∫ calculus

1. The problem states that the curve is given by $y = f(x)$ and the tangent line at any point $(x, y)$ on the curve is given by $y = g(x)$. 2. By definition, the tangent line $g(x)

1. The problem asks to identify which graph represents the derivative $f'(x)$ of the given function $y=f(x)$ shown in the top-right graph. 2. The original function $f(x)$ starts ne

1. The problem asks to determine the number of inflection points of the function $f(x)$ given the graph of its derivative $f'(x)$ on the interval $-1 \leq x \leq 3$. 2. Recall that

1. **Problem statement:** Given two differentiable functions $f(x)$ and $g(x)$ on the interval $[a,b]$, determine which of the following functions is always increasing on $[a,b]$:

1. The problem states that the top-right graph represents the first derivative $f'(x)$ of a function $f(x)$ defined on $\mathbb{R}$. We need to identify which of the four given gra

1. The problem asks to determine whether the function defined on the interval $x \in [0,2[$ has an absolute minimum and/or maximum value. 2. From the graph description, the functio

1. The problem states that the given curve represents the first derivative $f'$ of a continuous function $f$ on $\mathbb{R}$. We need to identify the wrong statement among the opti

1. The problem states that $f'(3)$ is undefined and $f''(x) > 0$ for all $x \neq 3$ on the interval $[1,5]$. 2. Since $f''(x) > 0$ for $x \neq 3$, the function $f$ is concave upwar

1. The problem states that $f$ is continuous with $f(0) = 3$, $f'(2) = f'(-2) = 0$, and $f'(x) > 0$ for $-2 < x < 2$. 2. The condition $f'(2) = f'(-2) = 0$ means the slope of the t

1. The problem asks which graph could represent the function $h(x) = f(x) - g(x)$ given the graphs of the derivatives $f'(x)$ and $g'(x)$. 2. Recall that the derivative of $h(x)$ i

1. **Problem statement:** We analyze the function $f$ defined on the interval $[1,5]$ based on the given graph and determine which statement among (a), (b), (c), and (d) is not cor

1. The problem asks to identify which graph (a, b, c, or d) could represent the original function $y = f(x)$ given that the opposite figure represents its first derivative $f'(x)$.

1. The problem asks which function among the options is increasing on the interval $]a,b[$ given that $f$ is a function defined on $[a,b]$ with values in $\overline{\mathbb{R}}$ an

1. **State the problem:** We want to compute the integral $$\int_1^\infty \frac{e^{-3x} - e^{-7x}}{x} \, dx.$$ This is a classic example where Frullani's integral formula can be ap

1. **State the problem:** We need to show that the curve given by the function $$y=\frac{2x+1}{3x+6}$$ is increasing for all $$x \neq -2$$. 2. **Find the derivative:** To determine

1. **State the problem:** We want to find the limit $$\lim_{n \to \infty} n \ln\left(1 - \frac{5}{n}\right).$$ 2. **Rewrite the expression inside the logarithm:** As $n$ becomes ve

1. **State the problem:** We need to find the limit as $n$ approaches infinity of the expression $$\frac{1 + (-1)^n}{2 + (-1)^n}.$$\n\n2. **Analyze the behavior of $(-1)^n$:** The

1. **Problem:** Find the derivative of the exponential function $y = e^{-3x}$. 2. **Recall the rule:** The derivative of $e^u$ with respect to $x$ is $$\frac{d}{dx} e^u = e^u \frac

1. The problem is to find the derivative $\frac{dy}{dx}$ given the parametric equations: $$x = t^3 + 1, \quad y = 4t^2 - 4t$$

1. **State the problem:** We want to analyze the function $$y = x \ln(4 + x^2) + 4 \arctan\left(\frac{x}{2}\right) - 2x$$ and understand its behavior. 2. **Rewrite the function:**

1. **State the problem:** Find the first derivative of the function $$y = x \ln(4 + x^2) + 4 \arctan\left(\frac{x}{2}\right) - 2x.$$\n\n2. **Differentiate each term separately:**\n