∫ calculus

1. The problem is to find the derivative of the function $$f(x) = (3x + 2)^2 + (3x + 2)^4$$. 2. We use the chain rule for differentiation: if $$f(x) = g(h(x))$$, then $$f'(x) = g'(

1. **Problem 1a:** Find the area bounded by the curves $f(x) = 2x - x^2$ and $g(x) = x - 2$. 2. **Find the points of intersection:** Solve $2x - x^2 = x - 2$.

1. **Problem statement:** Find the area of the region bounded by the graphs of the functions $f(x) = 2x - x^2$ and $g(x) = x - 2$.

1. **State the problem:** Evaluate the integral $$\int_0^\pi \frac{e^x - 1}{e^x - x} \, dx.$$\n\n2. **Analyze the integrand:** Let $$f(x) = \frac{e^x - 1}{e^x - x}.$$ We want to fi

1. The problem asks to find the value of $x$ where the function $h(x)$ is continuous but not differentiable. 2. A function is continuous at $x=a$ if the left-hand limit, right-hand

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. **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