∫ calculus

1. **Problem statement:** Find the number of points of inflection of the function $f$ on the interval $0 < x < 3$ given that $f''(x) = \sin(3x) - \cos(x^2)$. 2. **Recall:** Points

1. The problem asks to find the locations of discontinuities for four different graphs described. 2. Discontinuities in functions often occur where the function is undefined, such

1. The problem involves identifying the locations of removable discontinuities (holes) and vertical asymptotes in given graphs. 2. A removable discontinuity (hole) occurs where a f

1. **State the problem:** Find the derivative of the function $$y = \ln(\cos(x))^x$$ with respect to $$x$$. 2. **Rewrite the function:** The function can be written as $$y = (\ln(\

1. Énonçons le problème : Calculer le volume total donné par $$V = \int_{\pi/2}^{3\pi/2} -2\pi x \cos(x) \, dx$$

1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$$. 2. **Recall the formula and rules:** When direct substitution results in an indeterminate form li

1. The problem is to find the domain of the function $f(x,y) = \ln(xy)$.\n\n2. The natural logarithm function $\ln(z)$ is defined only for $z > 0$. This means the argument inside t

1. **State the problem:** We want to find the volume $V$ by integrating the volume element $dV = \pi r^2 dh$ where $r = y = \frac{2}{x}$ and $dh = dx$. 2. **Write the formula:** Th

1. The problem is to differentiate the function $$f(x) = \frac{3x^2 - x}{\sqrt{1-2x}}$$. 2. We use the quotient rule for derivatives: $$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}

1. **Problem 1: Find the limit of the sequence** $$a_n = \frac{(3n + 1)!}{(3n - 1)!}$$ The factorial expression can be expanded:

1. **State the problem:** Differentiate the function $$y = 2\cos x + \sin x$$ with respect to $$x$$. 2. **Recall differentiation rules:**

1. **State the problem:** We need to find the value of $$Q_\infty = \lim_{t \to \infty} \frac{t^2 + 2t + 1}{4t^2}$$. 2. **Recall the limit rule for rational functions:** When $$t$$

1. **State the problem:** Find the area of the shaded region enclosed by the curve $y = x^3 - 7x$ and the line $y = 2x$ between points $O(0,0)$ and $A(3,6)$.

1. **State the problem.** We need to evaluate $\lim_{x\to 3}\frac{x^2-9}{x-3}$.

1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$$. 2. **Recall the formula and rules:** When direct substitution results in an indeterminate form li

1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$$. 2. **Recall the formula and rules:** This is a limit of a rational function where direct substitu

1. The problem asks to evaluate the integral $$\iiint_V V \, dV$$ where $V$ ranges from 1 to 10. 2. Since the integral is with respect to $V$ and the integrand is $V$, this is a si

1. **State the problem:** We are given the distance function $$d(t) = 6t^3 - 12t^2 + 40t$$ where $t$ is time in hours, and we want to find Alan's speed 30 minutes after the start o

1. **State the problem:** Find the first and second derivatives of the function $$T(v) = \frac{d(1+k(v+w)^2)}{v}$$ with respect to $$v$$. 2. **Rewrite the function:**

1. **State the problem:** We need to evaluate the integral $$\int \cos(\ln x) \, dx$$. 2. **Recall the formula and substitution:** Let us use the substitution $$t = \ln x$$, so tha

1. **State the problem:** We are given that $$\int \frac{t(x)}{x^2 + 3x + 5} \, dx = \ln |x^2 + 3x + 5| + c.$$