies $\tan 2a = -4a$. 2. To find the

1. **State the problem:** We are given the function $F(x) = \frac{1}{x}$ defined on the interval $0 < x < 4$. We need to determine whether this function is bounded on this interval

1. The problem is to find the antiderivatives (indefinite integrals) of certain trigonometric-related functions. 2. The formulas for these antiderivatives are:

1. **Problem:** Find the critical points of the function $$f(x) = x^4 - 4x^3 + 6x^2$$ on its domain. 2. **Formula and rules:** Critical points occur where the first derivative $$f'

1. **Problem:** Find the critical points of the function $$f(x) = x^4 - 4x^3 + 6x^2$$ on its domain. 2. **Formula and rules:** Critical points occur where the first derivative $$f'

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $y = -12\sqrt{x^3 + 7x - 1}$. 2. **Recall the formula:** For $y = k\sqrt{u}$ where $u$ is a function o

1. **State the problem:** We need to find the length of the segment AB where A is the x-intercept and B is the y-intercept of the tangent line to the curve $y = e^x$ at the point $

1. **State the problem:** We are given the function $$y=\sqrt[3]{27-8x^3}$$ and asked to discuss its monotony, i.e., where it is increasing or decreasing. 2. **Recall the formula a

1. **Stating the problem:** We have the function $$z = \frac{\sqrt[3]{x}}{y} = \frac{x^{1/3}}{y}$$ and $$Z = x^{2/3}$$ with $$f(x) = y$$. We need to prove two differential identiti

1. **Stating the problem:** We are given the function $y = x^{2/3} - \frac{1}{3}$ and $Z = \sqrt[3]{x^2} = x^{2/3}$. We need to prove two differential identities:

1. **Define continuity of a function $f(x)$ at a point $x=c$: ** A function $f(x)$ is continuous at $x=c$ if the following three conditions hold: