∫ calculus

1. **Problem Statement:** Given two functions $F(x)$ and $G(x)$ with graphs described, define $P(x) = F(x)G(x)$ and $Q(x) = \frac{F(x)}{G(x)}$. We need to find the derivatives $P'(

1. **Problem Statement:** Given values $f(-3) = -8$, $g(-3) = 6$, $f'(-3) = 2$, and $g'(-3) = 8$, find $h'(-3)$ for each function $h(x)$. 2. **Recall the derivative rules:**

1. **Problem:** Find the volume of a solid with a circular base of radius 1, where cross sections perpendicular to the base are equilateral triangles. 2. **Formula and rules:** The

1. **State the problem:** We are given the second derivative of a curve as $$\frac{d^2y}{dx^2} = 45 \cos 3x + 2 \sin x$$ with initial conditions $$\frac{dy}{dx} = -2$$ at $$x=0$$ a

1. Bài toán yêu cầu tính tích phân \( \int_{2}^{+\infty} \frac{1}{x \ln x} \, dx \). 2. Đây là một tích phân không xác định trên khoảng vô hạn, ta cần kiểm tra tính hội tụ của nó.

1. **State the problem:** We need to find the indefinite integral $$\int \left(e^{2x} - 5 \sin 2x\right) \, dx$$. 2. **Recall the integral formulas:**

1. **State the problem:** We need to find the first derivative $\frac{dy}{dx}$ and the second derivative $\frac{d^2y}{dx^2}$ implicitly from the equation $$3xy + \sin(x) = 9.$$\n\n

1. **State the problem:** We need to find the second derivative $\frac{d^2y}{dx^2}$ implicitly from the equation $$x^2y - 4 = 9x + y.$$\n\n2. **Rewrite the equation:** $$x^2y - 4 =

1. **State the problem:** We need to write the integral to find the area of the region bounded by the curves $u(y) = 2\sqrt{y} + 2$ and $v(y) = \frac{2(y + 2)}{3}$, integrating wit

1. **State the problem:** We are given two functions $f(x) = x^2 - 7x + 12$ and $g(x) = \frac{3x}{4} + \frac{21}{4}$. We want to write the integral with respect to $y$ to find the

1. **Problem statement:** Find the limit $$\lim_{x \to 1} [f(x) + g(x)]$$ given the graphs of $$f$$ and $$g$$. 2. **Recall the limit sum rule:** $$\lim_{x \to a} [f(x) + g(x)] = \l

1. **State the problem:** We want to find the derivative with respect to $x$ of the function $$f(x) = \int \frac{(x^3 - x^8)^5}{x^{-3}} \, dx.$$ 2. **Simplify the integrand:** Reca

1. **State the problem:** Find the limit $$\lim_{x \to 0} (1-2x)^{\frac{1}{x}}.$$\n\n2. **Recall the formula:** Limits of the form $$\lim_{x \to 0} (1 + ax)^{\frac{1}{x}} = e^a$$ a

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{1 - 2x}{\frac{1}{x}}.$$\n\n2. **Rewrite the expression:** Dividing by a fraction is the same as multiplying by its r

1. El problema es encontrar el área bajo la curva de la función $$y = x^{2} + 2x + 2$$ entre los puntos $$x_1 = -2$$ y $$x_2 = 2$$ sobre el eje X. 2. Para hallar el área bajo la cu

1. We are given the function $y = \log_2(7x - 3)$ and asked to find $\frac{dy}{dx}$ at $x=1$. 2. Recall the derivative formula for logarithms with base $a$:

1. The user asked to teach full calculus, which is a very broad topic covering limits, derivatives, integrals, and more. 2. Calculus studies how things change and accumulate.

1. **State the problem:** Find the limit as $x \to 0$ of the expression $$\frac{\sqrt{1+x^2} - \sqrt{1+x}}{\sqrt{1+x^3} - \sqrt{1+x}}.$$\n\n2. **Recall the conjugate multiplication

1. **State the problem:** Find the limit as $x \to 0$ of the expression $$\frac{\sqrt{1 + x^2} - \sqrt{1 + x}}{\sqrt{1 + x^3} - \sqrt{1 + x}}$$

1. **State the problem:** Find the derivative of the function $$y = (x^2 - 7)(x^2 + 4x + 2)$$. 2. **Formula used:** To differentiate a product of two functions, use the product rul

1. **State the problem:** Evaluate the definite integral $$\int_0^{\frac{\pi}{2}} \frac{x^2}{(x-1)^2} \, dx.$$\n\n2. **Rewrite the integrand:** We have the function $$f(x) = \frac{