∫ calculus

1. **State the problem:** Find the derivative of the function $$f(x) = 10 \frac{1}{x^3} + x^2 + 14.$$\n\n2. **Rewrite the function:** Express the function with negative exponents f

1. **State the problem:** We are given the function $f(x) = 4x^2 - 3x + 6$ and asked to evaluate its derivative $f'(x)$ at $x=6$ and $x=-8$. 2. **Find the derivative formula:** The

1. **Problem Statement:** We need to determine which statements about the derivatives of functions $f$ and $g$ are true. 2. **Recall the derivative rules:**

1. The problem is to analyze the concavity and inflection points of the curve given by the function $$y=\frac{1}{x^2+4}$$. 2. To study concavity and inflection points, we use the s

1. **Problem Statement:** We want to analyze the concavity and inflection points of the curve given by the function $$y=\frac{1}{x^2}+4$$. 2. **Formula and Rules:**

1. **State the problem:** Find the local minimum vertex of the cubic function $$y = x^3 + 2x^2 - x - 2$$. 2. **Formula and rules:** To find local minima or maxima, we use the first

1. **Stating the problem:** We need to find the derivative of the function $$f(x) = \frac{x^3}{x+2}$$ and then multiply it by $$x + \frac{2}{x^3}$$. 2. **Formula used:** To differe

1. **State the problem:** Find the derivative of the function $f(x) = \ln\left(\frac{x^3}{x+2}\right)$.\n\n2. **Recall the formula:** The derivative of $\ln(u)$ with respect to $x$

1. **State the problem:** Find the derivative of the function $f(x) = \ln\left(\frac{x^3}{x+2}\right)$.\n\n2. **Recall the formula:** The derivative of $\ln(u)$ with respect to $x$

1. **State the problem:** Find the derivative of the function $$y=\frac{\ln x^3}{x+2}$$. 2. **Rewrite the function:** Use the logarithm power rule $$\ln x^3 = 3 \ln x$$, so the fun

1. **State the problem:** We want to find the limit $$\lim_{x \to 0} \frac{(1+x)^{\frac{1}{x}} - e}{x}$$ using l'Hopital's rule. 2. **Recall the formula and rules:** The expression

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{(1+x)^{\frac{1}{x}} - e}{x}$$. 2. **Recall the known limit:** We know that $$\lim_{x \to 0} (1+x)^{\frac{1}{x}} = e$

1. **Stating the problem:** We need to sketch and calculate the area bounded by the curves $y=\frac{1}{x}$ and $y=\frac{1}{x^2}$ at $x=2$.

1. **Stating the problem:** We need to find the area bounded by the curves $y = x^2$ and $y = -x$. 2. **Find the points of intersection:** Set $x^2 = -x$ to find where the curves i

1. **Problem Statement:** Find the area bounded by the curves $y=2$, $y=\cos(x)+1$, and the vertical lines $x=0$ and $x=\pi$.

1. **Problem Statement:** Given the equation $$2 \int_0^{10} f(x) \, dx + 8 \int_0^{11} f(x) \, dx + 10 \int_0^{8} f(x) \, dx = 9,$$ find the value of $$2 \int_0^{11} f(x) \, dx.$$

1. **State the problem:** We are given the rate of change of the area of a lamina as $$\frac{dA}{dt} = e^{-0.2t}$$ where $t$ is in seconds. The initial area at $t=0$ is 140 cm². We

1. **Problem Statement:** Given a curve $y=f(x)$ with second derivative $$\frac{d^2y}{dx^2} = ax + b,$$ where $a$ and $b$ are constants. The curve has an inflection point at $(0,2)

1. **Stating the problem:** We want to find the intervals where the function $f(x) = \tan x - x$ is increasing (croissante) and decreasing (décroissante). 2. **Formula and rules:**

1. **Problem Statement:** Find the function $y = f(x)$ whose slope of the tangent at any point $(x,y)$ is given by the derivative $\frac{dy}{dx} = -6 \cos^5 x \sin x$, and the curv

1. **Problem Statement:** We are given the slope of the tangent to the curve $f$ at any point $(x,y)$ as $g(x) = \frac{xe^x}{(x+1)^2}$.