∫ calculus

1. Problem statement: The graph shows $f(x)=0$ for $0\le x\le 1$ and a line from $(1,0)$ to $(10,3)$, and you are given $\int_4^{10} f(x)\,dx = 12$. Find the integrals (a) through

1. **State the problem:** Find the limit $$\lim_{x \to -\infty} \frac{x^2 + 6x + 5}{x^5 + 4}$$. 2. **Recall the rule for limits at infinity:** When evaluating limits of rational fu

1. **State the problem:** We need to find the limit \( \lim_{x \to 2} \frac{xe^{x-1} - 2e}{x-2} \). 2. **Recognize the form:** Substitute \(x=2\) directly:

1. **State the problem:** We need to find the limit $$\lim_{x \to 2} \frac{xe^x - 1 - 2e}{x - 2}$$. 2. **Recognize the form:** Substitute $x=2$ directly:

1. **Problem Statement:** We are given the graph of the derivative $f'$ of a twice-differentiable function $f$ on the domain $(-9,9)$, with points of inflection at $x=-5$, $x=-2$,

1. Énonçons le problème : Calculer l'intégrale $$\int \cos(t) e^t \, dt$$. 2. La méthode utilisée est l'intégration par parties, qui repose sur la formule :

1. **State the problem:** We have the cubic function $$y = x^3 - 8x^2 - 12x + 5$$ and need to find the coordinates of the two stationary points A and B, where the x-coordinate of A

1. **Stating the problem:** We need to find the limit $$\lim_{x \to \infty} \frac{\sqrt{x^2 + 2}}{x - 8}$$. 2. **Formula and rules:** When dealing with limits at infinity involving

1. **State the problem:** Find the limit as $x$ approaches 0 of the expression $$\frac{\sqrt{x + 9} - 3}{\sqrt{x + 16} - 4}.$$\n\n2. **Recall the formula and approach:** When direc

1. **State the problem:** We need to find the value of $k$ (either 1 or 2) such that the volume of the solid formed by rotating the region bounded by the curve $y = e^x - k$, the x

1. **State the problem:** We need to find the integral of the function $\left(e^x - k\right)^2$ with respect to $x$. 2. **Formula and rules:** Recall that the integral of a sum is

1. We are asked to find the limit: $$\lim_{x \to \frac{\pi}{8}} \frac{\sen \left( \frac{5\pi}{4} - 2x \right)}{2x - \frac{\pi}{4}}$$

1. The problem is to differentiate the function $$y = \left(\frac{x - 1}{3 + x^2}\right)^4$$ with respect to $x$. 2. This is a composite function where an inner function $$u = \fra

1. We are asked to evaluate the limit $$\lim_{x \to 0} \frac{1}{x^2}$$ and determine if it equals infinity. 2. Recall that $x^2$ is always positive except at $x=0$, and as $x$ appr

1. The problem asks to find the limit: $$\lim_{x \to 0} \frac{1}{x^2}$$. 2. Recall that $x^2$ is the square of $x$, so it is always positive except at $x=0$ where it is zero.

1. The problem is to find the derivative of the function given as $$f'(x) = 9\sqrt{x + 5} - 9\sqrt{x - 1}$$. 2. Recall the derivative rule for a square root function: $$\frac{d}{dx

1. **State the problem:** We need to evaluate the definite integral $$\int_{-\sqrt{6}}^{\pi} (2x + 2|x|) \sin(6x) \, dx.$$\n\n2. **Understand the absolute value:** The function ins

1. **Problem statement:** Perform a curve discussion (Kurvendiskussion) for the function $$f(x) = \frac{4x}{x^2 - 4}$$. 2. **Domain:** The denominator cannot be zero, so solve $$x^

1. **State the problem:** We are given gold production data $G = f(t)$ for years $t$ from 2014 to 2018 and asked to analyze the derivative $f'(t)$, which represents the rate of cha

1. The problem asks to identify the correct graph of the derivative of the function $f(x)$, where $f(x)$ is a downward-opening parabola with vertex at approximately $(1,4)$. 2. Rec

1. The problem is to understand the behavior of a function as the input approaches negative infinity. 2. When analyzing limits at negative infinity, we look at the value of the fun