∫ calculus

1. **Problem Statement:** Given the function $$y = x^4 - x^3 + 4x - 1,$$ find the second derivative $$\frac{d^2y}{dx^2}$$ and determine the values of $$x$$ for which $$\frac{d^2y}{

1. **State the problem:** Find the first derivative $f'(x)$ and the second derivative $f''(x)$ of the function $$f(x) = -2 \sin\left(\frac{1}{2}x - 1\right).$$ 2. **Recall the deri

1. **Problem statement:** Given the function $f(x) = x^3 - 3x^2 + 3$, (a) Find the equation of the tangent line at the inflection point.

1. **State the problem:** Evaluate the definite integral $$\int_1^3 x^2 \, dx$$. 2. **Recall the formula:** The integral of $$x^n$$ with respect to $$x$$ is $$\frac{x^{n+1}}{n+1} +

1. **State the problem:** Evaluate the definite integral of the function $6x^2 + 5$ from $x=1$ to $x=3$. 2. **Formula and rules:** The definite integral of a function $f(x)$ from $

1. **Problem statement:** Calculate the definite integrals using the Fundamental Theorem of Calculus. 2. **Formula:** The Fundamental Theorem of Calculus states:

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{\sqrt{x+4} - 2}{x}$$. 2. **Recall the formula and rules:** When a limit results in an indeterminate form like $$\fra

1. **Énoncé du problème :** Soit la fonction $f(x) = 2x^3 - 6x$.

1. **Problem statement:** A cyclist's velocity is given by the function $f(t) = (-t + 6) \cdot e^{t-3}$ for $t$ in seconds, describing acceleration and braking until stopping. The

1. **State the problem:** Find the limit as $x \to \pm \infty$ of the expression $$\sqrt{x^2 - 6x - 8} + x - 3.$$\n\n2. **Recall the formula and rules:** When dealing with limits i

1. **Stating the problem:** Calculate the integral $$\int x \sqrt{x^2+3} \, dx = \int x (x^2+3)^{1/2} \, dx$$. 2. **Formula and substitution:** Use substitution method. Let $$u = x

1. **State the points at which an extremum can occur, as well as the necessary condition for an extremum.** An extremum occurs where the first derivative $f'(x)$ is zero or undefin

1. **Problem statement:** Given the function $$f(x) = 0.15x^3 - 0.2x^2 - 1x,$$ we need to analyze its extrema and curvature. 2. **Formula and rules:** Extrema occur where the first

1. **Problem statement:** Zeichnen Sie die Tangenten an die Funktionen \(f(x) = 0,5x^2 - 1\) und \(g(x) = x^3 - 2x + 1\) an den angegebenen Punkten und bestimmen Sie die Tangenteng

1. **State the problem:** Find the limits as $x \to +\infty$ for the given rational functions. 2. **Recall the rule for limits at infinity of rational functions:** When $x \to +\in

1. **State the problem:** Calculate the integral $$\int \frac{\sqrt{a^2 - y^2}}{y} \, dy$$ where $a$ is a constant. 2. **Recall the formula and substitution:** This integral involv

1. **State the problem:** We want to solve the integral $$\int \frac{\sqrt{a^2 - y^2}}{y} \, dy$$ where $a$ is a constant. 2. **Recall the formula and substitution:** This integral

1. The problem is to understand and solve an integral, which is a fundamental concept in calculus used to find areas, volumes, central points, and many useful things. 2. The integr

1. **State the problem:** Find the derivative with respect to $x$ of the function $$f(x) = \frac{\ln x}{x^2}.$$\n\n2. **Formula used:** We use the quotient rule for derivatives, wh

1. **State the problem:** Find the slope of the tangent line to the graph of $f(x) = x e^x$ at the point $(1, e)$. 2. **Recall the formula:** The slope of the tangent line at a poi

1. **Problem:** Find the derivative of $y = \sin(3x)$. 2. **Formula:** The derivative of $\sin(u)$ is $\cos(u) \cdot \frac{du}{dx}$.